Ring Theory SeminarSpeaker: Sylvia Wiegand (University of Nebraska)
Title: Prime ideals in Noetherian rings
Date: 27/2/2025
Time: 11:00
Web: http://mat.uab.cat/web/ligat/
Abstract: We study partially ordered sets of prime
ideals in Noetherian rings.
In particular we give a general description of the partially ordered sets $U$ that occur as
Spec $B$, for some integral domain $B$ that is a homomorphic image of a
three-dimensional mixed polynomial-power series ring over a field or over a one-dimensional Noetherian integral domain.
Most of this work is joint with Ela Celikbas, Christina Eubanks-Turner, William Heinzer, Christel Rotthaus, and Roger Wiegand.
Ring Theory SeminarSpeaker: Roger Wiegand (University of Nebraska)
Title: TBA
Date: 27/2/2025
Time: 11:00
Web: http://mat.uab.cat/web/ligat/
Abstract:
Topology SeminarSpeaker: Ran Levi (University of Aberdeen)
Title: Foundations of Differential Calculus for modules over small categories
Date: 17/2/2025
Time: 12:00
Web: http://mat.uab.cat/~topalg
Abstract: Let $?$ be a field and let $\mathcal C$ be a small category. A $?$-linear representation of $\mathcal C$, or a $k\mathcal C$-module, is a functor from $\mathcal C$ to the category of finite dimensional vector spaces over $k$. Unsurprisingly, it turns out that when the category $\mathcal C$ is more general than a linear order, then its representation type is generally infinite and in most cases wild. Hence the task of understanding such representations in terms of their indecomposable factors becomes difficult at best, and impossible in general. In a joint project with Jacek Brodzki and Henri Rihiimaki we proposed a new set of ideas designed to enable studying modules locally. Specifically, inspired by work in discrete calculus on graphs, we set the foundations for a calculus type analysis of $k\mathcal C$-modules, under some restrictions on the category $\mathcal C$. In this talk I will review the basics of the theory and describe some more recent advances.
Topology SeminarSpeaker: Ran Levi (University of Aberdeen)
Title: Foundations of Differential Calculus for modules over small categories
Date: 17/2/2025
Time: 12:00
Web: http://mat.uab.cat/~topalg
Abstract: Let $k$ be a field and let $\mathcal{C}$ be a small category. A $?$-linear representation of $\mathcal{C}$, or a $k\mathcal{C}$-module, is a functor from $\mathcal{C}$ to the category of finite dimensional vector spaces over $k$ . Unsurprisingly, it turns out that when the category $\mathcal{C}$ is more general than a linear order, then its representation type is generally infinite and in most cases wild. Hence the task of understanding such representations in terms of their indecomposable factors becomes difficult at best, and impossible in general. In a joint project with Jacek Brodzki and Henri Rihiimaki we proposed a new set of ideas designed to enable studying modules locally. Specifically, inspired by work in discrete calculus on graphs, we set the foundations for a calculus type analysis of $k\mathcal{C}$-modules, under some restrictions on the category $\mathcal{C}$. In this talk I will review the basics of the theory and describe some more recent advances.
Geometry SeminarSpeaker: Alexey Balitskiy (Luxembourg)
Title: Widths, waists, and curvature
Date: 17/2/2025
Time: 12:00
Web: http://mat.uab.cat/
Abstract: The Urysohn width measures the "approximate dimension" of a Riemannian manifold by approximating it with a lower-dimensional simplicial complex. Positive scalar curvature conjecturally implies upper bounds on the width. Using this conjecture as a guiding light, I will overview several peculiar properties and applications of the width. Here's an example of a question that I will answer: If our manifold is sliced into chunks of small approximate dimension, does that imply that the manifold itself has controlled approximate dimension?
Ring Theory SeminarSpeaker: Daniel Smertnig (University of Ljubljana)
Title: A monoid-theoretical approach to infinite direct-sum decompositions of modules
Date: 29/1/2025
Time: 10:00
Web: http://mat.uab.cat/web/ligat/
Abstract: It has long been understood that many questions about finite direct-sum decompositions in a class of modules $C$ (closed under isomorphisms and direct sums) translate into factorization questions in the associated monoid $V(C)$. Recently there has been a lot of progress in studying countably generated projective modules, in which case it is also sensible to consider infinite direct sums. We introduce the notion of $\kappa$-monoids, with $\kappa$ a cardinal, to capture such infinite direct sums via a monoid-theoretical approach. A first question is then which $\kappa$-monoids actually arise in such a fashion. This leads to a universal extension property for $\kappa$-monoids and an equivalent braiding property for families of modules. In particular, we see that for a hereditary algebra, the monoid of finitely generated projective modules $V(R)$ completely determines the $\kappa$-monoid $V^{\kappa}(R)$ of $\kappa$-generated projective modules. Together with Bergman's classical realization result for $V(R)$, it follows that the possible $V^{\kappa}(R)$ for hereditary algebras are precisely the universal $\kappa$-extensions of reduced commutative monoids with order-unit.
The talk is based on the preprint 2401.08203 (joint with Z. Nazemian).
Geometry SeminarSpeaker: Alejandro Cabrera (UFRJ, Brazil)
Title: About an instanton-type PDE for Poisson geometry
Date: 27/1/2025
Time: 15:00
Web: http://mat.uab.cat/
Abstract: In this talk, I will present an instanton-type PDE associated with a Poisson manifold M. After reviewing its role in an underlying field theory, we present the main theorem showing existence and classification of its solutions. Finally, we discuss its geometric significance leading to a generating function for a symplectic groupoid, Lie-theoretic, integration of M.
Geometry SeminarSpeaker: Simon Vialaret (Orsay/Bochum)
Title: Systolic inequalities for S1-invariant contact forms in dimension three, and applications
Date: 20/1/2025
Time: 14:00
Web: http://mat.uab.cat/
Abstract: In contact geometry, a systolic inequality aims to give a uniform upper bound on the shortest period of a periodic Reeb orbit for contact forms with fixed volume on a given manifold. This generalizes a well-studied notion in Riemannian geometry. It is known that there is no systolic inequality valid for all contact forms on any given contact manifold. In this talk, I will state a systolic inequality for contact forms that are invariant under a circle action in the three-dimensional case, and will discuss applications to a class of Finsler geodesic flows and to a conjecture of Viterbo.
Geometry SeminarSpeaker: Thomas Richard (UPEC)
Title: Curvatura escalar i radi d'injectivitat
Date: 13/1/2025
Time: 14:00
Web: http://mat.uab.cat/
Abstract: Als anys 1960, L. Green va demostrar que el radi d'injectivitat d'una varietat amb curvatura escalar superior a n(n-1) està acotat per π, amb igualtat només per a l'esfera estàndard. Una pregunta natural és llavors si una varietat amb curvatura escalar superior a n(n-1) i un radi d'injectivitat gairebé igual a π s'assembla a l'esfera. Mostraré que a la dimensió 3, si una varietat amb curvatura escalar superior a n(n-1) té un radi d'injectivitat superior a 2π/3, llavors és un quocient de S^3 per un grup cíclic de cardinal senar. La prova utilitza superfícies mínimes i mu-bombolles. En dimensions superiors, aquests mètodes s'apliquen per donar millors límits al radi d'injectivitat de mètriques amb curvatura escalar positiva a S^2xT^kxR^l amb l≤2 i 2+k+l≤7.
Geometry SeminarSpeaker: Diego Artacho De Obeso (Imperial College of London)
Title: Geometria d’Espin i Generalitzacions
Date: 9/12/2024
Time: 11:45
Web: http://mat.uab.cat/
Abstract: Moltes propietats i estructures geomètriques es poden caracteritzar mitjançant l’existència d’espinors especials. Aquests resultats, però, són aplicables únicament a varietats amb estructura d’espin. En aquesta xerrada, explorarem una família d’estructures que generalitzen aquesta idea i permeten estudiar qualsevol varietat orientable, ampliant així l’abast de les tècniques tradicionals d’espin en geometria.
Ring Theory SeminarSpeaker: Eduard Ortega (NTNU, Norway)
Title: Spectral invariance of algebras associated to groups of subexponential growth
Date: 3/12/2024
Time: 10:00
Web: http://mat.uab.cat/web/ligat/
Abstract: Wiener's theorem says that the Fourier algebra of the free abelian group with $n$ generators is a dense and inverse closed subalgebra of the algebra of continuous functions on the $n$-torus. More generally, I will show how to construct dense spectral invariant subalgebras of algebras of convolution operators of groups of subexponential growth.
Ring Theory SeminarSpeaker: Laurent Cantier (UAB)
Title: Around the Nielsen-Thomsen sequence
Date: 27/11/2024
Time: 10:00
Web: http://mat.uab.cat/web/ligat/
Abstract: Thomsen established a link between traces and projections, the Hausdorffized algebraic K1-group, and the (topological) K1-group of a C*-algebra in a short exact sequence, via the de la Harpe-Skandalis determinant.
The functorial version of this split-extension has proven to be essential when classifying *-homomorphisms beyond the scope of the classical Elliott invariant, as firstly highlighted in Nielsen and Thomsen's classification of simple AT-algebras.
In this talk, we revisit the functorial properties of the Nielsen-Thomsen sequence introducing concepts such as bases of split and rotation maps. This framework enables us to generalize the notion of determinant for any unitary element, and to compare *-homomorphisms at the level of their Hausdorffized algebraic K1-group, and subsequently, at the level of their Hausdorffized unitary Cuntz group.
As an application, we present pairs of non-simple AT-algebras and pairs of *-homomorphisms, only distinguishable by their Hausdorffized unitary Cuntz semigroup via the newly defined metrics.
Ring Theory SeminarSpeaker: Ferran Cedó (UAB)
Title: Simple solutions of the Yang-Baxter equation of cardinality $p^n$
Date: 20/11/2024
Time: 10:00
Web: http://mat.uab.cat/web/ligat/
Abstract: In (3) it is proven that if $n>1$ is a square-free integer and $n$ is not prime, then there is no simple, involutive, non-degenerate, set-theoretic solution of the YBE of cardinality $n$. It is not known whether there exists a simple, involutive, non-degenerate, set-theoretic solution of the YBE of cardinality $m^2n$ for any integers $m,n>1$. This was claimed in (1, Theorem 4.12), but the proof was incorrect, though an example of a simple solution of non-square cardinality was given; see (4). On the other hand, all the simple solutions of the YBE of non-prime cardinality constructed in (1, 2) have square cardinality.
In (5, Theorem 5.3) it is proven that if $n>1$ is an integer and $p$ is a prime divisor of $q-1$ for every prime divisor $q$ of $n$, then there exists a simple, involutive, non-degenerate, set-theoretic solution of the YBE of cardinality $p^2n$.
For every prime number $p$ and integer $n>1$, we shall construct a simple, involutive, non-degenerate, set-theoretic solution $(X,r)$ of the Yang-Baxter equation of cardinality $|X|=p^n$.
This is a joint work with Jan Okninski, (6).
References:
(1) F. Cedó and J. Okninski, Constructing finite simple solutions of the Yang-Baxter equation, Adv. Math. 391 (2021), 107968.
(2) F. Cedó and J. Okninski, New simple solutions of the Yang-Baxter equation and solutions associated to simple left braces, J. Algebra 600 (2022), 125-151.
(3) F. Cedó and J. Okninski, Indecomposable solutions of the Yang-Baxter equation of square-free cardinality, Adv. Math. 430 (2023), 109221.
(4) F. Cedó and J. Okninski, Corrigendum to 'Constructing finite simple solutions of the Yang-Baxter equation' (Adv. Math. 391 (2021) 107968), Adv. Math. 451 (2024) 109807.
(5) F. Cedó and J. Okninski, New simple solutions of the Yang-Baxter equation and their permutation groups, arXiv:2401.12904v2.
(6) F. Cedó and J. Okninski, Simple solutions of the Yang-Baxter equation of cardinality $p^n$, arXiv:2407.07907v1.
Ring Theory SeminarSpeaker: Pere Ara (UAB)
Title: Munn trees on graphs
Date: 13/11/2024
Time: 10:00
Abstract: Inverse semigroups are an important tool in the construction of large classes of algebras. Given a combinatorial object, it is in many occasions possible to build an inverse semigroup
associated to the object which reflects relevant aspects of the combinatorial structure. One can then consider the universal groupoid as well as the tight groupoid associated to the inverse semigroup.
From these topological groupoids one can construct interesting algebras, using a construction introduced by Benjamin Steinberg. Using the notion of a Munn tree over a graph, we will develop the construction of an inverse semigroup associated to an arbitrary subshift, not necessarily of finite type.
This is joint work in progress with Alcides Buss and Ado Dalla Costa, both from the Universidade Federal de Santa Catarina (Brazil).
Ring Theory SeminarSpeaker: Dolors Herbera (UAB)
Title: Projective modules over right noetherian rings satisfying a Polynomial Identity
Date: 30/10/2024
Time: 10:00
Web: http://mat.uab.cat/web/ligat/
Abstract: In this talk we will explain that countably generated right or left projective modules over a right noetherian ring satisfying a polynomial identity have a relatively tame structure.
More precisely, we will show that if P is a projective right module over such a ring then there exists an idempotent ideal I of $R$ such that:
$P/PI$ is finitely generated,
$P\cong P\oplus Q^{(\omega)}$ for any countably generated projective module $Q$ with trace ideal $I$.
Even if the hypothesis seem to be asymmetric, a right noetherian PI ring is two-sided noetherian modulo the nilradical. Hence, a similar result holds for countably generated projective left $R$-modules.
The proof of the result depends on showing that these rings satisfy a certain kind of descending chain condition for ideals. The proof of this technical part is patterned on the one due to Small and Robson [RS] showing that these kind of rings only have a finite number of idempotent ideals.
This result is part of the on-going project [AHP2].
[AHP2] R. \'Alvarez, D. Herbera, P. P\v r\'\i hoda, \emph{Monoids of modules and infinite direct sums decompositions: relatively big modules.} In progress (2024).
[RS] L.~W.~Small, J.~C.~Robson, \emph{Idempotent ideals in P.I. rings}. J. London. Math. Soc. \textbf{14} (1976), 120--122.
Ring Theory SeminarSpeaker: Joan Claramunt (Universidad Carlos III)
Title: Mathematics of quantum spin systems
Date: 23/10/2024
Time: 10:00
Abstract: The study of phases of matter is a central research area in condensed matter physics, with many ramifications in numerical analysis, experimental physics and mathematics. Recently, topological and quantum phases have attracted the attention of leading researchers. In particular, the theory of topological and quantum phases in quantum spin systems on lattices (typically $\mathbb Z^n$) and their classification are nowadays a highly relevant research topic.
In this talk, I will introduce an operator-algebraic framework of quantum spin systems on $\mathbb Z^n$, which enables us to use tools from operator algebra theory to the study of phases of matter. I will also introduce the so-called Symmetry Protected Topological (SPT) phases and how one can attempt to characterize them.
This is an on-going project together with Fernando Lledó. Our motivation is to analyze to what extend the recent developments done can be extended beyond the lattice framework, for instance to Cayley graphs.
Ring Theory SeminarSpeaker: Francesc Perera (Universitat Autònoma de Barcelona)
Title: Pure C*-algebras (II)
Date: 16/10/2024
Time: 10:00
Web: http://mat.uab.cat/web/ligat/
Abstract: In this talk we will recall the notion of $(m,n)$-pureness and dwell on the topic of when functional divisibility becomes automatic, hence pureness depends only on having enough good comparison. As an application, some details on pureness of certain C*-algebras will be given. Namely, we will consider the simple, unital, non-elementary algebras with a unique quasitracial state, and we will mention what the situation is in low noncommutative dimensions.
We shall also discuss how our results fit into the framework of a possible non-simple Toms-Winter conjecture.
This is joint work with Ramon Antoine (UAB), Hannes Thiel, and Eduard Vilalta (Chalmers).
Ring Theory SeminarSpeaker: Ramon Antoine (UAB)
Title: Pure C*-Algebras (I)
Date: 9/10/2024
Time: 10:00
Abstract: In this talk we will recall the definition of pure C*-algebra, its relation to the conditions on the Toms-Winter conjecture, and its abstract definition for Cuntz Semigroups in the non necessarily simple setting.
We will show that the condition of being pure can be reduced to very mild regularity conditions on comparability and divisibility in the semigroup. As a consequence, we prove that every C*-algebra that is $(m,n)$-pure in the sense of Winter is already pure.
In a subsequent talk, we will see how this techniques apply in particular cases to show that certain classes of C*-algebras are pure.
This is a joint work with Francesc Perera, Hannes Thiel and Eduard Vilalta.
Geometry SeminarSpeaker: Roberto Rubio (UAB)
Title: New geometric structures on 3-manifolds: surgery and generalized geometry
Date: 7/10/2024
Time: 15:00
Web: http://mat.uab.cat/
Abstract: I will first give an introduction to standard generalized complex geometry, which encompasses complex and symplectic structures. I will then describe how a variant of generalized complex geometry can reach odd-dimensional manifolds and finish by describing recent results on 3-manifolds that are joint work with Joan Porti.
Geometry SeminarSpeaker: Juan Andrés Trillo Gómez (UAB)
Title: Tube formulas for valuations
Date: 30/9/2024
Time: 14:00
Web: http://mat.uab.cat/
Abstract: We establish the existence of tube formulas for smooth valuations on riemannian manifolds. Moreover, we explicitly compute these formulas for invariant valuations in real and complex space forms. The central notion introduced is the tubular operator, a family of linear endomorphisms acting on valuations, whose differentiation yields the derivative operator, which describes how the valuation evolves as the radius of the tube increases. Finally, we derive explicit tube formulas for Federer valuations in both the complex and quaternionic space forms, and we apply these results to compute the Hopf push-forward of valuations via the Hopf fibration, revealing new families of valuations.
Ring Theory SeminarSpeaker: Román Álvarez (UAB)
Title: Infinitely generated projective representations of groups
Date: 25/9/2024
Time: 10:00
Abstract: Let $G$ be a finite group, and let $R$ be a ring with quotient field $K$ such that $K$ is a splitting field for $G$. It is known that, if the group algebra $RG$ is semiperfect, then the irreducible characters of $G$ in $K$ can be written as linear combinations of the irreducible Brauer characters of $G$. This can be written in a matrix form, which is called the decomposition matrix. Here we give an inductive way to compute decomposition matrices from subgroups of the given group. These decomposition matrices are useful to compute minimal idempotent ideals of the group algebras. Due to the theory of fair-sized modules of P. P\v ríhoda, we know that over module-finite algebras $\Lambda$ over commutative noetherian rings, we can compute every countably generated projective $\Lambda$-module $P$ from finitely generated modules of the form $P/PI$ for $I$ an idempotent ideal of $\Lambda$. In our case, we will be able to compute the infinitely generated projective representations of the group.
Ring Theory SeminarSpeaker: Afshin Amini (College of Sciences, Shiraz University)
Title: Weakly perspective rings and modules
Date: 18/9/2024
Time: 10:00
Web: http://mat.uab.cat/web/ligat/
Abstract: A module is called weakly perspective if isomorphic direct summands with isomorphic complements have a common complement. This generalizes the concept of perspectivity (isomorphic summands have a common complement). Among other things, we show that when a module is quasi-injective and weakly perspective, it is directly finite. Then we conclude that every quasi-injective weakly perspective module is perspective.
Ring Theory SeminarSpeaker: Guillem Quingles Daví (UAB)
Title: The Cuntz Semigroup of $C([0, 1])$ viewed as a C*-algebra and as a ring
Date: 11/6/2024
Time: 14:30
Abstract: Very recently, a new invariant has been defined for any ring $R$, denoted by S$(R)$ and called the Cuntz semigroup of the ring $R$, a partially ordered abelian semigroup built from an equivalence relation on the class of countably generated projective modules. In the talk I will explore the relation between the Cuntz semigroup of $C([0, 1])$ viewed as a C*-algebra and as a ring, and also for other rings of continuous functions on locally compact second countable one-dimensional Hausdorff spaces. This results will allow us to study the trace ideals of countably generated ideals of these rings. We will see that the situation is different when considering real-valued or complex-valued functions.
This is work in progress of my Ph.D. done with my advisors, Pere Ara and Francesc Perera.
Geometry SeminarSpeaker: David Fisac Camara (Universitat Luxembourg - UAB)
Title: Comptant corbes sobre el tor punxat amb longitud de paraula acotada.
Date: 10/6/2024
Time: 12:00
Web: http://mat.uab.cat/web/ligat/
Abstract: Parlarem sobre el problema de trobar una fórmula tancada pel nombre de corbes tancades sobre el tor punxat (superfície de gènere 1 sense un punt) amb longitud de paraula (nombre de lletres necessàries per representar la corba al grup fonamental) i auto-intersecció donades; presentant una caracterització de totes les paraules que representen corbes amb auto-intersecció 1 (anàleg al cas ja sabut per corbes simples) i donant un mètode per trobar la fórmula quan la caracterització és sabuda. Després discutirem com traslladar aquests resultats a la mateixa superfície amb una mètrica hiperbòlica i com es podrien derivar resultats sobre conjectures obertes. Aquesta xerrada es basa en feina conjunta amb el Mingkun Liu.
Geometry SeminarSpeaker: Teo Gil Moreno de Mora i Sardà (UAB - UPEC)
Title: Descomposició de $3$-varietats de curvatura escalar positiva amb decreixement subquadràtic.
Date: 3/6/2024
Time: 12:00
Web: http://mat.uab.cat/web/ligat/
Abstract: Una qüestió central en l'estudi de les varietats de dimensió 3 consisteix a comprendre l'estructura topològica de les 3-varietats que admeten una mètrica riemanniana completa de curvatura escalar positiva, conegudes com a varietats PSC. A les darreries dels anys setanta, els resultats obtinguts per Schoen i Yau utilitzant la teoria de superfícies minimals i, paral·lelament, els mètodes basats en la teoria de l'índex desenvolupats per Gromov i Lawson permeteren classificar les 3-varietats PSC tancades i orientables: són exactament aquelles que es descomponen en suma connexa de varietats esfèriques i de productes S2xS1.
En aquesta xerrada presentarem un resultat de descomposició per a les 3-varietats no compactes: si la seva curvatura escalar presenta un decreixement subquadràtic, aleshores la varietat es descompon en suma connexa (possiblement infinita) de varietats esfèriques i de S2xS1. Discutirem també el caràcter òptim d'aquest resultat de descomposició.
Aquest resultat s'inscriu en la continuació de treballs recents de Gromov i Wang.
Treball en col·laboració amb en Florent Balacheff i en Stéphane Sabourau.
Topology SeminarSpeaker: Guillermo Carrión Santiago (Universidad de Málaga)
Title: Construcció plus relativa
Date: 24/5/2024
Time: 12:00
Web: http://mat.uab.cat/~topalg
Abstract: La construcció plus per a homologia entera va ser popularitzada per Quillen als anys 70 com a un mètode per a calcular la teoria K d'un anell. Consisteix en transformar un espai X adjuntant cel·les de dimensió 2 i 3 per a matar un subgrup perfecte H del seu grup fonamental però sense modificar la seva homologia. En el cas que H sigui el subgrup perfecte maximal, la construcció plus resulta ser una nul·lificació homològica. Arran d'aquesta propietat, es generalitza aquesta construcció per a qualsevol teoria d'homologia, però amb la pèrdua de poder escollir el subgrup H. Recentment, Broto, Levi i Oliver, defineixen una construcció plus a la Quillen, adjuntant cel·les de dimensió 2 i 3, per a homologia amb coeficients en un anell R relativa a un subgrup (fortament) R–perfecte qualsevol.
En aquesta xerrada, veurem que malgrat que la proposta de BLO no es pot descriure com a una nul·lificació, podem definir la construcció plus relativa com a una nul·lificació homològica a la categoria d'aplicacions entre espais.
Topology SeminarSpeaker: Guille Carrión Santiago (Universidad de Málaga)
Title: Construcció plus relativa.
Date: 24/5/2024
Time: 12:00
Web: http://mat.uab.cat/~topalg
Abstract: La construcció plus per a homologia entera va ser popularitzada per Quillen als anys 70 com a un mètode per a calcular la teoria K d'un anell. Consisteix en transformar un espai X adjuntant cel·les de dimensió 2 i 3 per a matar un subgrup perfecte H del seu grup fonamental però sense modificar la seva homologia. En el cas que H sigui el subgrup perfecte maximal, la construcció plus resulta ser una nul·lificació homològica. Arran d'aquesta propietat, es generalitza aquesta construcció per a qualsevol teoria d'homologia, però amb la pèrdua de poder escollir el subgrup H. Recentment, Broto, Levi i Oliver, defineixen una construcció plus a la Quillen, adjuntant cel·les de dimensió 2 i 3, per a homologia amb coeficients en un anell R relativa a un subgrup (fortament) R–perfecte qualsevol.
En aquesta xerrada, veurem que malgrat que la proposta de BLO no es pot descriure com a una nul·lificació, podem definir la construcció plus relativa com a una nul·lificació homològica a la categoria d'aplicacions entre espais.
Topology SeminarSpeaker: Thomas Jan Mikhail
Title: Type Theory for the Working Mathematician (part 2)
Date: 3/5/2024
Time: 12:00
Web: http://mat.uab.cat/~topalg
Abstract: In this talk we pick up where we left off last time (12/04/2024) and begin by first sketching out the relationship between type theories and logic, known as the Curry-Howard correspondence. In the remainder of the talk we will go through some applications, exemplifying the utility of type theory as a tool for proving statements in categories. Depending on the time these may include algebraic theories, topoi and homotopy type theory.
Geometry SeminarSpeaker: Antonin Guilloux (Sorbonne)
Title: Generalized Hilbert metrics
Date: 22/4/2024
Time: 12:00
Web: http://mat.uab.cat/web/ligat/
Abstract: Hilbert metrics on convex sets of the euclidean space give a wealth of interesting metric spaces and have been used for example in the study of representations of surface groups.
We propose to extend the definition to specific subsets of complex projective spaces. Early examples of this extension comprise bounded symmetric domains, for which we give a complete description of this new metric, and subset of the complex projective plane related to representations of surface groups.
Topology SeminarSpeaker: Thomas Jan Mikhail (UAB)
Title: Categorical Logic for the Working Mathematician
Date: 12/4/2024
Time: 12:00
Web: http://mat.uab.cat/~topalg
Abstract: According to the Curry-Howard-Lambek correspondence, logics, type theories and structured categories are intimately related. In its cleanest form, the relation between the latter two can be realized in terms of an adjunction, given by the internal language functor and the syntactic category functor. One way of understanding this relation is that type theories provide us with a convenient language for proving things about categories by presenting free structures in a particular way. This is the narrative undertaken by Shulman in his draft 'Categorical Logic from a Categorical Point of View' of which I will give an overview. In some sense this can be taken as an answer to the question of why anyone using category theory might also want to learn type theory.
Topology SeminarSpeaker: Thomas Jan Mikhail (UAB)
Title: Type Theory vs Category Theory
Date: 12/4/2024
Time: 12:00
Web: http://mat.uab.cat/~topalg
Abstract: According to the Curry-Howard-Lambek correspondence, logics, type theories and structured categories are intimately related. In its cleanest form, the relation between the latter two can be realized in terms of an adjunction, given by the internal language functor and the syntactic category functor. One way of understanding this relation is that type theories provide us with a convenient language for presenting free structured categories. This is the narrative undertaken by Shulman in his draft 'Categorical Logic from a Categorical Point of View' of which I could give an overview. In some sense this approach can be taken as answer the question of why anyone using category theory might also want to learn type theory.
Ring Theory SeminarSpeaker: Eduard Vilalta
Title: Pure C*-algebras and *-homomorphisms
Date: 19/3/2024
Time: 14:30
Abstract: The notion of (m,n)-pure C*-algebras was introduced by Winter in his seminal work on separable, simple, unital C*-algebras of finite nuclear dimension. Although a lot of effort has been put on understanding (0,0)-pureness (often simply called pureness), much less is known about the apparently weaker notion of (m,n)-pureness for m,n>0. This is especially the case in the non-simple setting.
I will begin the talk by recalling the Toms-Winter conjecture and the importance of pureness in its study. I will then discuss results from two different ongoing projects: Pure *-homomorphisms and their properties (joint with J. Bosa), and (m,n)-pure C*-algebras (joint with R. Antoine, F. Perera, and H. Thiel).
Topology SeminarSpeaker: Wolfgang Pitsch (UAB)
Title: Quillen's conjecture: first steps into the solvable case, II
Date: 15/3/2024
Time: 12:00
Web: http://mat.uab.cat/web/ligat/
Abstract: In this talk we will explain the Cohen-Macaulay properties of the poset for some extensions of a solvable group by uniquely divisible groups. This is the key result to prove Quillen's conjecture for finite solvable groups.
Ring Theory SeminarSpeaker: Simone Virili (UAB)
Title: Model theoretic methods for the study of rank functions (Part II)
Date: 8/3/2024
Time: 10:30
Abstract: Given a ring R (associative and unital), there is an homeomorphism between P(mod-R) — the space of (Sylvester module) rank functions on f.p. right R-modules mod-R — and L([R-mod,Ab]) — the space of normalized length functions on the category of additive functors to Abelian groups. After recalling the needed details of this correspondence, I will concentrate on applications of this formalism to the study of rank functions. Indeed, consider the following:
Theorem [Schofield,1985]. Let rk∈P(mod-R) be α-full, for all α in Σ, a given set of maps between f.g. projectives. Then, the universal localization π:R→RΣ is nonzero and rk∈π*(P(mod-RΣ)).
Schofield's original proof is based on a series of complicated computations with matrices. Recently, Hanfeng Li has given a simplified proof using the theory of bivariant length functions but his argument still takes five pages of technical matrix computations and it relies on the following result, whose proof takes a couple more pages:
Theorem [Li,2020]. Given a ring epimorphism π:R→S, the following are equivalent for rk∈P(mod-R):
(1) rk∈π*(P(mod-S));
(2) rk^(π)= rk^(idS)=1, where rk^ is the extended map rank function associated with rk.
The goal of this talk is to show that, passing to functor categories, it is possible to give an almost trivial proof of both results. Moreover, I will show that they naturally extend to all rings of definable scalars R→S, which are a class of ring homomorphisms containing all ring epimorphisms and, in particular, all universal localizations.
Ring Theory SeminarSpeaker: Simone Virili (UAB)
Title: Model theoretic methods for the study of rank functions (Part II)
Date: 8/3/2024
Time: 10:30
Abstract: Given a ring R (associative and unital), there is an homeomorphism between P(mod-R) — the space of (Sylvester module) rank functions on f.p. right R-modules mod-R — and L([R-mod,Ab]) — the space of normalized length functions on the category of additive functors to Abelian groups. After recalling the needed details of this correspondence, I will concentrate on applications of this formalism to the study of rank functions. Indeed, consider the following:
Theorem [Schofield,1985]. Let rk∈P(mod-R) be α-full, for all α in Σ, a given set of maps between f.g. projectives. Then, the universal localization π:R→RΣ is nonzero and rk∈π*(P(mod-RΣ)).
Schofield's original proof is based on a series of complicated computations with matrices. Recently, Hanfeng Li has given a simplified proof using the theory of bivariant length functions but his argument still takes five pages of technical matrix computations and it relies on the following result, whose proof takes a couple more pages:
Theorem [Li,2020]. Given a ring epimorphism π:R→S, the following are equivalent for rk∈P(mod-R):
(1) rk∈π*(P(mod-S));
(2) rk^(π)=rk^(idS)=1, where rk^ is the extended map rank function associated with rk.
The goal of this talk is to show that, passing to functor categories, it is possible to give an almost trivial proof of both results. Moreover, I will show that they naturally extend to all rings of definable scalars R→S, which are a class of ring homomorphisms containing all ring epimorphisms and, in particular, all universal localizations.
Ring Theory SeminarSpeaker: Simone Virili (UAB)
Title: Model theoretic methods for the study of rank functions
Date: 5/3/2024
Time: 14:30
Abstract: Given a ring R (associative and unital), there is an homeomorphism between P(mod-R) — the space of (Sylvester module) rank functions on f.p. right R-modules mod-R — and L([R-mod,Ab]) — the space of normalized length functions on the category of additive functors to Abelian groups. After recalling the needed details of this correspondence, I will concentrate on applications of this formalism to the study of rank functions. Indeed, consider the following:
Theorem [Schofield,1985]. Let rk∈P(mod-R) be α-full, for all α in Σ, a given set of maps between f.g. projectives. Then, the universal localization π:R→RΣ is nonzero and rk∈π*(P(mod-RΣ)).
Schofield's original proof is based on a series of complicated computations with matrices. Recently, Hanfeng Li has given a simplified proof using the theory of bivariant length functions but his argument still takes five pages of technical matrix computations and it relies on the following result, whose proof takes a couple more pages:
Theorem [Li,2020]. Given a ring epimorphism π:R→S, the following are equivalent for rk∈P(mod-R):
(1) rk∈π*(P(mod-S));
(2) rk^(π)=rk^(idS)=1, where rk^ is the extended map rank function associated with rk.
The goal of this talk is to show that, passing to functor categories, it is possible to give an almost trivial proof of both results. Moreover, I will show that they naturally extend to all rings of definable scalars R→S, which are a class of ring homomorphisms containing all ring epimorphisms and, in particular, all universal localizations.
Topology SeminarSpeaker: Wolfgang Pitsch (UAB)
Title: Quillen's conjecture: first steps into the solvable case
Date: 1/3/2024
Time: 12:00
Web: http://mat.uab.cat/~topalg
Abstract: In this talk we will explain the Cohen-Macaulay properties of the poset $\mathcal{A}_p(G)$ for some extensions of a $p$-solvable group by uniquely $p$-divisible groups. This is the key result to prove Quillen's conjecture for finite solvable groups.
Ring Theory SeminarSpeaker: Laurent Cantier
Title: Fraïssé Theory for Cuntz semigroups
Date: 27/2/2024
Time: 14:30
Abstract: We introduce a (categorical) Fraïssé theory applied to the category of abstract Cuntz semigroups, drawing inspiration from Kubiś’ approach for metric-enriched categories. Throughout the discussion, we will explore metric properties available within the Hom-sets of Cu-morphisms, including concepts like Cauchy sequences and approximate intertwinings. Lastly, we will illustrate instances of Cu-semigroups that naturally emerge as Fraïssé limits.
Ring Theory SeminarSpeaker: Fernando Lledó (Univ. Carlos III, Madrid)
Title: Foelner type approximations and canonical commutation relations
Date: 20/2/2024
Time: 14:30
Abstract: The dichotomy amenable/paradoxical was first observed in the context of groups by von Neumann in 1929. Since then it has pervaded many areas of mathematics, including, for example, metric spaces, pure algebra or operator theory in Hilbert spaces.
Motivated by the importance of the quantum mechanical canonical commutation relations, I will present new ideas concerning Foelner type (finite-dimensional) approximations in the context of unbounded operators in Hilbert spaces.
These approximations correspond to an operator theoretic version of amenability.
Geometry SeminarSpeaker: Samir Bedrouni (Université des Sciences et de la Technologie Houari Boumediene, Alger)
Title: Pre-foliations of co-degree one on $\mathbb C P^2$ with a flat Legendre transform
Date: 19/2/2024
Time: 12:00
Web: http://mat.uab.cat/web/ligat/
Abstract: A holomorphic pre-foliation of co-degree $1$ and degree $d$ on $\mathbb C P^2$ is the data of a line $L$ of $\mathbb C P^2$ and a holomorphic foliation $F$ on $\mathbb C P^2$ of degree $d-1$. In this talk, I will present the main results of a recent paper on pre-foliations of co-degree $1$ on $\mathbb C P^2$ with a flat Legendre transform (dual web), cf. arXiv:2309.12837. First, I will explain the outline of the proof of the result which states that the dual web of a reduced convex pre-foliation of co-degree $1$ on $\mathbb C P^2$ is flat. Second, I will give a description of pre-foliations of co-degree $1$ and degree $3$ on $\mathbb C P^2$ whose associated foliation has only non-degenerate singularities and whose dual $3$-web is flat.
Ring Theory SeminarSpeaker: Pavel Příhoda
Title: Iterated power intersections of ideals in universal enveloping algebras
Date: 13/2/2024
Time: 14:30
Abstract: If I is an ideal of a ring R, I(1) denotes the intersection of all powers of I. If m is a positive integer we inductively define I(m) as (I(m-1))(1).
Puninski noticed that if R is noetherian, and for every proper ideal I of R there exists a positive integer m such that I(m)=0, then every projective R-module is either finitely generated or free.
In the talk I will explain an elementary method showing that this property holds if R is a universal enveloping algebra of a completely solvable Lie algebra of finite dimension.
Ring Theory SeminarSpeaker: Guillem Quingles Daví (UAB)
Title: The Cuntz Semigroup of $C([0, 1])$ viewed as a C*-algebra and as a ring
Date: 11/2/2024
Time: 14:30
Abstract: Very recently, a new invariant has been defined for any ring $R$, denoted by S$(R)$ and called the Cuntz semigroup of the ring $R$, a partially ordered abelian semigroup built from an equivalence relation on the class of countably generated projective modules. In the talk I will explore the relation between the Cuntz semigroup of $C([0, 1])$ viewed as a C*-algebra and as a ring, and also for other rings of continuous functions on locally compact second countable one-dimensional Hausdorff spaces. This results will allow us to study the trace ideals of countably generated ideals of these rings. We will see that the situation is different when considering real-valued or complex-valued functions.
This is work in progress of my Ph.D. done with my advisors, Pere Ara and Francesc Perera.
Topology SeminarSpeaker: Gabriel Martínez de Cestafe Pumares (UAB)
Title: Quillen’s conjecture via finite topological spaces
Date: 9/2/2024
Time: 12:15
Web: http://mat.uab.cat/~topalg
Abstract: Given a finite group $G$ and a prime $p$, Quillen studied the poset $S_p(G)$ of non-trivial $p$-subgroups of $G$ from an homotopical point of view. For this, he considered the geometric realization of the order complex of $S_p(G)$ and conjectured that it is contractible if and only if $G$ has a non-trivial normal $p$-subgroup. In this talk, we will follow an alternative approach, which exploits the close relation between the notions of finite poset and finite $T_0$ topological space. This will allow us to view $S_p(G)$ itself as a topological space and to reformulate Quillen’s conjecture in a purely homotopical language.
Ring Theory SeminarSpeaker: Dolors Herbera
Title: Finite free resolutions (part 2)
Date: 5/2/2024
Time: 16:00
Abstract: Let R be a ring. Given a bounded complex of finitely generated free right R-modules we want to determine for which left R-modules L, C⊗L is exact except for one term. In particular, if C is the free resolution of a module M, we are interested in determining all left modules L such that TorRi(M,L)=0, for all i>0.
For general rings this is a very difficult question but for commutative rings there is a very nice answer.
In this talk we will explain how to use an old theorem due to McCoy (1948) for maps between finitely generated free modules over commutative rings to solve the problem in that case. McCoy's result was extended by Buchsbaum and Eisenbud (1973) to general bounded complexes of finitely generated free modules over commutative noetherian rings and, finally, Northcott (1976) proved it for general commutative rings. These results allow us to give an answer to the initial question for general commutative rings.
As a consequence, we can give some new light to the classification of tilting classes over commutative rings done by Hrbek and Stovicek (2019).
This is work in progress with Giovanna LeGros.
Geometry SeminarSpeaker: Jaime Pedregal Pastor (Utrecht)
Title: Lie algebroid holonomy
Date: 5/2/2024
Time: 12:00
Web: http://mat.uab.cat/web/ligat/
Abstract: Lie algebroids can be considered as “adapted tangent bundles” for specific geometric situations. As such, it makes sense to consider Lie algebroid connections and Lie algebroid holonomy. In this talk, after a very brief recap of classical holonomy, we will introduce the notion of Lie algebroids, with examples, and give some intuition on their usefulness. We will then, following the “adapted tangent bundle” philosophy, introduce Lie algebroid holonomy. Two remarkable properties distinguish Lie algebroid holonomy from classical holonomy: the Ambrose–Singer theorem must be enlarged beyond curvature and holonomy can jump from leaf to leaf, with not much control over these jumps. Depending on time we will give examples of both features.
Ring Theory SeminarSpeaker: Guido Arnone (Universidad de Buenos Aires)
Title: Hacia una clasificación graduada de álgebras de Leavitt
Date: 29/1/2024
Time: 16:00
Abstract: En esta charla daremos una introducción a la K-teoría algebraica bivariante graduada y su relación con la conjetura de clasificación graduada para álgebras de Leavitt. Comentaremos resultados recientes y futuras direcciones de investigación.
Ring Theory SeminarSpeaker: Dolors Herbera (UAB)
Title: Finite free resolutions
Date: 22/1/2024
Time: 16:00
Abstract: Let R be a ring. Given a bounded complex of finitely generated free right R-modules we want to determine for which left R-modules L, C⊗L is exact except for one term. In particular, if C is the free resolution of a module M, we are interested in determining all left modules L such that TorRi(M,L)=0, for all i>0.
For general rings this is a very difficult question but for commutative rings there is a very nice answer.
In this talk we will explain how to use an old theorem due to McCoy (1948) for maps between finitely generated free modules over commutative rings to solve the problem in that case. McCoy's result was extended by Buchsbaum and Eisenbud (1973) to general bounded complexes of finitely generated free modules over commutative noetherian rings and, finally, Northcott (1976) proved it for general commutative rings. These results allow us to give an answer to the initial question for general commutative rings.
As a consequence, we can give some new light to the classification of tilting classes over commutative rings done by Hrbek and Stovicek (2019).
This is work in progress with Giovanna LeGros.
Ring Theory SeminarSpeaker: Giovanna Le Gros (UAB)
Title: Serre's conditions and the finite type of classes of modules of bounded projective dimension
Date: 15/1/2024
Time: 16:00
Abstract: The class of modules of projective dimension at most n, denoted P_n, is said to be of finite type when its right Ext-orthogonal is exactly the right Ext-orthogonal of the subclass of strongly finitely presented modules in P_n (the strongly finitely presented modules are the modules with a projective resolution consisting of finitely generated modules). In particular, the finite type of P_n is equivalent to P_n being deconstructible: every module in P_n is a direct summand of a module filtered by strongly finitely presented modules in P_n.
The classes P_n which are of finite type enjoy many additional properties with respect to those which are not, thus, we are interested in characterising the rings over which P_n is of finite type for some n. In this talk, we give a complete answer to this question for commutative noetherian rings. Explicitly, over a commutative noetherian ring, the class P_n is of finite type if and only if Serre's condition (S_n) holds.
This talk is based on joint work with Michal Hrbek.
Topology SeminarSpeaker: Jesper M. Møller (Universitat de Copenhaguen)
Title: The non-generating and the Quillen simplicial complex of a finite group
Date: 12/1/2024
Time: 12:15
Web: http://mat.uab.cat/~topalg
Abstract: The Quillen complex of a finite group $G$ is a $G$-simplicial complex simple homotopy equivalent to the order complex of the poset of nontrivial $p$-subgroups. The non-generating complex is a $G$-collapsible simplicial complex containing the Quillen complex as a subcomplex. The orbit $CW$-complexes are contractible. The $f$-vectors are unimodal and even, in most cases, $\log$-concave. The Quillen conjecture can be formulated simplicially in terms of the Quillen simplicial complex.
The non-generating and the Quillen simplicial complex are associated to subgroup posets. There are similar simplicial complexes associated to coset posets.
Topology SeminarSpeaker: Carles Broto (UAB)
Title: Homotopy properties of the poset of non-trivial $p$-subgroups of a group V
Date: 15/12/2023
Time: 12:15
Web: http://mat.uab.cat/~topalg
Abstract: Given a map of posets $f\colon X\to Y$, we analyse a spectral sequence converging to the homology of $Y$. This allows one to show that certain posets are Cohen-Macaulay.
Geometry SeminarSpeaker: Pablo Montealegre (Univ. Montpellier)
Title: On the stable norm of flat surfaces
Date: 12/12/2023
Time: 11:00
Web: http://mat.uab.cat/web/ligat/
Abstract: On a Riemannian manifold, it is known that the systole provides informations on the global geometry of the manifold. Since the shortest length of a non-homologically trivial curve is interesting, it is natural to ask what is the shortest length of a curve inside a fixed homology class, and how it depends on the chosen homology class. This is called the stable norm of the manifold, and to this day there are very few explicit examples.
In this presentation I will be interested in the stable norm of flat surfaces. More precisely, I will show that it is possible to compute the stable norm of flat slit tori. Then, I will glue those tori together to construct half-translation surfaces on which we are able to compute the stable norm. Finally, I will show that on those surfaces the number of homology classes that are minimized by simple curves of length less than x grows sub-quadratically in x.
Geometry SeminarSpeaker: Graham Andrew Smith (Pontifícia Universidade Católica do Rio de Janeiro)
Title: Plateau problems and asymptotic counting of surfaces subgroups
Date: 12/12/2023
Time: 10:00
Web: http://mat.uab.cat/web/ligat/
Abstract: We adapt the asymptotic counting result of Calegari-Marques-Neves to the cas of constant extrinsic curvature (CEC) surfaces. In particular, following recent work of Labourie, we show how this result is expressed in a natural manner in terms of an equidistribution property of a certain class of measures over the space of pointed CEC surfaces. This is joint work with Ben Lowe and Sébastien Alvarez.
Geometry SeminarSpeaker: Anna Roig (Institut de mathématiques de Jussieu – Paris Rive Gauche)
Title: L'espectre de longituds de varietats tridimensionals hiperbòliques aleatòries
Date: 5/12/2023
Time: 11:00
Web: http://mat.uab.cat/web/ligat/
Abstract: Per poder conèixer millor les varietats tridimensionals hiperbòliques, podem mirar el comportament dels seus invariants geomètrics, com la longitud de les seves geodèsiques. Una forma d'encarar aquestes questions és utilitzant mètodes probabilístics. És a dir, considerem un conjunt de varietats hiperbòliques, l'equipem amb una mesura de probabilitat, i ens preguntem questions de la forma: quina és la probabilitat de que una varietat aleatòria tingui un certa propietat? Existeixen diferents models de varietats aleatòries. En aquesta xerrada, explicaré un del principals models probabilístics que existeixen en dimensió 3 i presentaré un resultat relatiu a l'espectre de longituds- el conjunt de longituds de totes les geodèsiques tancades- d'una variedad tridimensional construida sota aquest model.
Geometry SeminarSpeaker: Kostiantyn Drach (UB)
Title: Reverse isoperimetric inequality under curvature constraints
Date: 28/11/2023
Time: 11:00
Web: http://mat.uab.cat/web/ligat/
Abstract: What is the smallest volume a convex body $K$ in ${\mathbb R}^n$ can have for a given surface area? This question is in the reverse direction to the classical isoperimetric problem and, as such, has an obvious answer: the infimum of possible volumes is zero. One way to make this question highly non-trivial is to assume that $K$ is uniformly convex in the following sense. We say that $K$ is $\lambda$-convex if the principal curvatures at every point of its boundary are bounded below by a given constant $\lambda>0$ (considered in the barrier sense if the boundary is not smooth). By compactness, any smooth strictly convex body in ${\mathbb R}^n$ is $\lambda$-convex for some $\lambda>0$. Another example of a $\lambda$-convex body is a finite intersection of balls of radius $1/\lambda$ (sometimes referred to as ball-polyhedra). Until recently, the reverse isoperimetric problem for $\lambda$-convex bodies was solved only in dimension $2$. In a recent joint work with Kateryna Tatarko, we resolved the problem also in dimension $3$. We showed that the lens, i.e., the intersection of two balls of radius $1/\lambda$, has the smallest volume among all $\lambda$-convex bodies of given surface area. For $n>3$, the question is still widely open. I will outline the proof of our result and put it in a more general context of reversing classical inequalities under curvature constraints in various ambient spaces.
Ring Theory SeminarSpeaker: Michal Hrbek
Title: Some new results about the Telescope Conjecture in D(X)
Date: 20/11/2023
Time: 16:00
Web: http://mat.uab.cat/
Abstract: In the generality of a big tt-category, the Telescope Conjecture (TC) asks if every smashing ideal is compactly generated. This has been a conjecture in the case of the stable homotopy category of spectra until the announcement of the negative answer this year. For the derived category D(X) of a qcqs scheme, (TC) is a property which sometimes holds (namely, for noetherian schemes) and sometimes does not. Balmer and Favi showed that (TC) is an affine-local property, and thus the question reduces to affine schemes. With Hu and Zhu, we recently showed that (TC) is even stalk-local, and thus reduces to local rings.
In the present work (arXiv:2311.00601), we show a stronger locality proposition, which reduces (TC) to inspection of the definable ideal generated by the residue field of a local ring. This ties (TC) very strongly with the properties of the m-adic topology on the ring. We apply this to recover most known examples of validity or failure of (TC) in D(X), as well as to construct some new ones. Moreover, we show that certain restriction of (TC) can be characterized in terms of pseudoflat ring epimorphisms over R, yielding an interesting example of a non-surjective pseudoflat local ring morphism.
Topology SeminarSpeaker: Natàlia Castellana
Title: Homotopy properties of the poset of non-trivial $p$-subgroups of a group II
Date: 10/11/2023
Time: 12:15
Web: http://mat.uab.cat/~topalg
Abstract: In this talk we follow the study of the homotopy properties of the simplicial complex associated to the poset of non-trivial $p$-subgroups of a group. Bouc proved the equivalence to the subposet of the Bouc family of subgroups. In the work of Thévenaz and Webb, taking into account the action of the group by conjugation, it is shown that all these inclusions induce $G$-homotopy equivalences.
Topology SeminarSpeaker: Roger Bergadà (UAB)
Title: Homotopy properties of the poset of non-trivial $p$-subgroups of a group I
Date: 3/11/2023
Time: 12:15
Web: http://mat.uab.cat/web/ligat/
Abstract: In this talk we review the first part of Quillen’s paper "Homotopy properties of the poset of non-trivial $p$-subgroups of a group” where the author describes homotopy properties of the simplicial complex associated to the poset of non-trivial $p$-subgroups of a group. One of the main results is the equivalence to the subposet of non-identity elementary abelian $p$-subgroups.
Geometry SeminarSpeaker: Filip Moucka - UAB/Czech Technical University (Prague)
Title: Cartan calculus, symmetric Poisson geometry, and $C_n$-generalized geometry
Date: 31/10/2023
Time: 11:00
Web: http://mat.uab.cat/web/ligat/
Abstract: We introduce analogues of the exterior derivative, the Lie derivative, and the Lie bracket of vector fields, on the algebra of completely symmetric covariant tensor fields. Then we discuss the basic properties and geometrical interpretation of these objects. Using the correspondence between the Cartan calculus and its symmetric counterpart, we introduce a symmetric version of Poisson geometry and generalized geometry.
Topology SeminarSpeaker: Thomas Jan Mikhail (UAB)
Title: Lawvere's Fixed Point Theorem and its Applications
Date: 27/10/2023
Time: 12:15
Web: http://mat.uab.cat/~topalg
Abstract: In 1969 Lawvere published a paper called \emph{Cartesian Closed Categories and Diagonal Arguments}. In this paper, he identifies Cartesian closed categories as a suitable general framework unifying known diagonal arguments. He spells out a fixed point theorem with a remarkably short proof (especially when using the internal language of CCCs) and of which the contrapositive may be interpreted as an abstract diagonal argument. Applications include Cantor's diagonal argument (in any elementary topos in fact), the halting paradox, Russell's paradox as well as Gödel's (first) incompleteness theorem.
Topology SeminarSpeaker: Carles Broto (UAB)
Title: Quillen’s conjecture on subgroup complexes, I
Date: 20/10/2023
Time: 12:15
Web: http://mat.uab.cat/~topalg
Abstract: We will state the conjecture and discuss what is known about it.
Geometry SeminarSpeaker: David Físac (UAB)
Title: Desigualtat de Basmajian per superfícies Riemannianes compactes.
Date: 17/10/2023
Time: 11:00
Web: http://mat.uab.cat/web/ligat/
Abstract: La identitat de Basmajian en superfícies hiperbòliques és un resultat clàssic que descriu la longitud de la vora d'una superfície compacta hiperbòlica donades les longituds de les seves ortogeodèsiques (arcs geodèsics amb extrems a la vora). En un treball conjunt amb en Florent Balacheff, hem estudiat aquest fenomen de rigidesa de l'ortoespectre quan es permet a la mètrica tenir curvatura variable. Presentaré una desigualtat per una certa família de grafs, que codificarà més endavant l'espectre d'ortogeodèsiques d'una superfície Riemanniana, permetent-nos trobar una desigualtat d'estil Basmajian per mètriques amb curvatura variable, tot utilitzant l'invariant conegut com a entropia volúmica de la superfície (o el graf).
Ring Theory SeminarSpeaker: Pace Nielsen (Brigham Young University)
Title: Connections between elementwise properties in rings
Date: 3/7/2023
Time: 16:00
Abstract: We illustrate some recent connections (and disconnections) discovered between some standard properties of rings, and their elements. Of particular importance is the discovery of "better" inner inverses for von Neumann regular elements. This has some surprising consequences in module theory, regarding direct sum decompositions.
Ring Theory SeminarSpeaker: Laurent Cantier (UAB-Czech Academy of Sciences)
Title: Webbing transformations and C*-algebras
Date: 12/6/2023
Time: 16:00
Abstract: In the recent light of the emergence of new invariants for non-simple C*-algebras, we expose a categorical construction that we refer to as the webbing transformation, allowing to generically merge distinct C*-invariants together. E.g. the Cuntz semigroup together with K-theoretical data. One of the benefits is to naturally incorporate the data encoded within any (closed two-sided) ideals. In this talk, we will first define our categorical framework and study properties of these webbed objects, including an ideal-quotient theory, to then venture into their possible impact on the classification of non-simple C*-algebras.
Ring Theory SeminarSpeaker: Roozbeh Hazrat (University of Western Sydney)
Title: Sandpile Graphs and Graph Algebras
Date: 29/5/2023
Time: 16:00
Abstract: We give a down to earth introduction to seemingly two very different topics, one about sandpile models (a model about spreading objects along networks) and the other is how to associate interesting algebras to graphs. We then relate these two topics, via the concept of monoids.
Ring Theory SeminarSpeaker: Manuel Saorín (Universidad de Murcia)
Title: On an overlooked conjecture
Date: 29/5/2023
Time: 15:00
Abstract: The concept of flat object can be defined in any Grothendieck category. In 2007 Juan Cuadra and Daniel Simson conjectured that any locally finitely presented Grothendieck with enough flats has enough projectives. Since by (an extended version of) Gabriel-Popescu's theorem, any Grothendieck category is equivalent to the quotient $({\rm Mod}-\mathcal{A})/\mathcal{T}$, where $\mathcal{A}$ is a preadditive category and $\mathcal{T}$ is a hereditary torsion class of ${\rm Mod}-\mathcal{A}$, in order to tackle the conjecture one needs to ask first what are the conditions on $\mathcal{T}$ for the mentioned quotient to: 1) be locally finitely presented; 2) have enough flats; 3) have enough projectives. In this talk we will identify those conditions in the particular case when $\mathcal{T}$ is also a torsion free class (i.e. it is a TTF class) in which case, due to a generalization of Jan's bijection, there is a uniquely determined idempotent ideal $\mathcal{I}$ of $\mathcal{A}$ such that $\mathcal{T}$ consists of the right $\mathcal{A}$-modules (=additive functors $\mathcal{A}^{op}\to {\rm Ab}$) that vanish on $\mathcal{I}$. It turns out that $({\rm Mod}-\mathcal{A})/\mathcal{T}$ has enough projectives in this case if, and only, if $\mathcal{I}$ is the trace of a projective right $\mathcal{A}$-module. From this point of view, as the much more popular telescope conjecture of Ravenel, this restricted version of Cuadra-Simson's conjecture is a particular case of the general question of when a given idempotent ideal $\mathcal{I}$ of a (pre)additive category, e.g. of a ring, is the trace of a (special type of) projective $\mathcal{A}$-module. We will give some partial positive answers to Cuadra-Simson's conjecture when $\mathcal{T}$ is a TTF class.
Geometry SeminarSpeaker: Juan Andrés Trillo (UAB)
Title: Tube formulas for valuations in complex space forms
Date: 25/5/2023
Time: 12:30
Web: http://mat.uab.cat/web/ligat/
Abstract: Given an isometry invariant valuation on a complex space form we compute its value on the tubes of sufficiently small radii around a set of positive reach. This generalizes classical formulas of Weyl, Gray and others about the volume of tubes. We also develop a general framework on tube formulas for valuations in riemannian manifolds.
Ring Theory SeminarSpeaker: Raimund Preusser (Nanjing University of Information Science and Technology)
Title: Graded Bergman algebras
Date: 22/5/2023
Time: 16:00
Abstract: This talk is about an ongoing research project with Roozbeh Hazrat and Huanhuan Li. Recall that for a (unital and associative) ring R, the V-monoid of R is the set of isomorphism classes of finitely generated projective left R-modules. It becomes an abelian monoid with direct sum. George Bergman has shown that any conical finitely generated abelian monoid with an order unit can be realised as the V-monoid of a hereditary algebra. We want to obtain a graded version of this result as follows. For an abelian group G, a G-monoid is an abelian monoid M together with an action of G on M via monoid homomorphisms. Our goal is to show that any conical finitely presented G-monoid with an order unit can be realised as the graded V-monoid of a G-graded algebra which is hereditary.
Geometry SeminarSpeaker: Joan Porti (UAB)
Title: Lema de Morse en espais simètrics.
Date: 18/5/2023
Time: 12:30
Web: http://mat.uab.cat/web/ligat/
Abstract: El lema de Morse afirma que tota quasi-geodèsica a l'espai hiperbòlic està uniformement a prop d'una geodèsica. Aquest lema és fals al pla euclidià i per a un espai simètric de tipus no compacte (que pot contenir plans euclidians) cal donar un enunciat que ho tingui en compte. Donarem l'enunciat correcte, la idea de la demostració i algunes aplicacions. Treball en col·laboració amb M. Kapovich i B. Leeb.
Ring Theory SeminarSpeaker: Fernando Lledó (UC3M-ICMAT)
Title: Finite dimensional approximations in two classes of operator algebras
Date: 15/5/2023
Time: 16:00
Abstract: In this talk I will present finite dimensional matrix approximations in two classes of operator algebras:
\begin{itemize}
\item The resolvent algebra introduced by Buchholz and Grundling in 2008 to give an alternative bounded operator approach to the canonical commutation relations (CCR) in quantum mechanics.
\item The uniform Roe algebras of an inverse semigroup, where the inverse semigroup is viewed as a metric space.
\end{itemize}
The results presented are included in the recent publications
\begin{enumerate}
\item F. Lledó and D. Martínez, {\textit A note on commutation relations and finite dimensional approximations}, Expositiones Mathematicae {\textbf 40} (2022) 947–960.
\item F. Lledó and D. Martínez, {\textit The uniform Roe algebra of an inverse semigroup}, Journal of Mathematical Analysis and Applications {\textbf 499} (2021) 124996.
\end{enumerate}
Ring Theory SeminarSpeaker: Eduard Vilalta (UAB)
Title: Nowhere scattered multiplier algebras
Date: 8/5/2023
Time: 16:00
Abstract: A natural assumption that ensures sufficient noncommutativity of a C*-algebra is nowhere scatteredness, which in one of its many formulations asks the algebra to contain no nonzero elementary ideal-quotients. This notion enjoys many good permanence properties, but fails to pass to certain unitizations. For example, no minimal unitization of a non-unital C*-algebra (nowhere scattered or not) can ever be nowhere scattered. However, it is unclear when a nowhere scattered C*-algebra has a nowhere scattered multiplier algebra.
In this talk, I will give sufficient conditions under which this happens. It will follow from the main result of the talk that a $\sigma$-unital C*-algebra of finite nuclear dimension, or of real rank zero, or of stable rank one and $k$-comparison, is nowhere scattered if and only if its multiplier algebra is. I will also give some examples of nowhere scattered C*-algebras whose multiplier algebra is not nowhere scattered.
Ring Theory SeminarSpeaker: Joachim Zacharias (University of Glasgow)
Title: On a finite section method to approximate exact C*-algebras
Date: 24/4 /2023
Time: 16:00
Abstract: Exact C*-algebras are an important class of C*-algebras which is closed under subalgebras and contains all nuclear C*-algebras. A basic result due to Kirchberg asserts that any such separable C*-algebra is a sub-quotient of a UHF-algebra.
We give a short survey on exact C*-algebras, indicating a simplified 'finite-section' approach to Kirchberg's basic result and outline possible applications, including a Stone-Weierstrass type Theorem for exact C*-algebras.
Geometry SeminarSpeaker: Pavao Mardesic (Institut de Mathématiques de Bourgogne, Dijon, Francia y Universidad de Zagreb, Croacia)
Title: Deformaciones de foliaciones de Darboux genéricas y integrales pseudo-abelianas.
Date: 20/4/2023
Time: 12:30
Web: http://mat.uab.cat/
Abstract: Presentamos un trabajo realizado con Colin Cristopher. En éste estudiamos deformaciones $\omega+\epsilon \eta$ de sistemas de Darboux genéricas y la parte principal $M_1$ de la función de desplazamiento, dada por una integral pseudo-abeliana.
Introducimos la noción de forma Darboux relativamente exacta y mostramos que $M_1$ es idénticamente nula si y solo si $\eta$ es Darboux relativamente exacta. Obtenemos tres corolarios:
1) El estrato de centros con una integral primera de Darboux es una componente algebraica irreducible de la variedad de centros.
2) Un algoritmo para calcular la primera función de Melnikov $M_k$ no nula,
3) Una cota inferior para el número de ciclos que se pueden crear desde un centro genérico. Esos resultados generalizan resultados clásicos de Ilyashenko y Françoise sobre deformaciones de sistemas Hamiltonianos.
Ring Theory SeminarSpeaker: Francesc Perera (UAB)
Title: The dynamical Cuntz semigroup and crossed products
Date: 17/4/2023
Time: 16:00
Abstract: In this talk I shall discuss the definition of dynamical subequivalence for open subsets of a compact topological space and its natural counterpart involving Cuntz subequivalence. This will lead to the definition of the dynamical Cuntz semigroup. I will mention how this semigroup is related to the construction of crossed products in various categories. This is part of joint work with J. Bosa, J. Wu, and J. Zacharias, and also R. Antoine and H. Thiel.
Geometry SeminarSpeaker: Sylvain Maillot (Université de Montpellier)
Title: Mean curvature flow and Heegaard Surfaces in Lens Spaces
Date: 13/4/2023
Time: 12:30
Web: http://mat.uab.cat/web/ligat/
Abstract: Lens spaces $L(p,q)$ are a family of closed 3-manifolds indexed by two coprime integers. They can be described as quotients of the 3-sphere by free isometric actions of cyclic groups; hence they carry riemannian metrics of constant positive sectional curvature. Alternatively, they can be obtained by gluing together two solid tori along their common boundary, which is called a Heegaard torus.
Our main theorem is as follows: fix a metric of constant sectional curvature 1 on $L(p,q)$, and denote by $\mathcal{M}_{H>0}(p,q)$ the moduli space of Heegaard tori in $L(p,q)$ that have positive mean curvature. If $q\cong \pm 1 \mod p$, then $\mathcal{M}_{H>0}(p,q)$ is path-connected. Otherwise it has exactly two path-components.
This is work in progress, joint with Reto Buzano.
Geometry SeminarSpeaker: Thiziri Moulla (Université Montpellier)
Title: On finitely presented groups
Date: 30/3/2023
Time: 12:30
Web: http://mat.uab.cat/
Abstract: For any finitely presented group G, we consider a 2-simplicial complex K of fundamental group G. In this talk I will define a new invariant of combinatorial type on G from the numbers of vertices of all the 2-simplicial complexes of fundamental group G then we will see this invariant on some families of groups. If we have time, I will give links between this invariant and other invariants of different type.
Ring Theory SeminarSpeaker: Guillem Quingles (UAB)
Title: Finiteness properties of local cohomology modules
Date: 27/3/2023
Time: 16:00
Web: http:// mat.uab.cat/web/ligat/
Abstract: Local cohomology modules were introduced by Grothendieck in 1961 and they quickly became an important tool in commutative algebra. They have been studied by a number of authors, but the structure of these modules is still quite unknown. When a local cohomology module $H^i (\Gamma_I (M^*) )$ is nonzero, it is rarely finitely generated, even if $M$ is. So it is not clear whether they satisfy finiteness properties that finitely generated modules do. Huneke proposed a list of problems on local cohomology which guided the study of local cohomology modules. One of the questions on the list asks the following: Is the number of associated primes of $H^i (\Gamma_I (R^*) )$ finite? Are all the Bass numbers of $H^i (\Gamma_I (R^*) )$ finite? Lyubeznik conjectured that the answer is affirmative when $R$ is a Noetherian regular commutative ring with unit. Substantial progress has been made on this conjecture. If the regular ring has prime positive characteristic $p$, then the conjecture was completely settled by Huneke and Sharp. Lyubeznik proved the conjecture for regular rings containing a field of characteristic zero. For complete unramified regular local rings of mixed characteristic, the conjecture was also settled by Lyubeznik. The finiteness of associated primes of local cohomology was also proved by Bhatt, Blickle, Lyubeznik, Singh and Zhang for smooth $Z$-algebras. The conjecture is still open when $R$ is a ramified regular local ring of mixed characteristic.
In this talk I will explain the concepts and tools needed to understand the problem of the finiteness of the set of associated primes and Bass numbers of local cohomology modules $H^i (\Gamma_I (M^*) )$, and the techniques that lead to the proof of the cases where $R$ has positive characteristic and where $R$ is a $K$-algebra, with $K$ a field of characteristic 0.
Geometry SeminarSpeaker: Ara Basmajian (City University of New York)
Title: Homogeneous Riemann surfaces
Date: 23/3/2023
Time: 12:30
Web: http://mat.uab.cat/
Abstract: We are interested in spaces that look the same from any point of the space (that is, ``homogeneous spaces"). Of course the notion of looking the same is dependent on the category of objects one works within. For this talk, the category of objects we work with are Riemann surfaces. Now, the Riemann sphere, complex plane, and unit disc are conformally homogeneous Riemann surfaces. In fact, along with the punctured plane and the torus these are the only ones. On the other hand, given any surface it is not difficult to cook up a diffeomorphism between any two points of the surface. Hence one needs a notion that is not as strong as conformality and not as weak as differentiability. The key observation is that while smooth maps can distort infinitesimal circles to ellipses with unbounded eccentricity (the ratio of the major to the minor axis can be arbitrarily large), conformal maps do not distort infinitesimal circles at all. This leads to the notion of a homeomorphism being $K$-quasiconformal (has eccentricity bounded by $K$). Conformal homeomorphisms are 1-quasiconformal.
A Riemann Surface X is said to be {\it K-quasiconformally homogeneous} if for any two points x and y on it, there exists a K-quasiconformal self-mapping taking x to y. If such a K exists we say that X is a QCH Riemann surface. After introducing the basics, the focus of this talk will be on Riemann surface structures that are QCH, and their connections to the topology and hyperbolic geometry of the surface. The new results of this talk are joint work with Nick Vlamis.
Ring Theory SeminarSpeaker: Martin Mathieu (Queen's University Belfast)
Title: A contribution to Kaplansky's problem
Date: 20/3/2023
Time: 16:00
Abstract: A Jordan homomorphism between two unital, complex algebras $A$ and $B$ is a linear mapping $T$ such that $T(x^2)=(Tx)^2$ for all $x\in A$. Equivalently, $T$ preserves the Jordan product $xy+yx$. Every surjective unital Jordan homomorphism preserves invertible elements. In 1970, Kaplansky asked whether the following converse is true: Suppose $T\colon A\to B$ is a unital surjective invertibility-preserving linear mapping between unital (Jacobson) semisimple Banach algebras $A$ and $B$. Does it follow that $T$ is a Jordan homomorphism?
In the past 50 years a lot of progress has been made towards a positive solution to Kaplansky's problem, however, as it stands, it is still open. We will report on some recent joint work with Francois Schulz (University of Johannesburg, SA) which gives a positive answer if $B$ is a C*-algebra with faithful tracial state. Until recently, the existence of traces had been a major obstacle to a solution. Moreover, in our approach, no assumption on the existence of projections (such as real rank zero) is necessary.
I will further discuss a sharpening of Kaplansky's problem in which the assumption on $T$ is reduced to the preservation of the spectral radius only (a spectral isometry).
Geometry SeminarSpeaker: Antonia Jabbour (Université Gustave Eiffel)
Title: Sharp bounds on the length of the shortest closed geodesic.
Date: 16/3/2023
Time: 12:30
Web: http://mat.uab.cat/
Abstract: In this talk, I will demonstrate how we can obtain sharp universal upper bounds on the length of the shortest closed geodesic on a punctured sphere with three or four ends endowed with a complete Riemannian metric of finite area. In both cases, I will describe the extremal metrics, which are modeled on the Calabi-Croke sphere or the tetrahedral sphere. Note that the Calabi-Croke sphere is obtained by gluing two copies of an equilateral triangles along their boundaries, and the tetrahedral sphere is given by the regular tetrahedron.
Geometry SeminarSpeaker: Teo Gil Moreno de Mora (UAB/Univ. Paris-Est Créteil)
Title: An isosystolic inequality for Finsler reversible torii and the Busemann-Hausdorff area
Date: 2/3/2023
Time: 12:30
Web: http://mat.uab.cat/
Abstract: In 1949, Loewner discovered a much celebrated inequality: the systole of any Riemannian torus of dimension 2 is controlled by its area. The key step in his proof was the reduction to the flat case by means of the conformal representation theorem.
The generalization of this inequality to the Finsler framework results in a wide variety of results. In this talk I will survey the different known isosystolic inequalities on two-dimensional Finsler torii involving the two main notions of area in Finsler geometry: the Busemann-Hausdorff area and the Holmes-Thompson area. I will also complete the picture presenting a new isosystolic inequality on reversible Finsler 2-torii for the Busemann-Hausdorff area, which is obtained again by reduction to the flat case. It is a joint work with Florent Balacheff.
Ring Theory SeminarSpeaker: Ado Dalla Costa (Universidade Federal de Santa Catarina)
Title: Free actions of groups on separated graphs and their associated C*-algebras
Date: 27/2/2023
Time: 16:00
Abstract: I will report on joint work with Alcides Buss and Pere Ara on the study of free actions of groups on separated graphs and explain how this structure reflects on the level of their associated C*-algebras. We prove a version of the Gross-Tucker theorem in this context and show how this can be used to describe the various C*-algebras attached to separated graphs carrying a free action. All this leads to certain Landstad-type duality theorems involving these algebras.
Ring Theory SeminarSpeaker: Pere Ara (UAB)
Title: The inverse semigroup of a separated graph
Date: 20/2/2023
Time: 16:00
Abstract: For a directed graph $E$, the graph semigroup $S(E)$ was defined by Ash and Hall in 1975. The graph semigroup $S(E)$ is an inverse semigroup, and has been studied by many authors in connection with the theories of graph $C^*$-algebras, Leavitt path algebras, and topological groupoids. For a separated graph $(E,C)$, the direct analogue of $S(E)$ is not an inverse semigroup in general. However, we will introduce an inverse semigroup $IS(E,C)$ for each separated graph, which produces the same graph semigroup $S(E)$ as above in the non-separated case. We will develop a normal form of the elements of $IS(E,C)$ in close analogy to the Scheiblich normal form for elements of the free inverse semigroup.
This is joint work in progress with Alcides Buss and Ado Dalla Costa, both from Universidade Federal de Santa Catarina (Brazil).
Ring Theory SeminarSpeaker: Dolors Herbera (UAB)
Title: Torsion free modules over commutative domains of Krull dimension 1
Date: 13/2/2023
Time: 16:00
Abstract: Let $R$ be a commutative domain. Let $\mathcal{F}$ be the class of $R$ modules that are infinite direct sums of finitely generated torsion-free modules. In the talk we will discuss the question whether $\mathcal{F}$ is closed under direct summands.
If $R$ is local of Krull dimension $1$, $\mathcal{F}$ being closed under direct summands is equivalent to say that any indecomposable, finitely generated torsion-free module has local endomorphism ring.
For the global case, we show also in the case of Krull dimension $1$ that the property on $\mathcal{F}$ is inhereted by the localization at a maximal ideal. Moreover, there is an interesting relation between ranks of indecomposable modules over such localizations.
The machinery we use to prove these results was explained in Roman Álvarez's talk, in the previous session of the seminar.
Time permitting, we will also discuss the property `being locally a direct summand' versus `being a direct summand' in the setting of our problem. The results we obtain allows us to give a complete answer to the initial problem in some particular cases.
The talk is based on a joint work with Roman Álvarez and Pavel Příhoda.
Ring Theory SeminarSpeaker: Román Álvarez (UAB)
Title: Package Deal Theorems for Localizations over h-local Domains
Date: 6/2/2023
Time: 16:00
Abstract: Let $R$ be a commutative ring with total ring of fractions $Q$, let $\Lambda$ be a (not necessarily commutative) $R$-algebra, and let $M$ be a finitely generated right $\Lambda$-module. For each maximal ideal $m$ of $R$, consider a (not necessarily finitely generated) $\Lambda_m$-submodule $X(m)$ of $M_m$. For which such families is there a $\Lambda$-submodule $N$ of $M$ such that $N_m=X(m)$? This question was answered by Levy-Odenthal for $R$ a commutative Noetherian ring of Krull dimension $1$ under two consistency hypotheses:
1. $X(m)=M_m$ for almost all maximal ideals $m$ of $R$;
2.$ X(m)\otimes Q=M\otimesQ$ for all maximal ideals $m$ of $R$.
They called these kinds of results Package Deal Theorems.
In this talk, I will give a version of this result for a larger class of domains, namely $h$-local domains, which were introduced by Matlis in the 1960s. $H$-local domains are commutative domains with the property that each non-zero element is contained in only finitely many maximal ideals of the ring and each non-zero prime ideal is contained in a unique maximal ideal of the ring. From results of previous work of Herbera-Příhoda and the original techniques of Levy-Odenthal, I will conclude with a Package Deal Theorem for traces of projective modules over $h$-local domains.
Geometry SeminarSpeaker: Dmitry Faifman (Tel-Aviv University)
Title: A unique extension property of integral transforms on higher grassmannians.
Date: 20/12/2022
Time: 15:00
Web: http://mat.uab.cat/web/ligat/
Abstract: We will consider certain integral operators on higher grassmannians that appear naturally e.g. in convex geometry: the Radon and cosine transforms. The image of such operators is often a rather small subspace of all functions, and can be explicitly described in terms of harmonic analysis. We will describe a quasianalytic-type property exhibited by those images, allowing to uniquely determine a function from its values on a small set. This allows to sharpen classical uniqueness theorems of Funk and Alexandrov in geometric tomography, and of Klain and Schneider in valuation theory.
A key component in the proof is a new support-type uncertainty principle for distributions on grassmannians.
Ring Theory SeminarSpeaker: Eduard Vilalta (UAB)
Title: Low-dimensional C*-algebras
Date: 19/12/2022
Time: 16:00
Abstract: A common theme in the theory of C*-algebras is that meaningful topological notions should translate to useful C*-analogues. This approach has produced many key concepts, one particular instance being the many C*-analogues of Lebegue's covering dimension.
C*-algebras that attain the lowest possible value of one of these dimensions enjoy many nice permanence properties, and the understanding of these algebras brings new insights to the structure of C*-algebras. I will begin the talk by giving a brief overview on some of the most relevant C*-dimensions. I will then report on current work with H. Thiel (and, if time allows, on work with A. Asadi-Vasfi and H. Thiel) on the structure of C*-algebras that are of Cuntz covering dimension zero.
Ring Theory SeminarSpeaker: Jan Šťovíček (Charles University)
Title: Decomposition properties of modules via structure theory for topological rings
Date: 12/12/2022
Time: 16:00
Abstract: Given a module M over a ring, it is a classical question whether (and to what extent uniquely) it decomposes into a direct sum of indecomposable modules. Based on a recent work with Leonid Positselski (arXiv:1909.12203 and arXiv:2201.03488), I will explain how this question is controlled by structural properties of the topological ring End(M) - the ring of endomorphisms of M together with a natural (so-called finite) right linear topology. This naturally leads to generalization of the concepts of semisimple, perfect and semiperfect rings to topological rings.
Geometry SeminarSpeaker: Andreas Bernig (Goethe Universität Frankfurt)
Title: Integral geometry of complex space forms and Tutte's sequence 1,3,13,68,399,...
Date: 29/11/2022
Time: 11:00
Web: http://mat.uab.cat/web/ligat/
Abstract: I will present a connection between integral geometry and combinatorics. More precisely, the kinematic formulas on complex space forms are related to the famous sequence 1,3,13,68,... called Tutte sequence. This connection was conjectured by Joseph Fu in 2008. The solution is based on some very recent results on valuations on Kaehler manifolds as well as some algebraic manipulations using holonomic functions.
Ring Theory SeminarSpeaker: Carles Casacuberta (UB)
Title: Homotopy reflectivity is equivalent to the weak Vopenka principle
Date: 25/11/2022
Time: 12:30
Abstract: It is well known that the existence of homotopical localization with respect to every (possibly proper) class of maps between spaces or spectra is implied by suitable large-cardinal axioms. However, no concluding evidence had been given that the existence of such localizations could not be proved in ZFC. Using a recent result of Trevor Wilson, we prove that the existence of localizations with respect to classes of maps of spaces or spectra is equivalent to the weak Vopenka principle, stating that there is no full embedding of the opposite category of ordinals into any locally presentable category. In fact we prove that the weak Vopenka principle is equivalent to the claim that every colocalizing subcategory of the homotopy category of any stable locally presentable model category is reflective. This is joint work with Javier Gutiérrez.
Ring Theory SeminarSpeaker: Ferran Cedó (UAB)
Title: Indecomposable solutions of the Yang-Baxter equation of square-free cardinality
Date: 21/11/2022
Time: 16:00
Abstract: Let $p_1,\ldots ,p_n$ be $n$ distinct prime numbers. Let $m_1,\ldots , m_n$ be positive integers such that $m_1+\ldots +m_n\gt n$. In previous joint work with J. Okni\'{n}ski, we proved that there exist simple involutive non-degenerate set-theoretic solutions $(X,r)$ of the Yang-Baxter equation with $\vert X\vert = p_1^{m_1}\cdots p_n^{m_n}$. A natural question is asked: If $n\gt1$, is there a simple involutive non-degenerate set-theoretic solution $(X,r)$ of the Yang-Baxter equation with $\vert X\vert = p_1\cdots p_n$?
In this talk, I will answer this question.
This is joint work with J. Okni\'{n}ski
Geometry SeminarSpeaker: Jesús Yepes Nicolás (Univ. Murcia)
Title: On mass-distributions of partitions of convex bodies by hyperplanes.
Date: 15/11/2022
Time: 11:00
Web: http://mat.uab.cat/web/ligat/
Abstract: A classical result by Grünbaum provides the extremal mass ratio (in terms of Lebesgue measure) for the portions obtained when cutting a convex body by a hyperplane passing through its centroid.
In this talk we will discuss, on the one hand, some extensions of this result to the case of different cuts (by hyperplanes) through a whole uniparametric family of points. On the other hand, we will show how this result allows us to connect some different well-known inequalities involving the centroid of a convex body, such as a classical result due to Minkowski (in dimension 3) and Radon (for arbitrary dimension), or a more recent one by Fradelizi.
This is about joint work with D. Alonso-Gutiérrez, F. Marín Sola and J. Martín Goñi.
Ring Theory SeminarSpeaker: Wolfgang Pitsch (UAB)
Title: Witt group and Maslov index
Date: 14/11/2022
Time: 15:00
Abstract: The main subjects of this talk will be $W(k)$, the Witt group over a field $k$, and the Maslov index of three Lagrangians in a symplectic space, which is an invariant, originally introduced in topology, taking values in $W(k)$. I will show how the machinery of Sturm sequences and Sylvester matrices developed by Barge-Lannes can be used to prove that the equivalence class of Maslov's $2$-cocycle, associated to the homonymous index, is trivial modulo $I^2$, with $I$ being the fundamental ideal of $W(k)$.
Geometry SeminarSpeaker: Joana Cirici (UB)
Title: Hodge-de Rham numbers of almost complex 4-manifolds
Date: 22/3/2022
Time: 12:00
Web: http://mat.uab.cat/web/ligat/
Abstract: I will introduce a Frölicher-type spectral sequence that is valid for all almost complex manifolds, yielding a natural Dolbeault cohomology theory for non-integrable structures. As an application, I will focus on the case of almost complex 4-manifolds, showing how the Frölicher-type spectral sequence gives rise to Hodge-de Rham numbers with very special properties. This is joint work with Scott Wilson.
Geometry SeminarSpeaker: Arturo Fernández (ICEX - Universidade Federal de Minas Gerais - Brasil)
Title: Number of Milnor and Tjurina of Foliations
Date: 21/10/2021
Time: 13:00
Web: http://mat.uab.cat/web/ligat/
Abstract: In this lecture, I will show the relationship between the Milnor and Tjurina numbers of a foliation in the complex plane. Such numbers are similar to the classic Milnor and Tjurina numbers for singular curves. This work is in collaboration with Evelia García Barroso (Universidad de la Laguna - Spain) and Nancy Saravia Molina (PUCP-Peru).
Ring Theory SeminarSpeaker: Eduard Ortega (NTNU Trondheim)
Title: Left cancellative small categories and their associated algebras
Date: 13/10/2021
Time: 16:00
Abstract: In this talk I will explain how to associate an étale groupoid to a left cancellative small category. We will show that certain categories with a length function can be written as a Zappa-Zsép product of a free subcategory and the groupoid of invertible elements. This talk is based in a common project with Enrique Pardo.
Topology SeminarSpeaker: Luca Pol (University of Regensburg)
Title: Local Gorenstein duality in chromatic group cohomology
Date: 23/9/2021
Time: 10:30
Web: http://mat.uab.cat/~topalg
Abstract: Many algebraic definitions and constructions can be made in a derived or homotopy invariant setting and as such make sense for ring spectra. Dwyer-Greenlees-Iyengar (followed by Barthel-Heard-Valenzuela) showed that one can make sense of local Gorenstein duality for ring spectra. In this talk, I will show that cochain spectra C*(BG;R) satisfy local Gorenstein duality surprisingly often, and explain some of the implications of this. When R=k is a field this recovers duality properties in modular representation theory conjectured by Benson and later proved by Benson-Greenlees. However, the result also applies to more exotic coefficients R such as Lubin-Tate theories, K-theory spectra or topological modular forms, showing that chromatic analogues of Benson’s conjecture also hold. This is joint work with Jordan Williamson.
Topology SeminarSpeaker: Paolo Salvatore
Title: Multi-simplicial operations, equivariance and effective homology
Date: 28 /6/2021
Time: 16:00
Abstract: We define a cup product on multi-simplicial cochain complexes, and more generally an E-infinity algebra structure. We then show how to apply this, together with the equivariance with respect to a group action on the complexes, to improve substantially the effectiveness of the algorithms appearing in formality problems.
Ring Theory SeminarSpeaker: Joan Bosa (Universitat Autònoma de Barcelona)
Title: Stable Elements and Property (S)
Date: 27/5/2021
Time: 16:00
Abstract: We study the relation (and differences) between stability and Property (S) in the
simple and stably finite framework. This leads us to characterize stable elements in terms of its
support, and study these concepts from different sides : hereditary subalgebras, projections in
the multiplier algebra and order properties in the Cuntz semigroup. We use these approaches
to show both that cancellation at infinity on the Cuntz semigroup just holds when its Cuntz
equivalence is given by isomorphism at the level of Hilbert right-modules, and that different
notions as Regularity, $\omega$-comparison, Corona Factorization Property, property R, etc.. are
equivalent under mild assumptions.
Ring Theory SeminarSpeaker: Román Alvarez Arias (Universitat Autònoma de Barcelona)
Title: Non-Finitely Generated Projective Modules over Integral Group Rings
Date: 13/5/2021
Time: 16:00
Abstract: We introduce a relative version of the big projective modules introduced by Bass, which is an example of a non-finitely generated projective module. We develop the general theory of I-big projective modules introduced by Pavel Príhoda (2010). We inquire more deeply in a correspondence between countably generated projective modules over a ring R and finitely generated projective modules over a ring R modulo an ideal I and generalize it into an equivalence of categories as it is done by Herbera-Príhoda-Wiegand in a recent preprint (2020). Finally, we approach I-big projective modules over well-known rings in order to give an explicit example of the construction of non-finitely generated projective modules over the integral group ring ZA5, where A5 denotes the alternating group on 5 letters.
Ring Theory SeminarSpeaker: Eduard Vilalta (Universitat Autònoma de Barcelona)
Title: The range problem for the Cuntz semigroup of AI-algebras
Date: 18/3/2021
Time: 15:00
Abstract: A C*-algebra A is said to be a (separable) AI-algebra if A is isomorphic to an inductive limit of the form $lim_n (C[0,1]\otimes F_n)$ with $F_n$ a finite dimensional C*-algebra for every n. Whenever A is unital and commutative, A is isomorphic to C(X) with X an inverse limit of finite disjoint copies of unit intervals.
In this 2-session talk, we will study the range problem for the Cuntz semigroup of AI-algebras. That is, we will study whether or not one can determine a natural set of properties that an abstract Cuntz semigroup must satisfy in order to be isomorphic to the Cuntz semigroup of an AI-algebra.
During the first part of the talk, we will focus on unital commutative AI-algebras. In this case, one is able to solve the range problem for this class, thus giving a list of properties that an abstract Cuntz semigroup S satisfies if and only if S is isomorphic to the Cuntz semigroup of such an algebra. In order to prove this result, we first introduce the notion of almost chainable spaces and prove that a compact metric space X is almost chainable if and only if C(X) is an AI-algebra. We also characterize when S is isomorphic to the Cuntz semigroup of lower-semicontinuous functions $X-amp;gt{0,1,...,\infty }$ for some T1-space X. The results in this first session will appear in [4].
In the second session, we will present a local characterization for the Cuntz semigroup of any AI-algebra resembling Shen's local characterization of dimension groups[3], later used in the celebrated Effros-Handelman-Shen theorem[2]. One of the key features in the proof of our result will be the notion of Cauchy sequences for Cu-morphisms (with respect to the distance introduced in [1]) and the fact that, under the right assumptions, they have a unique limit; see [5].
$[1]$ Ciuperca, A. and Elliott, G. "A remark on invariants for C*-algebras of stable rank one", Int. Math. Res. Not. IMRN(2008)
$[2]$ Effros, E. G. and Handelman, D. E. and Shen, C. L. "Dimension groups and their affine representations", Amer. J. Math.102(1980), 385–407.
$[3]$ Shen, C. L. "On the classification of the ordered groups associated with the approximately finite dimensional C*-algebras" ,Duke Math. J.46(1979), 613–633.
$[4]$ Vilalta, E. "The Cuntz semigroup of unital commutative AI-algebras", in preparation.
$[5]$ Vilalta, E. "A local characterization for the Cuntz semigroup of AI-algebras", (preprint) arXiv:2102.13557 [math.OA]
Ring Theory SeminarSpeaker: Laurent Cantier (Universitat Autònoma de Barcelona)
Title: The Cu1 semigroup as an invariant for K1-obstruction cases
Date: 11/3/2021
Time: 15:00
Abstract: The aim of this talk is to explicitly shows that the unitary Cuntz semigroup, defined using the Cuntz semigroup and the K1-group, strictly contains more information than the latter invariants alone. To that end, we construct two C*-algebras, distinguished by their unitary Cuntz semigroup, whose K-Theory and Cu-semigroup are isomorphic. Both A and B, constructed as inductive limits of NCCW 1-algebras, are non-simple unital separable C∗-algebras of stable rank one with K1-obstructions. This shows that a likewise invariant is necessary in order to extend classification results of C*-algebras by means of Cuntz semigroup to the non trivial K1 group case.
Ring Theory SeminarSpeaker: Giovanna Le Gros (Università di Padova)
Title: Enveloping and n-tilting classes over commutative rings
Date: 2/2/2021
Time: 15:00
Abstract: In this talk, we will discuss approximations by tilting classes over commutative rings, where tilting classes are generated by infinitely generated tilting modules. Recently, all the commutative rings over which 1-tilting classes are enveloping were classified. This used the classification of 1-tilting classes over commutative rings by faithful finitely generated Gabriel topologies proved by Hrbek. This classification was extended to the general tilting case by Hrbek-Stovicek, where instead the correspondence is with suitable finite sequences of Gabriel topologies. In this talk we will discuss some results toward the classification of tilting cotorsion pairs that provide approximations, using Hrbek and Stovicek's classification. In particular, we will discuss how considering an induced tilting class in suitable factor rings retains useful properties of the original tilting class and their approximations.
This talk is based on current work with Dolors Herbera.
Geometry SeminarSpeaker: Daniel Räde (Augsburg)
Title: Macroscopic band width inequalities
Date: 2/4/2020
Time: 13:00
Web: http://mat.uab.cat/web/ligat/
Abstract: If $M^{n-1}$ is a closed smooth manifold and $g$ is a smooth Riemannian metric on $V:=M\times[0,1]$, we call $(V,g)$ a Riemannian band over $M$. The width of the Riemannian band $(V,g$ is defined to be the distance between the two boundary components.
In a recent paper M. Gromov conjectured that if $M$ does not admit a metric with positive scalar curvature and if we assume that $Sc(V,g)\geq\sigma>0$, then $width(V,g)$ is bounded from above by a sharp constant only depending on the dimension $n$ and $\sigma$. He proved this conjecture for several classes of manifolds including the torus $T^{n-1}$.
In this talk we want to discuss some results regarding band width estimates under a different condition on the metric. Instead of a lower scalar curvature bound we assume that unit balls in $(V,g)$ or in the universal cover $(\tilde{V},\tilde{g})$ have very small volume. This is based on L. Guth's notion of macroscopic scalar curvature and relates to systolic geometry.
Geometry SeminarSpeaker: Dan Agüero (IMPA)
Title: New invariants and a splitting theorem for complex Dirac structures
Date: 26/3/2020
Time: 12:00
Web: http://mat.uab.cat/web/ligat/
Abstract: In this talk, we start by recalling the relationship between Poisson and Dirac geometry. We use this as motivation for studying complex Dirac structures with constant real index. Then we introduce a new invariant, the order and redefine the previously known invariant: the type. Finally we give a local description for complex Dirac structures with constant real index and order via a splitting theorem.
Geometry SeminarSpeaker: Jérôme Los (Aix-Marseille)
Title: Volume entropy for surface groups via dynamics.
Date: 12/3/2020
Time: 12:00
Web: http://mat.uab.cat/web/ligat/
Abstract: In this talk, I will explain how we can use the dynamics of Bowen-Series Like maps to compute explicitly the volume entropy for a class of presentations of surface groups. In particular the « classical » presentations are covered by this class and it gives a way to make the computation explicite for all surfaces.
Ring Theory SeminarSpeaker: Joachim Zacharias (University of Glasgow)
Title: AF-embeddings and quotients of the Cantor set
Date: 5/3/2020
Time: 11:30
Abstract: The classical Aleksandrov-Uryson Theorem says that every compact metric space X is a quotient of the Cantor set S, hence the C*-algebra C(X) of continuous functions on X embeds into C(S), an AF algebra, i.e. an inductive limit of finite dimensional C*-algebras. Thus every separable commutative C*-algebra is AF-embeddable. Whilst this cannot be true for arbitrary separable non-commutative C*-algebras such embeddings into AF-algebras have been established in many cases. We explore how the proof of the classical A-U-Theorem can be mimicked to obtain AF-embeddings and related results for classes of non-commutative C*-algebras.
Ring Theory SeminarSpeaker: Ferran Cedó (Universitat Autònoma de Barcelona)
Title: Construcció de noves braces finites simple
Date: 27/2/2020
Time: 11:30
Abstract: Aquest és un treball conjunt amb l'Eric Jespers i el Jan Okninski. Donat un grup abelià finit $A$ qualsevol, explicaré com construir braces simples finites amb grup multiplicatiu metabelià (és a dir, amb longitud derivada 2) tals que $A$ és isomorf a un subgrup del seu grup additiu. Abans d'aquest treball, cap de les braces simple finites conegudes contenia elements amb ordre additiu 4. En un treball anterior (junt amb David Bachiller, Eric Jespers i Jan Okninski), s'havien construït braces finites simples tals que el seu grup additiu contenia qualsevol grup abelià prefixat d'ordre senar, però el grup multiplicatiu d'aquestes braces era de longitud derivada 3.
Geometry SeminarSpeaker: Jérôme Los (Aix Marseille)
Title: Geometrization of some piecewise homeomorphisms of the circle.
Date: 20/2/2020
Time: 12:00
Web: http://mat.uab.cat/web/ligat/
Abstract: In this talk I will describe a class of piecewise homeomorphisms of the circle from which we can construct a subgroup of the group Home^+ (S^1).
In this particular class we show that the group in question is a surface group and the map is a Bowen-Series-Like map for that group.
Geometry SeminarSpeaker: Dmitry Novikov (Weizmann Institute)
Title: Complex Cellular Structures (joint with Gal Binyamini)
Date: 13/2/2020
Time: 15:00
Web: http://mat.uab.cat/web/ligat/
Abstract: Real semialgebraic sets admit so-called cellular decomposition, i.e. representation as a union of convenient semialgebraic images of standard cubes.
The Gromov-Yomdin Lemma (latter generalized by Pila and Wilkie) proves that the maps could be chosen of $C^r$-norm at most one, and the number of such maps is uniformly bounded for finite-dimensional families.
This number was not bounded by Yomdin or Gromov, but it necessarily grows as $r\to\infty$.
We explained the obstruction to complexification of this result in terms of the inner hyperbolic metric properties of complex holomorphic sets.
Further, we accidentally proved a new simple lemma about holomorphic functions in annulii, a quantitative version of great Picard theorem. This lemma allowed us to construct a proper holomorphic version of the above results in all dimensions, effective and with explicit polynomial bounds on complexity for families of complex (sub)analytic and semialgebraic sets, combining best properties of both aforementioned results.
As first corollaries we effectively bound the Yomdin-Gromov number (thus proving a long-standing Yomdin conjecture about tail entropy) and prove a bound on the number of rational points in $\log$-sets in the spirit of Wilkie conjecture.
Ring Theory SeminarSpeaker: Eric Jespers (Vrije Universiteit Brussel)
Title: Associative structures associated to set-theoretic solutions of the Yang--Baxter equation
Date: 5/2/2020
Time: 11:00
Abstract: Let $(X,r)$ be a set-theoretic solution of the YBE, that is $X$ is a set and $r\colon X\times X \to X\times X$ satisfies
$$(r \times id)\circ (id \times r)\circ (r \times id) = (id \times r)\circ (r \times id)\circ (id \times r)$$ on $X^{3}$. Write $r(x,y)=(\lambda_x (y), \rho_y (x))$, for $x,y\in X$. Gateva-Ivanova and Majid showed that the study of such solutions is determined by solutions $(M,r_M)$, where
\[M=M(X,r) =\langle x\in X\mid xy=\lambda_x(y) \rho_y(x), \text{ for all } x,y\in X \rangle\]
is the structure monoid of $(X,r)$, and $r_M$ restricts to $r$ on $X^2$. For left non-degenerate solutions, i.e. all $\sigma_x$ are bijective, it has been shown that $M(X,r)$ is a regular submonoid of $A(X,r)\times \mathcal{G}(X,r)$, where $\mathcal{G}(X,r)=\langle \lambda_x\mid x\in X\rangle$ is the permutation group of $(X,r)$, and
\[A(X,r) =\langle x\in X \mid x\ lambda_{x}(y) =\lambda_{x}(y) \lambda_{\sigma_{x}(y)}(\rho_{y}(x) \rangle\]
is the derived monoid of $(X,r)$. It also is the structure monoid of the rack solution $(X,r')$ with
$$r'(x,y)=(y,\lambda_y\rho_{\lambda^{-1}_x(y)}(x)).$$
This solution ``encodes'' the relations determined by the map $r^{2} \colon X^{2} \to X^{2}$. The elements of $A=A(X,r)$ are normal, i.e. $aA=Aa$ for all $a\in A$. It is this ``richer structure'' that has been exploited by several authors to obtain information on the structure monoid $M(X,r)$ and the structure algebra $kM(X,r)$.
In this talk we report on some recent investigations for arbitrary solutions, i.e. not necessarily left non-degenerate nor bijective.
This is joint work with F. Ced\'o and C. Verwimp. We prove that there is a unique $1$-cocycle $M(X,r)\to A(X,r)$ and we determine when this mapping is injective, surjective, respectively bijective. One then obtains a monoid homomorphism $M(X,r) \to A(X,r)\times \langle \sigma_x \mid x\in X\ rangle$. This mapping is injective when all $\sigma_x$ are injective. Further we determine the left cancellative congruence $\eta$ on $M(X,r)$ and show that $(X,r)$ induces a set-theoretic solution in $M(X,r)/\eta$ provided $(X,r)$ is left non-degenerate.
Geometry SeminarSpeaker: Lucas Ambrozio (Warwick)
Title: Min-max width and volume of Riemannian three-dimensional spheres
Date: 30/1/2020
Time: 12:00
Web: http://mat.uab.cat/web/ligat/
Abstract: By the work of Simon and Smith, every Riemannian three-dimensional sphere contains an embedded minimal two-dimensional sphere. Their method of construction is a min-max method for the area functional and the area of this minimal sphere is bounded from above by a number depending only on the ambient geometry, known as the width. In this talk, we will discuss upper bounds for the width among certain classes of metrics with the same volume. This is joint work with Rafael Montezuma (UMass-Amhrest).
Ring Theory SeminarSpeaker: Maria Stella Adamo(University of Rome "Tor Vergata)
Title: Cuntz-Pimsner algebras associated to C*-correspondences over commutative C*-algebras
Date: 23/1/2020
Time: 11:00
Web: http://mat.uab.cat/web/ligat/
Abstract: In this talk, structural properties of Cuntz-Pimsner algebras arising by full, minimal, non-periodic, and finitely generated C*-correspondences over commutative C*-algebras will be discussed. A broad class of examples is provided considering the continuous sections $\Gamma(V,\varphi)$ of a complex locally trivial vector bundle $V$ on a compact metric space $X$ twisted by a minimal homeomorphism $\varphi: X\to X$.
In this case, we identify a "large enough" C*-subalgebra that captures the fundamental properties of the containing Cuntz-Pimsner algebra. Lastly, we will examine conditions when these C*-algebras can be classified using the Elliott invariant.
This is joint work in progress with Archey, Forough, Georgescu, Jeong, Strung, Viola.
Geometry SeminarSpeaker: Florent Balacheff (UAB)
Title: Entropia de volum i longituds de corbas homotòpicament independents
Date: 16/1/2020
Time: 12:00
Web: http://mat.uab.cat/web/ligat/
Abstract: Presentaré una desigualtat per les varietats riemannianes tancades que impliqua l'entropía del volum i el conjunt de longituds de qualsevol família de corbes homotòpicament independents basats en un mateix punt. Aquesta desigualitat implica un teorema del collaret universal, és a dir sense restricció de curvatura. És un treball en col.laboració amb el Louis Merlin.
Topology SeminarSpeaker: Rune Haugseng (Trondheim)
Title: The universal property of bispans
Date: 10/1/2020
Time: 12:00
Web: http://mat.uab.cat/~topalg
Abstract:
Geometry SeminarSpeaker: Andrea Sambusetti (Roma)
Title: Topological rigidity and finiteness for non-geometric 3-manifolds
Date: 19/12/2019
Time: 12:00
Web: http://mat.uab.cat/web/ligat/
Abstract: The Riemannian geometry of non-geometric 3-manifolds (that is, those which do not admit any of the eight complete maximal homogeneous 3-dimensional geometries) deserved considerably less attention than their geometric counterparts, with a few remarkable exceptions. In this seminar, we will explain some peculiar topological rigidity and finiteness properties of the class of non-geometric Riemannian 3-manifolds with bounded entropy and diameter, with respect to the Gromov-Hausdorff distance. The talk is based on the papers https://arxiv.org/abs/1705.06213 and https://arxiv.org/abs/1711.06210 in collaboration with F. Cerocchi"
Geometry SeminarSpeaker: Sorin Dumitrescu (Université Côte d'Azur, Nice)
Title: Rational parallelisms and generalized Cartan geometries on complex manifolds
Date: 5/12/2019
Time: 12:00
Web: http://mat.uab.cat/web/ligat/
Abstract: This talk deals with (generalized) holomorphic Cartan geometries on compact complex manifols. The concept of holomorphic Cartan geometry encapsulates many interesting geometric structures including holomorphic parallelisms, holomorphic Riemannian metrics, holomorphic conformal structures, holomorphic affine connections or holomorphic projective connections. A more flexible notion is that of a generalized Cartan geometry which allows some degeneracy of the geometric structure. This encapsulates for example some interesting rational parallelisms.
We discuss classification and uniformization results for compact complex manifolds bearing (generalized) holomorphic Cartan geometries. This is joint work with Indranil Biswas (TIFR, Mumbai).
Ring Theory SeminarSpeaker: Eduard Vilalta (UAB)
Title: The real rank of uniform Roe algebras"
Date: 28/11/2019
Time: 11:00
Abstract: The aim of this 2-session seminar is to introduce the relation that has recently been found between the asymptotic dimension of a bounded geometry metric space X and the real rank of its associated uniform Roe algebra $C^*u(X)$ [1].
During the first session, I will give the definitions and results that will be needed for the second part. These include the real and stable rank of a C*-algebra [2], the asymptotic dimension of both a topological space and a group[3], and the uniform Roe algebra of a bounded geometry metric space[4].
In the second session, I will follow [1] to prove that, given a bounded geometry metric space X, the real rank of $C^*u(X)$ is 0 whenever the asymptotic dimension of X is 0. I will also explain the involvement of the first Chern class in the computation of the k0-group of $C^*u(Z^2)$, which is used in [1] to prove that the real rank of this algebra is non-zero.
$[1]$ K. Li and R. Willet. "Low Dimensional Properties of Uniform Roe Algebras". Journal of the London Mathematical Society, 97:98–124, 2018.
$[2]$ L.G. Brown and G.K. Pedersen. "C*-Algebras of Real Rank Zero". Journal of Functional Analysis, 99:131–149, 1991.
$[3]$ G. Bell and A. Dranishnikov. "Asymptotic dimension". Topology and its Applications, (155):1265–1296, 2008.
$[4]$ N.P. Brown and N.Ozawa. "C*-Algebras and Finite-Dimensional Approximations", volume 88 of Graduate Studies in Mathematics. American Mathematical Society, 2008.
Geometry SeminarSpeaker: Pallavi Panda (Lille)
Title: Parametrisation of the deformation spaces of ideal polygons and punctured polygon
Date: 21/11/2019
Time: 12:00
Web: http://mat.uab.cat/web/ligat/
Abstract: To every hyperbolic surface, one can associate a simplicial complex called the arc complex whose 0-th skeleton is the set of isotopy classes of non-trivial embedded arcs and there is a k-simplex for every (k+1)-tuple of pairwise disjoint and distinct isotopy classes. The arc complexes of ideal polygons and punctured polygons are finite and are homeomorphic to spheres. The deformation spaces of these two hyperbolic surfaces can be completely parametrised via their arc complexes using strip deformations along the arcs. This is a partial generalisation of a result by Dancinger-Guéritaud-Kassel.
Ring Theory SeminarSpeaker: Joan Claramunt (Universitat Autònoma de Barcelona)
Title: A correspondence between dynamical systems and separated graphs
Date: 14/11/2019
Time: 11:00
Abstract: In 1992 Herman, Putnam and Skau established (following the work of Versik) a bijective correspondence between essentially simple ordered Bratteli diagrams and essentially minimal dynamical systems. This correspondence enable the authors to study a particular subfamily of C*-crossed products (i.e. C(X) x Z given by a single homeomorphism f : X -amp;gt X; here X is the Cantor set).
In these 2-session seminars I would like to present the work obtained so far in extending the above correspondence between dynamical systems (not necessarily minimal) and (a special class of) separated graph algebras. In the first session I will introduce the basic definitions, concepts and known results which will be used throughout the 2-session seminar. In the second session I will concentrate on presenting the work obtained so far, which is joint work in progress with P. Ara and M. S. Adamo.
Topology SeminarSpeaker: Luis Javier Hernández Paricio (Universidad de La Rioja)
Title: Endomorphisms of the Hopf fibration and numerical methods
Date: 13/11/2019
Time: 10:00
Web: http://mat.uab.cat/~topalg
Abstract: We have developed and implemented in Julia language a collection of algorithms for the iteration of a rational function that avoids the problem of overflows caused by denominators close to zero and the problem of indetermination which appears when simultaneously the numerator and denominator are equal to zero. This is solved by working with homogeneous coordinates and the iteration of a homogeneous pair of bivariate polynomials.
This homogeneous pair induces in a canonical way a self-map of the pointed Hopf fibration. Moreover, if the homogenous pair is irreducible, we also have a self-map of the standard Hopf fibration. We study the points of indeterminacy evaluating a canonical map associated with a homogeneous pair on the orbit of a point of the Riemann sphere.
These algorithms can be applied to any numerical method that builds a rational map to find the roots of an univariate polynomial equation. In particular with these procedures we analyze the existence of multiple roots for the Newton method and the relaxed Newton method.
This project is being developed together with J.I. Extremiana, J. M. Guti\'errez and M. T. Rivas (University of La Rioja).
Geometry SeminarSpeaker: Marcos Cossarini (UPEM)
Title: Discrete surfaces with length and area and minimal fillings of the circle.
Date: 7/11/2019
Time: 12:00
Web: http://mat.uab.cat/web/ligat/
Abstract: We propose to imagine that every Riemannian metric on a surface is discrete at the small scale, made of curves called walls. The length of a curve is its number of crossings with the walls, and the area of the surface is the number of crossings between the walls themselves. We show how to approximate a Riemannian or self-reverse Finsler metric by a wallsystem.
This work is motivated by Gromov's filling area conjecture (FAC) that the hemisphere has minimum area among orientable Riemannian surfaces that fill isometrically a closed curve of given length. (A surface fills its boundary curve isometrically if the distance between each pair of boundary points measured along the surface is not less than the distance measured along the boundary.) We introduce a discrete FAC: every square-celled surface that fills isometrically a 2n-cycle graph has at least n(n-1)/2 squares. This conjecture is equivalent to the FAC extended to surfaces with self-reverse Finsler metric.
If the surface is a disk, the discrete FAC follows from Steinitz's algorithm for transforming curves into pseudolines. This gives a new, combinatorial proof that the FAC holds for disks with Riemannian or self-reverse Finsler metric.
If time allows, we also discuss how to discretize a directed metric on a surface using a triangulation with directed edges. The length of each edge is 1 in one way and 0 in the other way, and the area of the surface is the number of triangles. These discrete surfaces are dual to Postnikov's plabic graphs.
Geometry SeminarSpeaker: Adrien Boulanger (Aix-Marseille Université)
Title: Counting problems in infinite measure
Date: 31/10/2019
Time: 12:00
Abstract: Given a group $\Gamma$ acting properly discontinuously and by isometries on a metric space $X$, one can wonder how grows the orbit of a given point. More precisely, given two points $x,y \in X$ and $\rho > 0$, we define the orbital function as
$$ N_{\Gamma}(x,y,\rho) := \sharp ( \Gamma \cdot y \cap B(x,\rho)) \ , $$
where $B(x,\rho)$ denotes the ball centred at $x$ of radius $\rho$. A counting problem consists to estimate the counting function when $\rho \to \infty$.
In the setting of groups acting on hyperbolic spaces this question was widely investigated for decades, with mainly two different approaches: an analytical one relying on Selberg's pre-trace formula due to Huber in the 50's and a dynamical one relying on the mixing of the geodesic flow due to Margulis in the late 60's.
During the talk, we shall describe Margulis' dynamical method in order to motivate the introduction of the Brownian motion. Combined with the use of the pre-trace formula, we shall establish a counting theorem linking the heat kernel of the quotient manifold and the orbital function. If the time allows it, we shall also review a couple of corollaries of the approach.
Topology SeminarSpeaker: Guillem Sala (UPC)
Title: Topological Cyclic Homology and L-functions
Date: 25/10/2019
Time: 12:00
Web: http://mat.uab.cat/~topalg
Abstract: It has already been noted in the past that there is a deep connection between number theory, algebraic geometry and algebraic topology. An example of this was Grothendieck's proof of the rationality part of the Weil conjectures, where he provided an étale cohomological interpretation of the Hasse-Weil zeta function for "nice" varieties over finite fields.
The goal of this talk is to follow the work of Lars Hesselholt and extend this result to the realm of homotopy theory, providing a cohomological interpretation of the Hasse-Weil zeta function using the cohomology associated to a certain spectrum, namely the Topological Periodic Cyclic Homology spectrum.
Geometry SeminarSpeaker: Ignasi Mundet Riera (UB)
Title: Subgrups compactes de grups de difeomorfismes i propietat Jordan
Date: 24/10/2019
Time: 12:00
Web: http://mat.uab.cat/web/ligat/
Abstract:
Ring Theory SeminarSpeaker: Joan Bosa (Universitat Autònoma de Barcelona)
Title: Villadsen algebras : Projections and Vector Bundles
Date: 24/10/2019
Time: 11:00
Abstract: Les Villadsen àlgebres són un tipus de C*-àlgebres que van ser utilitzades per trobar contraexemples a la conjectura de Classificació d'Elliott. Per provar que la conjectura fallava van utilitzar que l'ordre de les projeccions sobre espais topologics s'associa a l'ordre entre els vector bundles d'aquests. Així, en les Villadsen àlgebres s'utilitza fortament la teoria de vector bundles per tal de construir l'exemple dessitjat. En aquesta xerrada explicarem una mica la història de la classificació de C*-àlgebres, i donarem algunes pinzellades sobre com utilitzar la teoria de vector bundles i les classes de Chern al món de les C*-àlgebres.
Geometry SeminarSpeaker: Gil Solanes (UAB)
Title: Valoracions de Lipschitz-Killing en varietats pseudo-riemannianes
Date: 10/10/2019
Time: 12:00
Web: http://mat.uab.cat/web/ligat/
Abstract: Les curvatures de Lipschitz-Killing són invariants riemannians que apareixen en situacions tan diverses com la fórmula dels tubs de Weyl i l'espectre del laplacià de les formes diferencials. Les valoracions de Lipschitz-Killing són l'extensió d'aquests invariants als subconjunts compactes suficientment regulars d'una varietat riemanniana.
Presentarem un treball conjunt amb A. Bernig i D. Faifman on estenem les valoracions de Lipschitz-Killing al context de les varietats pseudo-riemannianes.
Ring Theory SeminarSpeaker: Giovanna Le Gros (University of Padova)
Title: Minimal approximations and 1-tilting cotorsion pairs over commutative rings
Date: 10/10/2019
Time: 11:00
Abstract: Minimal approximations of modules, or covers and envelopes of modules, were introduced as a tool to approximate modules by classes of modules which are more manageable. For a class C of R-modules, the aim is to characterise the rings over which every module has a C-cover or C-envelope. Moreover A-precovers and B-preenvelopes are strongly related to the notion of a cotorsion pair (A,B).
In this talk we are interested in the particular case that (P_1,B) is the cotorsion pair generated by the modules of projective dimension at most one (denoted P_1) over commutative rings. More precisely, we investigate over which rings these cotorsion pairs admit covers or envelopes. Furthermore, we interested in Enochs' Conjecture in this setting, that is if P_1 is covering necessarily implies that it is closed under direct limits. The investigation of the cotorsion pair (P_1,B) splits into two cases: when the cotorsion pair is of finite type and when it is not. In this talk I will outline some results for the case that the cotorsion pair is of finite type, where we consider more generally a 1-tilting cotorsion pair over a commutative ring.
Ring Theory SeminarSpeaker: Cristóbal Gil (Universidad de Málaga)
Title: Representations of relative Cohn path algebras
Date: 19/9/2019
Time: 10:30
Abstract: In this talk we study relative Cohn path algebras, also known as Leavitt-Cohn path algebras. Given any graph E we define E-relative branching systems and prove how they induce representations of the associated relative Cohn path algebra. We give necessary and su cient conditions for faithfulness of the
representations associated to E-relative branching systems (this improves previous results known to Leavitt path algebras of row-finite graphs with no sinks).
Topology SeminarSpeaker: Michelle Strumila (University of Melbourne)
Title: Infinity operads and their generalisations
Date: 9/9/2019
Time: 15:00
Web: http://mat.uab.cat/~topalg
Abstract: Infinity categories are a way of taking categories up to homotopy. This talk is about how this can be extended to infinity operads, along with generalisations to the non-directional and higher genus cases.
Topology SeminarSpeaker: Assaf Libman (University of Aberdeen)
Title: Selfmaps of equivariant spheres
Date: 6/9/2019
Time: 12:00
Web: http://mat.uab.cat/~topalg
Abstract: We describe a stabilization property of the homotopy groups of space of equivariant self maps of spheres with action of a finite group G. This result gives an extension (of a special case of) tom-Dieck's splitting theorem to incomplete universes.
Topology SeminarSpeaker: Eva Belmont (Northwestern University)
Title: The motivic Adams spectral sequence
Date: 6/9/2019
Time: 11:00
Web: http://mat.uab.cat/~topalg
Abstract: The Adams spectral sequence is one of the main tools for computing stable homotopy groups of spheres. In this talk, I will give an introduction to the Adams spectral sequence in motivic homotopy theory over C and over R, and describe some connections with classical and C_2-equivariant homotopy theory. I will describe joint work with Dan Isaksen to compute the motivic Adams spectral sequence over R and obtain applications to the Mahowald invariant.
Topology SeminarSpeaker: Mark Penney (MPIM Bonn)
Title: Dijkgraaf-Witten invariants of 2-knots
Date: 26/7/2019
Time: 12:00
Web: http://mat.uab.cat/~topalg
Abstract: The aim of this talk is to introduce a family of invariants of 2-knots which generalize the Dijkgraaf-Witten (DW) knot invariants. I will begin with a casual review of the DW knot invariants, making connections to Fox n-colourings in the process. The naive generalization to 2-knots yields much weaker invariants and so I will discuss a homotopy-theoretic generalization which has the potential to yield finer results.
Topology SeminarSpeaker: Philip Hackney
Title: Right adjoints to operadic restriction functors
Date: 18/7/2019
Time: 09:30
Web: http://mat.uab.cat/~topalg
Abstract: If f : P → Q is a morphism of operads, then there is a restriction functor from Q-algebras to P-algebras. This restriction functor generally admits a left adjoint. This restriction may or may not admit a right adjoint: if G → H is a group homomorphism, then the forgetful functor from H-sets to G-sets has a right adjoint, while there is no right adjoint to the functor from commutative algebras to associative algebras. In this talk, we provide a concise necessary and sufficient condition for the existence of a right adjoint to the restriction functor, phrased in terms of the operad map f. We give a simple formula for this right adjoint, and examine the criterion in special cases. All of this is applicable over quite general ground categories. (Joint work with Gabriel C. Drummond-Cole)
Topology SeminarSpeaker: Matt Feller (University of Virginia)
Title: New model structures on simplicial sets
Date: 5/7/2019
Time: 12:00
Web: http:// mat.uab.cat/~topalg
Abstract: In the way Kan complexes and quasi-categories model up-to-homotopy groupoids and categories, can we find model structures on simplicial sets which give up-to-homotopy versions of more general objects? We investigate this question, with the particular motivating example of 2-Segal sets. Cisinski's work on model structures in presheaf categories provides abstract blueprints for these new model structures, but turning these blueprints into a satisfying description is a nontrivial task. As a first step, we describe the minimal model structure on simplicial sets arising from Cisinski's theory.
Ring Theory SeminarSpeaker: Roozbeh Hazrat(Western Sydney University)
Title: The talented monoid of a Leavitt path algebra
Date: 4/7/2019
Time: 12:00
Abstract: There is a tight relation between the geometry of a directed graph and the algebraic structure of a Leavitt path algebra associated to it. We show a similar connection between the geometry of the graph and the structure of a certain monoid associated to it. This monoid is isomorphic to the positive cone of the graded K0-group of the Leavitt path algebra which is naturally equipped with a Z-action. As an example, we show that a graph has a cycle without an exit if and only if the monoid has a periodic element. Consequently a graph has Condition (L) if and only if the group Z acts freely on the monoid. We go on to show that the algebraic structure of Leavitt path algebras (such as simplicity, purely infinite simplicity, or the lattice of ideals) can be described completely via this monoid. Therefore an isomorphism between the monoids (or graded K0’s) of two Leavitt path algebras implies that the algebras have similar algebraic structures. These all confirm that the graded Grothendieck group could be a sought-after complete invariant for the classification of Leavitt path algebras.
This is joint work with Huanhuan Li.
Ring Theory SeminarSpeaker: Huanhuan Li (Western Sydney University)
Title: The injective Leavitt complex
Date: 4/7/2019
Time: 11:00
Abstract: For a finite graph E without sinks, we consider the corresponding finite dimensional algebra A with radical square zero. We construct an explicit compact generator for the homotopy category of acyclic complexes of injective A-modules. We call such a generator the injective Leavitt complex of E. This terminology is justified by the following result: the differential graded endomorphism algebra of the injective Leavitt complex of E is quasi-isomorphic to the Leavitt path algebra of E. Here, the Leavitt path algebra is naturally Z-graded and viewed as a differential graded algebra with trivial differential.
Geometry SeminarSpeaker: Samir Bedrouni (USTHB, Alger)
Title: Convex foliations of degree four on the complex projective plane
Date: 27/6/2019
Time: 12:00
Web: http://mat.uab.cat/web/ligat/
Abstract: In this talk, I will present the main results of a recent paper in collaboration with D. Marín, cf. arXiv:1811.07735. First, I will explain the outline of the proof of the result which states that up to automorphism there are 5 homogeneous convex foliations of degree four on the complex projective plane. Second, we will see how to use this result to obtain a partial answer to a question posed in 2013 by D. Marín and J. Pereira about the classification of reduced convex foliations on the complex projective plane.
Geometry SeminarSpeaker: Nabil Kahouadji (Northeastern Illinois University)
Title: Isometric Immersions of Pseudo-Spherical Surfaces via PDEs.
Date: 27/6/2019
Time: 10:45
Web: http://mat.uab.cat/web/ligat/
Abstract: Pseudo-spherical surfaces are surfaces of constant negative Gaussian curvature. A way of realizing such a surface in 3d space as a surface of revolution is obtained by rotating the graph of a curve called tractrix around the z-axis (infinite funnel). There is a remarkable connection between the so- lutions of the sine-Gordon equation uxt = sin u and pseudo-spherical surfaces, in the sense that every generic solution of this equation can be shown to give rise to a pseudo-spherical surface. Furthermore, the sine-Gordon equation has the property that the way in which the pseudo-spherical surfaces corresponding to its solutions are realized geometrically in 3d space is given in closed form through some remarkable explicit formulas. The sine-Gordon equation is but one member of a very large class of differential equations whose solutions likewise define pseudo-spherical surfaces. These were defined and classified by Chern, Tenenblat [1] and others, and include almost all the known examples of "integrable" partial differential equations. This raises the question of whether the other equations enjoy the same remarkable property as the sine-Gordon equation when it comes to the realization of the corresponding surfaces in 3d space. We will see that the answer is no, and will provide a full classification of hyper- bolic and evolution equations [2, 3, 4]. The classification results will show, among other things, that the sine-Gordon equation is quite unique in this regard amongst all integrable equations.
Ring Theory SeminarSpeaker: Eduard Ortega (NTNU Norwegian University of Science and Technology)
Title: Group topologic del goupoide d'accions auto-similars en un graf
Date: 20/6/2019
Time: 11:45
Abstract: En aquesta xerrada definiré què és una acció auto-similar en un graf (Exel-Pardo) i definiré el seu grupoide. Després calcularem el seu grup topològic i el relacionarem amb la homologia del grupoide.
Topology SeminarSpeaker: Marc Stephan (Max Planck Institute for Mathematics, Bonn)
Title: A multiplicative spectral sequence for free p-group actions
Date: 24/5/2019
Time: 12:00
Web: http://mat.uab.cat/~topalg
Abstract: Carlsson conjectured that if a finite CW complex admits a free action by an elementary abelian p-group G of rank n, then the sum of its mod-p Betti numbers is at least $2^n$. In 2017, Iyengar and Walker constructed equivariant chain complexes that are counterexamples to an algebraic version of Carlsson’s conjecture. This raised the question if these chain complexes can be realized topologically by free G-spaces to produce counterexamples to Carlsson’s conjecture. In this talk, I will explain multiplicative properties of the spectral sequence obtained by filtering the mod-p cochains of a space with a free p-group action by powers of the augmentation ideal and deduce that the counterexamples can not be realized topologically. This is joint work with Henrik Rüping.
Topology SeminarSpeaker: Sune Precht Reeh (BGSMath-UAB)
Title: A formula for p-completion by way of the Segal conjecture
Date: 10/5/2019
Time: 10:00
Web: http://mat.uab.cat/~topalg
Abstract: A variant of the Segal conjecture (theorem by Carlsson) gives a correspondence between homotopy classes of stable maps from BG to BH and the module of (G,H)-bisets that are H-free and where the module is completed with respect to the augmentation ideal I(G) in the Burnside ring of G. The details of this correspondence change depending on whether you add a disjoint basepoint to BG, BH, or both, and it is also not a priori clear what algebraic consequences the I(G)-adic completion has for the module of (G,H)-bisets. Separately, we have the functor of p-completion for spaces or spectra. We can apply p-completion to each classifying space BG, and according to the Martino-Priddy conjecture (theorem by Oliver) the p-completed classifying space depends only on the saturated fusion system $\mathcal F_p(G)$ of G at the prime p.
Saturated fusion systems also have modules of bisets, and so it is not unreasonable to ask how p-completion interacts with the Segal conjecture: Suppose we are given a (G,H)-biset, we can interpret the biset as a stable map from BG to BH. Apply p-completion to get a stable map from $B\mathcal F_p(G)$ to $B\mathcal F_p(H)$. By the Segal conjecture for fusion systems, that stable map corresponds to an $(\mathcal F_p(G), \mathcal F_p(H))$-biset -- up to p-adic completion. Which $(\mathcal F_p(G), \mathcal F_p(H))$-biset do we get?
This innocent question was the starting point for a joint paper with Tomer Schlank and Nathaniel Stapleton, and in my talk I will give an overview of all the categories involved and how they fit together with functors. If time permits, we will even see how p-completion and fusion systems can help us understand the I(G)-adic completion for any finite group -- and I suppose we might even consider that "a formula for the Segal conjecture by way of p-completion".
Geometry SeminarSpeaker: Robert Cardona (UPC)
Title: A contact topology approach to Euler flows universality
Date: 9/5/2019
Time: 12:00
Web: http://mat.uab.cat/web/ligat/
Abstract: There has been several steps towards establishing universality of Euler flows in the last years, especially in two papers by Terence Tao in 2017 and 2019. In this talk, we propose a new approach to this question. After presenting a correspondence theorem between some steady Euler flows and Reeb vector fields in contact geometry, we provide new results in the direction of Tao’s programme. By means of high dimensional contact topology, we prove some realization theorems on Reeb dynamics. We deduce some new universality properties for steady Euler flows, for instance “Turing universality” which was one of the suggested open problems in Tao’s papers. This is a joint work in progress with Eva Miranda, Daniel Peralta-Salas and Francisco Presas.
Topology SeminarSpeaker: Matthew Gelvin (Bilkent University, Ankara)
Title: Fusion-minimal groups
Date: 26/4/2019
Time: 12:00
Web: http://mat.uab.cat/~topalg
Abstract: Every saturated fusion system $\mathcal{F}$ on the $p$-group $S$ has an associated collection of characteristic bisets. These are $(S,S)$-bisets that determine $\mathcal{F}$, and are in turn determined by $\mathcal{F}$ up to a more-or-less explicit parameterization. In particular, there is always a unique minimal $\mathcal{F}$-characteristic biset, $\Omega_\mathcal{F}$. If $G$ is a finite group containing $S$ as a Sylow $p$-subgroup and realizing $\mathcal{F}$, then $G$ is itself, when viewed as an $(S,S)$-biset, $\mathcal{F}$-characteristic. If it happens that $_SG_S=\Omega_\mathcal{F}$ is the minimal biset for its fusion system, we say that $G$ is \emph{fusion-minimal}.
In joint work with Sune Reeh, it was shown that any strictly $p$-constrained group (i.e., one that satisfies $C_G(O_p(G))\leq O_p(G)$) is fusion minimal. We conjecture that converse implication holds. In this talk, based on joint work with Justin Lynd, we prove this to be the case when $p$ is odd and describe the obstruction to a complete proof. Along the way, we will draw a connection with the module structure of block algebras and how this relates to the question at hand.
Geometry SeminarSpeaker: Martin Henk (Technische Universität Berlin)
Title: The dual Minkowski problem
Date: 25/4/2019
Time: 12:00
Web: http://mat.uab.cat/web/ligat/
Abstract: The (classical) Minkowski problem asked for sufficient and necessary conditions such that a finite Borel measure on the sphere is the surface area measure of a convex body. Its solution, based on works by Minkowski, Aleksandrov and Fenchel&Jessen, is one of the centerpieces of the classical Brunn-Minkowski theory.
There are two far-reaching extensions of the classical Brunn-Minkowski theory, the $L_p$-Brunn-Minkowski theory and the dual Brunn-Minkowski theory. In the talk we will discuss the analog of the (classical) Minkowski problem within the dual Brunn-Minkowski theory, i.e., the characterization problem of the dual curvature measures of a convex body.
These measures were recently introduced by Huang, Lutwak, Yang and Zhang and they are the counterparts to the area measures within the dual theory.
(Based on joint works wit Karoly Böröczky Jr. and Hannes Pollehn)
Topology SeminarSpeaker: Joshua Hunt (University of Copenhagen)
Title: Lifting G-stable endotrivial modules
Date: 12/4/2019
Time: 12:00
Web: http://mat.uab.cat/~topalg
Abstract: Endotrivial modules of a finite group G are a class of modular representations that is interesting both because endotrivial modules have enough structure to allow us to classify them and because such modules give structural information about the stable module category of G. They form a group T(G) under tensor product, and Carlson and Thévenaz have classified the endotrivial modules of a p-group. We examine the restriction map from T(G) to T(S), where S is a Sylow p-subgroup of G, and provide an obstruction to lifting an endotrivial module from T(S) to T(G). This allows us to describe T(G) using only local information and to provide a counterexample to some conjectures about T(G). This is joint work with Tobias Barthel and Jesper Grodal.
Ring Theory SeminarSpeaker: Ferran Cedó (Universitat Autònoma de Barcelona)
Title: "Braided algebras of Gelfand-Kirillov dimension one"
Date: 11/4/2019
Time: 11:30
Abstract: Let $X$ be a finite set of cardinality $n>1$. Gateva-Ivanova posed interesting questions concerning the lower bound of $dim(A(K,X,r)_2)$, where $A(K,X,r)$ is the structure K-algebra of a non-degenerate square-free braided set (i.e. set-theoretic solution $(X,r)$ of the Yang-Baxter equation). In this talk I will explain recent results obtained in a joint work with Eric Jespers and Jan Okninski that answer some of these questions. Our main result is the following:
Theorem. Let $(X,r)$ be a finite non-degenerate square-free SD braided set. Suppose that $|X|>1$. Let K be a field. Then $dim(A(K,X,r)_2)\geq 2|X|-1$. Furthermore $dim(A(K,X,r)_2)=2|X|-1$ if and only if, up to isomorphism, one of the following holds.
1. $|X|$ is an odd prime and $(X,r)$ is the braided set associated to the dihedral quandle.
2. $|X|=2$ and $(X,r)$ is the trivial braided set.
3. $X=\{ 1,2,3\}$ and $r(x,y)=(\sigma_x(y),x)$, for $x,y\in X$, with $\sigma_1=\sigma_2=id$ and $\sigma_3=(1,2)$.
Geometry SeminarSpeaker: Cédric Oms (UPC)
Title: Hamiltonian Dynamics on Singular Symplectic Manifolds
Date: 4/4/2019
Time: 12:00
Web: http://mat.uab.cat/web/ligat/
Abstract: The study of singular symplectic manifolds was initiated by the work of Radko, who classified stable Poisson structures on surfaces. It was observed by Guillemin—Miranda—Pires that stable Poisson structures can be treated as a generalization of symplectic geometry by extending the de Rham complex. Since then, a lot has been done to understand the geometry, dynamics and topology of those manifolds.
We will explore the odd-dimensional case of those manifolds in this talk by extending the notion of contact manifolds to the singular setting. We prove existence of singular contact structures. We will prove that singularities allow do construct Reeb plugs and thereby disproving Weinstein conjecture in this setting. In particular, this yields the existence of proper Hamiltonians on singular symplectic manifolds without periodic Hamiltonian orbits.
This is joint work with Eva Miranda.
Ring Theory SeminarSpeaker: Joan Bosa (Universitat Autònoma de Barcelona)
Title: Ideals a les C*-algebres $O_\infty$ estables
Date: 4/4/2019
Time: 11:30
Abstract: Comentarem una nova técnica desenvolupada per Bosa-Gabe-Sims-White que permet seguir el comportament del conjunt d'ideals de les C*-algebres $O_\infty$ estables.
Topology SeminarSpeaker: Antonio Díaz (Universidad de Málaga)
Title: Fusion systems for profinite groups
Date: 29/3/2019
Time: 09:00
Web: http://mat.uab.cat/~topalg
Abstract: For both finite groups and compact Lie groups, there exist algebraic structures that encode their fusion patterns as well as their classifying spaces at a given prime. In this talk, I will introduce similar ideas for profinite groups and, in particular, for compact p-adic analytic groups. In particular, we will study classifying spaces and stable elements theorem for continuous cohomology. We will provide some concrete continuous cohomology computations.
This is an ongoing joint work with O. Garaialde, N. Mazza and S. Park.
Geometry SeminarSpeaker: Stefan Suhr (University of Bochum)
Title: A Morse theoretic Characterization of Zoll metrics
Date: 28/3/2019
Time: 11:00
Web: http://mat.uab.cat/web/ligat/
Abstract: From the Morse theoretic point of view Zoll metrics are rather peculiar. All critical sets of the energy on the loop space are nondegenerate critical manifolds diffeomorphic to the unit tangent bundle. This especially implies that min-max values associated to certain homology classes coincide. In my talk I will explain that the coincidence of these min-max values characterises Zoll metrics in any dimension. A specially focus will lie on the case of the 2-sphere. This is work in collaboration with Marco Mazzucchelli (ENS Lyon).
Geometry SeminarSpeaker: Clemens Huemer (UPC)
Title: Carathéodory's theorem in depth
Date: 21/3/2019
Time: 11:00
Web: http://mat.uab.cat/web/ligat/
Abstract: Let $S$ be a finite set of points in $\mathbb{R}^d$. Carathéodory's theorem states that if a point $q$ is inside the convex hull of $S$, then there exist points $p_1, \dots, p_{d+1}$ of $S$ such that $q$ is contained in the convex hull of $\{p_1,\dots,p_{d+1}\}$; that is $q$ is contained in the simplex defined by $\{p_1,\dots,p_{d+1}\}$.
We present a depth version of this theorem. Informally, we prove that if $q$ is "deep inside" the convex hull of $S$ then there exist "large" pairwise disjoint subsets $S_1,\dots,S_{d+1}$ of $S$, such that every simplex having a vertex
from each $S_i$ contains $q$.
We also prove depth versions of Helly’s and Kirchberger’s theorems.
This is a joint work with Ruy Fabila-Monroy.
Geometry SeminarSpeaker: Gilles Courtois (Institut de Mathématiques de Jussieu - Paris)
Title: Minimal entropy of manifolds and their fundamental group
Date: 14/3/2019
Time: 11:00
Web: http://mat.uab.cat/web/ligat/
Abstract: The Milnor-Svarc theorem says that the entropy of a closed Riemannian manifold is non zero if and only if its fundamental group has exponential growth but does not give an explicit relation between the minimal entropy of a manifold and the minimal entropy of its fundamental group. The goal of this talk is to explain this through examples and state that such relations hold for manifolds with Gromov-hyperbolic fundamental group.
Geometry SeminarSpeaker: Erika Pieroni (Università La Sapienza - Roma)
Title: Minimal Entropy of 3-manifolds
Date: 7/3/2019
Time: 11:00
Web: http://mat.uab.cat/web/ligat/
Abstract: We compute the minimal entropy of every closed, orientable 3-manifold, proving that the cube of this invariant behaves additively both with respect to the prime decomposition and the JSJ decomposition.
As a consequence, we deduce that the cube of the minimal entropy, in restriction of closed, orientable 3-manifolds, is proportional to the simplicial volume, where the proportionality constant only depends on the dimension n=3.
Geometry SeminarSpeaker: Laurent Meersseman (Université d'Angers)
Title: Kuranishi and Teichmüller
Date: 21/2/2019
Time: 11:00
Web: http://mat.uab.cat/web/ligat/
Abstract: Let $X$ be a compact complex manifold. The Kuranishi space of $X$ is an analytic space which encodes every small deformation of $X$. The Teichmüller space is a topological space formed by the classes of compact complex manifolds diffeomorphic to $X$ up to biholomorphisms smoothly isotopic to the identity. F. Catanese asked when these two spaces are locally homeomorphic. Unfortunatly, this almost never occurs. I will reformulate this question replacing these two spaces with stacks. I will then show that, if $X$ is Kähler, this new question has always a positive answer. Finally, I will discuss the non-kähler case.
Geometry SeminarSpeaker: Matias del Hoyo (Universidade Federal Fluminense, i visitante del CRM dentro de la Convocatoria Lluís Santaló financiada por l'Institut d'Estudis Catalans)
Title: Deformations of compact foliations
Date: 14/2/2019
Time: 11:00
Web: http://mat.uab.cat/web/ligat/
Abstract: In a recent work with R. Fernandes we show that a compact Hausdorff foliation over a compact connected manifold is rigid, in the sense that every one-parameter deformation of it must be trivial. We study the foliation by using the Lie theory of Lie groupoids and Lie algebroids. A foliation is an example of a Lie algebroid and it can always be integrated to its so-called holonomy groupoid. In this talk I will present the theorem and illustrate with examples, provide a glimpse on Lie groupoids and Lie algebroids, and discuss a sketch of our proof.
Geometry SeminarSpeaker: Alfredo Hubard (Paris Est-Marne La vallée)
Title: The branch-waist of Riemannian two-spheres.
Date: 31/1/2019
Time: 11:00
Web: http://mat.uab.cat/web/ligat/
Abstract: I will explain an equality between a min max quantity related to the diastole (or waist) of a map from a two sphere to a tree, and the largest antipodality of the sphere. The insights come from the theory of graph minors which we import to the Riemannian world. From these insights we improve some fundamental inequalities in curvature free Riemannian geometry. Joint work with Arnaud de Mesmay and Francis Lazarus.
Topology SeminarSpeaker: Jesper M. Møller (Københavns Universitet)
Title: Counting $p$-singular elements in finite groups of Lie type
Date: 25/1/2019
Time: 11:00
Web: http://mat.uab.cat/web/ligat/
Abstract: Let $G$ be a finite group and $p$ a prime number. We say that an element of $G$ is $p$-singular if its order is a power of $p$. Let $G_p$ be the set of $p$-singular elements in $G$, i.e. the union of the Sylow $p$-subgroups of $G$. In 1907, or even earlier, Frobenius proved that $|G|_p \mid |G_p|$: The number of $p$-singular elements in $G$ is divisible by the $p$-part of the order of $G$. The number of $p$-singular elements in a symmetric group is known. In this talk we discuss the number of $p$-singular elements in a finite (untwisted) group of Lie type in characteristic $p$. The situation in the cross-characteristic case will maybe also be considered.
Geometry SeminarSpeaker: Marco Mazzucchelli (ENS Lyon)
Title: On the boundary rigidity problem for surfaces
Date: 24/1/2019
Time: 11:00
Web: http://mat.uab.cat/web/ligat/
Abstract: The classical boundary rigidity problem asks whether, or to what extent, the inner geometry of a compact Riemannian manifold with boundary can be determined by means of boundary measurements, such as the distance function among boundary points, or the geodesic scattering map. In my talk I will review this problem and some of the known results that are valid for "simple" Riemannian manifolds. I will then sketch the proof of some recent boundary rigidity results for non-simple Riemannian surfaces, including surfaces with trapped geodesics or with non-convex boundary. The talk is based on joint work with Colin Guillarmou and Leo Tzou.
Topology SeminarSpeaker: Letterio Gatto (Politecnico di Torino)
Title: Hasse-Schmidt Derivations on Exterior Algebras and how to use them
Date: 18/1/2019
Time: 11:00
Web: http://mat.uab.cat/~topalg
Abstract: In the year 1937, Hasse \& Schmidt introduced the so-called higher derivations in Commutative Algebra, to generalize the notion of Taylor polynomial to positive characteristic. Exactly the same definition can be phrased in the context of exterior algebras, by replacing the ordinary associative commutative multiplication by the wedge product. Hasse-Schmidt derivations on exterior algebras embody a surprisingly rich theory that candidates itself to propose a unified framework for a number of theories otherwise considered distincts, such as, e.g., (quantum, equivariant) Schubert Calculus for complex Grassmannians. In the talk we shall focus on one of the simplest but most powerful tools of the theory, the integration by parts formula. It will enable us to guess the shape of the vertex operators arising in the representation theory of certain infinite dimensional Lie algebras. In spite of the fancy vocabulary used in the abstract, the talk shall be entirely self-contained and no special prerequisite, but elementary multi-linear algebra, will be required.
Geometry SeminarSpeaker: Carles Sáez (UAB)
Title: Which finite groups act smoothly on a given 4-manifold ?
Date: 17/1/2019
Time: 11:00
Web: http://mat.uab.cat/web/ligat/
Abstract: The problem of determining which finite groups (up to abstract isomorphism) can act smoothly and effectively on a given closed 4-manifold is in general a very difficult one. However, one can hope to prove theorems that impose restrictions on the finite groups that can act on closed $4$-manifolds. In particular, we are interested in the following property: an (infinite) group $G$ is called Jordan if there exists a constant $C>0$ such that every finite subgroup $H$ of $G$ has an abelian subgroup $A$ satisfying $[H:A] < C$. E. Ghys asked in the 90's if the diffeomorphism groups of closed manifolds are always Jordan. This is true for dimensions $2$ and $3$, but it was proved by Csikós, Pyber and Szabó that there are closed $4$-manifolds with non Jordan diffeomorphism group, the simplest example being $T^2 \times S^2$.
In this talk we prove that a slight generalization of the Jordan property (substituting abelian by nilpotent of nilpotency class at most 2) holds for all closed $4$-manifolds, and we use this result to give a partial answer to the question of which closed $4$-manifolds have Jordan diffeomorphism group. Finally, we will also discuss the Jordan property for groups of automorphisms of almost complex $4$-manifolds and for groups of symplectomorphisms of symplectic $4$-manifolds.
This is joint work with Ignasi Mundet i Riera.
Ring Theory SeminarSpeaker: Pere Ara (Universitat Autònoma de Barcelona)
Title: The Realization Problem
Date: 17/1/2019
Time: 09:00
Abstract:
Geometry SeminarSpeaker: Jean Raimbault (Universitat de Toulouse)
Title: The topology of arithmetic hyperbolic three-manifolds
Date: 10/1/2019
Time: 11:00
Web: http://mat.uab.cat/web/ligat/
Abstract: I will discuss various recent results that demonstrate the "particular beauty" of arithmetic congruence manifolds among the family of all hyperbolic manifolds of finite volume. In particular I will talk about when cusped arithmetic manifolds can be link complements.
Geometry SeminarSpeaker: Cyril Lecuire (Toulouse)
Title: Quasi-isometric rigidity of 3-manifold groups
Date: 13/12/2018
Time: 11:00
Web: http://mat.uab.cat/web/ligat/
Abstract: I will discuss the quasi-isometric rigidity of 3-manifold groups: A finitely generated group that roughly (when viewed from far away) looks like the fundamental group of a compact 3-manifold is in fact the fundamental group of a compact 3-manifold (up to taking a finite index subgroup). I will first introduce some definitions to turn the previous sentence into a precise statement. Then I will explain how we use characteristic decompositions, the Perelman-Thurston's Geometrization Theorem and previous works on quasi-isometric rigidity to prove it. This is a joint work with Peter Haissinsky.
Ring Theory SeminarSpeaker: Lukasz Grabowski (Lancaster University)
Title: Approximation of groups with respect to the rank metric.
Date: 4/12/2018
Time: 11:00
Abstract: I'll talk about an ongoing joint work with Gabor Elek about approximation of groups with respect two the rank metric. The basic question is the following variant of the Halmos problem about commuting matrices: if A and B are large matrices such that the rank of the image of the commutator is small, is it true that A and B can be perturbed with small rank matrices in such a way that the resulting matrices commute? There are interesting connections to classical notions of commutative algebra, in particular we develop what are perhaps some new (or forgotten) variants of Nullstellensatz for primary ideals.
Geometry SeminarSpeaker: Bram Petri (University of Bonn)
Title: Short geodesics on random hyperbolic surfaces
Date: 29/11/2018
Time: 09:00
Web: http://mat.uab.cat/web/ligat/
Abstract: Random Surfaces can be used to study the geometric properties of typical (hyperbolic) surfaces of large genus. Moreover, they can sometimes be used in existence proofs. That is, sometimes the easiest way of proving that surfaces with certain properties exist is to prove that the probability that a random surface has these properties is non-zero. Of course there are multiple possible models of random surfaces. In this talk, a random surface will be a surface that is picked at random using the Weil-Petersson volume form on the moduli space of hyperbolic surfaces of a given genus. I will speak about the length spectrum of these random surfaces. This is joint work with Maryam Mirzakhani.
Ring Theory SeminarSpeaker: Joan Claramunt (Universitat Autònoma de Barcelona)
Title: The lamplighter group algebra and the Atiyah problem
Date: 29/11/2018
Time: 09:00
Abstract: In 1976 Atiyah introduced a certain kind of homology and invariant associated to it, while studying actions of groups on Riemannian manifolds. These invariants are nowadays called $l^2$-Betti numbers. It is possible to give a purely algebraic definition of such numbers, and I will do so in this seminar.
After computing several $l^2$-Betti numbers ,which all turned out to be rational, Atiyah asked the natural question of whether it is possible or not to obtain irrational values. That was the beginning of the Atiyah problem, which I will explain in detail.
After that, I will talk about the lamplighter group G, and explain how the construction explained in the previous seminar is used to compute $l^2$-Betti numbers for G. I will also give some examples of computations.
Geometry SeminarSpeaker: Roberto Rubio
Title: From classical structures to Dirac structures and generalized complex geometry
Date: 22/11/2018
Time: 11:00
Web: http://mat.uab.cat/web/ligat/
Abstract: This is an introductory talk to motivate the definition of Dirac structures, which encompass presymplectic and Poisson geometry, and generalized complex structures, which encompass complex and symplectic geometry. We will start by reviewing all these classical structures on a smooth manifold $M$, and then propose alternative ways to look at them. For instance, the graph of both a presymplectic or a Poisson structure can be seen as a subbundle of $T \, M+T^*M$. This subbundle, by skew-symmetry, is maximally isotropic for the canonical symmetric pairing in $T \, M+T^*M$. Such a maximally isotropic subbundle is an almost Dirac structure. In order to talk about Dirac structures, we will need to introduce the suitable analogue to the Lie bracket on vector fields: the Dorfman bracket on sections of $T \, M+T^*M$. No previous knowledge about Dirac structures or generalized geometry will be assumed for this talk.
Geometry SeminarSpeaker: Joan Porti (UAB)
Title: Holomorphic forms on the $SL(N,{\mathbb C})$ moduli spaces of surfaces with boundary
Date: 15/11/2018
Time: 11:00
Web: http://mat.uab.cat/web/ligat/
Abstract: For an oriented surface of finite type, we consider the moduli space of representations in a simply connected reductive Lie group (eg $SL(N,{\mathbb C})$), and also the moduli space relative to the boundary. Find a relationship between complex valued volume forms in those moduli spaces, the relative and the absolute one. This is joint work with M. Heusener.
Ring Theory SeminarSpeaker: Jorge Castillejos (KU Leuven)
Title: The Toms-Winter conjecture
Date: 15/11/2018
Time: 09:00
Abstract: The classification programme of C*-algebras seeks to classify all separable simple unital nuclear C*-algebras using K-theory and traces. After enjoying tremendous success during several years, some exotic examples were found and certain regularity properties emerged as necessary conditions in the classification programme. The Toms-Winter conjecture asserts that these regularity properties are all equivalent. In this talk, I will discuss the current state of the Toms-Winter conjecture and the classification program.
Geometry SeminarSpeaker: Constantin Vernicos (Montpellier)
Title: Volume growth in Hilbert Geometry
Date: 8/11/2018
Time: 11:00
Web: http://mat.uab.cat/web/ligat/
Abstract: Hilbert geometries are metric spaces defined in the interior of a convex set thanks to the cross-ratio as in the so called projectif model of the hyperbolic geometry (also referred as Beltrami, Cayley or Klein model). While becoming acquainted with these geometries, we will survey what is nowadays known about the volume growth of metric balls with respect to they radius and in particular the volume entropy.
Ring Theory SeminarSpeaker: Laurent Cantier (Universitat Autònoma de Barcelona)
Title: Introducing the $Cu_1$ semigroup and its properties towards classification of $C^*$-algebras
Date: 25/10/2018
Time: 10:00
Abstract:
Geometry SeminarSpeaker: Simon Allais (ENS Lyon)
Title: Application of generating functions to symplectic and contact rigidity
Date: 18/10/2018
Time: 12:00
Web: http://mat.uab.cat/web/ligat/
Abstract: In 1992, Viterbo introduced new means to study the Hamiltonian dynamics of ${\mathbb R}^{2n}$ by applying Morse-theoretical methods to generating functions. Among his results, he gave a new proof of Gromov's non-squeezing theorem (1985) and sketched a proof of the more subtle symplectic camel theorem. A part of this work was generalized to the contact case by Sandon (2011) who provided a new way to derive the contact non-squeezing theorem of Eliashberg, Kim and Polterovich (2006).
We will recall the main points of this theory and show how it allows us to derive a proof of the symplectic camel theorem which can easily be extended to the contact case.
Geometry SeminarSpeaker: Carlos Florentino (Universitat de Lisboa)
Title: Geometry, Topology and Arithmetic of Character Varieties
Date: 11/10/2018
Time: 12:00
Web: http://mat.uab.cat/web/ligat/
Abstract: Character varieties are spaces of representations of finitely presented groups F into Lie groups G. For some choices of F and G, these play important roles in theories such as Chern-Simons or 2d conformal field theory, as well as in hyperbolic geometry and knot theory. Some character varieties are also interpreted as moduli spaces of G-Higgs bundles over Kähler manifolds. When G is a complex algebraic group, character varieties are algebraic and have interesting geometry and topology. We can also consider more refined invariants such as Deligne's mixed Hodge structures, which are typically very difficult to compute, but also provide relevant arithmetic information.
In this seminar, we present some explicit computations of polynomial invariants of character varieties, concentrating on a generalization of the Euler-Poincaré characteristic - the so-called E-polynomial - and when G is a group such as SL(n,C), (P)GL(n,C)or Sp(n,C). We also show a remarkable relation between these invariants and those for the corresponding irreducible loci inside the character varieties. All concepts will be motivated with several examples, and we will give an overview of known calculations of E-polynomials, as well as some conjectures and open problems.
This is joint work with A. Nozad, J. Silva and A. Zamora
Ring Theory SeminarSpeaker: University of Muenster
Title: Bivariant K-theory - Categorical Algebra Approach
Date: 11/10/2018
Time: 10:00
Abstract:
Geometry SeminarSpeaker: Teresa García (UAB)
Title: Actions on products of CAT(-1) spaces
Date: 4/10/2018
Time: 12:00
Web: http://mat.uab.cat/web/ligat/
Abstract: In this talk I will discuss the main result of my PhD thesis: for X a proper CAT(-1) space there is a maximal open subset of the horofunction compactification of $X \times X$ with respect to the maximum metric that compacti fies the diagonal action of an infinite quasi-convex group of the isometries of $X$.
I will also discuss briefly the case of a product action of two quasi-convex representations of an infinite hyperbolic group on the product of two different proper CAT(-1) spaces.
Ring Theory SeminarSpeaker: Francesc Perera (Universitat Autònoma de Barcelona)
Title: Existence of Infima in Cuntz semigroups and applications to the structure of C*-algebras with stable rank one
Date: 4/10/2018
Time: 10:00
Abstract: Let $A$ be a C*-algebra with stable rank one. We show that the Cuntz semigroup of $A$ satisfies Riesz interpolation. If $A$ is also separable, it follows that the Cuntz semigroup of $A$ has finite infima. This has several consequences:
(i) A conjecture of Blackadar and Handelman from 1982 is proved in the case of unital C*-algebras with stable rank one. This conjecture predicts that the normalized dimension functions on such a C*-algebra form a Choquet simplex.
(ii) We confirm the global Glimm halving conjecture for unital C*-algebras with stable rank one. This conjecture may be stated as follows: For each natural number $k$, the C*-algebra $A$ has no nonzero representations of dimension less than $k$ if and only if there exists a morphism from the cone over the algebra of $k\times k$ matrices to $A$ with full range.
(iii) The rank problem for separable, unital (not necessarily simple) C*-algebras with stable rank one that have no finite-dimensional quotients is solved, in the following sense: For every lower semicontinuous, strictly positive, affine function $f$ on the Choquet simplex of normalized $2$-quasitraces on $A$, there exists a positive element in the stabilization of $A$ whose rank is precisely $f$.
This is joint work with Ramon Antoine, Leonel Robert, and Hannes Thiel.
Topology SeminarSpeaker: Branislav Jurco (Charles University)
Title: Quantum L-infinity Algebras and the Homological Perturbation Lemma
Date: 17/9/2018
Time: 12:00
Web: http://mat.uab.cat/~topalg
Abstract: Quantum homotopy Lie algebras are a generalization of homotopy Lie algebras with a scalar product and with operations corresponding to higher genus graphs. We construct a minimal model of a given quantum homtopy Lie algebra algebra via the homological perturbation lemma and show that it is given by a Feynman diagram expansion, computing the effective action in the finite-dimensional Batalin-Vilkovisky formalism. We also construct a homotopy between the original and this effective quantum homotopy Lie algebra.
Topology SeminarSpeaker: Thomas Poguntke (Bonn)
Title: Higher Segal structures in algebraic K-theory
Date: 14/9/2018
Time: 12:00
Web: http://mat.uab.cat/~topalg
Abstract: We start with a brief recollection on the algebraic K-theory of exact categories. The first goal is to introduce a new simplicial delooping of it, which consists of higher dimensional analogues of Waldhausen's S-construction, where short exact sequences are replaced by longer extensions. Our main theorem provides a precise relation to the original Waldhausen construction, and in particular implies the delooping statement. Another consequence is the generalisation of one of the main results of Dyckerhoff-Kapranov's work on higher Segal spaces, which concerns the fibrancy properties of the (higher) S-construction; this is the first example of (> 2)-Segal objects explicitly appearing in the literature.
Topology SeminarSpeaker: Louis Carlier (UAB)
Title: Hereditary species as monoidal decomposition spaces
Date: 7/9/2018
Time: 12:00
Web: http://mat.uab.cat/~topalg
Abstract: Schmitt constructed an important family of combinatorial bialgebras from what he called hereditary species: they are combinatorial structures with three different functorialities. The species of simple graphs is an example. These bialgebras do not fit into the standard theory of incidence algebras of posets or categories. We show Schmitt's hereditary species induce decomposition spaces, the more general homotopical framework for incidence algebras and Möbius inversion introduced recently by Gálvez, Kock, and Tonks, and we show that the bialgebra associated to a hereditary species is the incidence bialgebra of the corresponding monoidal decomposition space.
Topology SeminarSpeaker: Nitu Kitchloo (Johns Hopkins University)
Title: Stability for Kac-Moody Groups
Date: 20/7/2018
Time: 12:00
Web: http://mat.uab.cat/~topalg
Abstract: In the class of Kac-Moody groups, one can extend all the exceptional families of compact Lie groups yielding infinite families $(E_n, F_n, G_n)$, as well as other infinite families. We will show that these exceptional families stabilize in a homotopical sense and that the (co)homology of their classifying spaces is torsion free for all but a finite set of primes that is determined by the family (and not the individual groups in the family).
Topology SeminarSpeaker: Marithania Silvero (BGSMath-UB)
Title: Strongly quasipositive links and Conway polynomial
Date: 20/7/2018
Time: 10:45
Web: http://mat.uab.cat/~topalg
Abstract: Strongly quasipositive links are those links which can be seen as closures of positive braids in terms of band generators. We give a necessary condition for a link with braid index 3 to be strongly quasipositive, by proving that they have positive Conway polynomial (that is, all its coefficients are non-negative). We also show that this result cannot be extended to a higher number of strands, as we provide a strongly quasipositive braid on 5 strands whose closure has non-positive Conway polynomial.
Ring Theory SeminarSpeaker: Jan Trlifaj (Charles University, Prague)
Title: Faith's problem on $R$-projectivity is not decidable in ZFC
Date: 16/7/2018
Time: 10:00
Abstract: In [1], Faith asked for a characterization of the rings R such that each R-projective module is projective, that is, the Dual Baer Criterion holds in Mod-R. Such rings were called right testing. Sandomierski [3] proved that each right perfect ring is right testing. Puninski et al. [2] have recently shown for a number of non-right perfect rings that they are not right testing (in ZFC), and noticed that [4] proved consistency with ZFC of the statement ‘each right testing ring is right perfect’ (the proof used Shelah’s uniformization).
We prove the complementing consistency result: the existence of a right testing, but non-right perfect ring is also consistent with ZFC (our proof uses Jensen-functions, and the K-algebra of all eventually constant sequences over a field K). Thus the answer to the Faith’s question above is not decidable in ZFC, [5]. Moreover, for each cardinal κ, we provide examples of non-right perfect rings R, such that the Dual Baer Criterion holds (in ZFC) for all ≤ κ-generated R-modules.
1.- C. Faith, Algebra II. Ring Theory, GMW 191, Springer-Verlag, Berlin 1976.
2.- H. Alhilali, Y. Ibrahim, G. Puninski, M. Yousif, When R is a testing module for projectivity?, J. Algebra 484 (2017), 198-206.
3.- F. Sandomierski, Relative Injectivity and Projectivity, PhD thesis, Penn State University,1964.
4.- J. Trlifaj: Whitehead test modules, Trans. Amer. Math. Soc. 348 (1996), 1521-1554.
5.- J. Trlifaj: Faith’s problem on R-projectivity is undecidable, to appear in Proc. Amer. Math. Soc.
Topology SeminarSpeaker: David Spivak (MIT)
Title: A higher-order temporal logic for dynamical systems
Date: 6/7/2018
Time: 12:00
Web: http://mat.uab.cat/~topalg
Abstract: We consider a very general class of dynamical systems---including discrete, continuous, hybrid, deterministic, nondeterministic, etc.---based on sheaves. We call these sheaves behavior types: they tell us the set of possible behaviors over any interval of time. A machine can be construed as a wide span of such sheaves, and these machines can be composed as morphisms in a hypergraph category. The topos of sheaves has an internal language, which we use as a new sort of higher-order internal logic for talking about behaviors. We can use this logic to prove properties about a composite system of systems from properties of the parts and how they are wired together.
Ring Theory SeminarSpeaker: Ferran Cedó(Universitat Autònoma)
Title: Introductory course : Left Braces
Date: 4/7/2018
Time: 10:00
Abstract: In 2007 Rump introduced braces as a generalization of Jacobson radical rings to study non-degenerate involutive set-theoretic solutions of the Yang--Baxter equation.
Bachiller in his Ph. D. thesis (2016) showed the power of the theory of braces solving difficult open problems. He also found interesting links with other algebraic structures such as Hopf--Galois extensions. After this, the study and development of the theory of left braces has increased quickly. I this course I will explain an introduction to the brace theory.
Ring Theory SeminarSpeaker: Diego Martínez (Universidad Carlos III de Madrid)
Title: Amenability in semigroups and C*-algebras
Date: 25/6/2018
Time: 15:00
Abstract: Amenability in the group case is a well studied field, relating several different notions in mathematics. In this talk, we will study amenability notions in the more general semigroup case, trying to recover some classical equivalences, such as the Følner condition or the non-paradoxicality of the semigroup. Furthermore, we will restrict ourselves to the inverse semigroup case, and prove that this case is very much the same as the classical group one. In particular, we will study the relation between the amenability of an inverse semigroup, its reduced C*-algebra being Følner and 0 and 1 being not equal in the K_0 group of the reduced C*-algebra.
Geometry SeminarSpeaker: Dmitry Faifman (University of Toronto)
Title: Curvature in contact manifolds and integral geometry
Date: 12/6/2018
Time: 12:00
Abstract: Valuations are finitely additive measures on nice subsets, for example the Euler characteristic, volume and surface area are valuations. During the 20th century, valuations have been studied predominantly on convex bodies and polytopes, in linear spaces and lattices. Valuations on manifolds were introduced about 15 years ago by S. Alesker, with contributions by A. Bernig, J. Fu and others, and immediately brought under one umbrella a range of classical results in Riemannian geometry, notably Weyl's tube formula and the Chern-Gauss-Bonnet theorem. These results circle around the real orthogonal group.
In the talk, the real symplectic group will be the central player. Drawing inspiration from the Lipschitz-Killing curvatures in the Riemannian setting, we will construct some natural valuations on contact and dual Heisenberg manifolds, which generalize the Gaussian curvature. We will also construct symplectic-invariant distributions on the grassmannian, leading to Crofton-type formulas on the contact sphere and symplectic space.
Ring Theory SeminarSpeaker: Joan Bosa (Universitat Autònoma de Barcelona)
Title: Realization problem and Steinberg Algebras
Date: 11/6/2018
Time: 15:00
Abstract: The realization problem for von Neumann (vN) regular rings asks whether all conical refinement monoids arise from monoids induced by the projective modules over a vN regular ring. In this article we show the last developments on this problem and relate it to the Steinberg algebras associated to a separated graph.
Topology SeminarSpeaker: Rémi Molinier (Université de Grénoble)
Title: Cohomology with twisted coefficients of linking systems and stable elements
Date: 8/6/2018
Time: 12:00
Web: http://mat.uab.cat/~topalg
Abstract: A theorem of Boto, Levi and Oliver describes the cohomology of the geometric realization of a linking system, with trivial coefficients, as the submodule of stable elements in the cohomology of the Sylow. When we are looking at twisted coefficients, the formula can not be true in general as pointed out by Levi and Ragnarsson but we can try to understand under which condition it holds. In this talk we will see some conditions under which we can express the cohomology of a linking system as stable elements.
Geometry SeminarSpeaker: Florent Balacheff (Univ. Lille/UAB)
Title: Producte de longituds de geodèsiques tancades homòlogicament independents
Date: 5/6/2018
Time: 12:00
Abstract: En aquesta xerrada considerarem generalitzacions del segon teorema de Minkowski per a varietats Riemannianes. Per exemple, explicarem per què els tors, les superfícies i la suma connexa de dos espais projectius reals de dimensió 3, amb una mètrica Riemanniana de volum normalitzat, sempre admeten una base d’homologia mòdul 2 induïda per geodèsiques tancades, per a les quals el producte de les longituds està acotat per sobre. Basat en un treball conjunt amb S. Karam i H. Parlier.
Ring Theory SeminarSpeaker: Christian Bonicke (University of Muenster)
Title: The Baum-Connes conjecture for ample group bundles
Date: 4/6/2018
Time: 15:00
Abstract: In this talk I will discuss how the Baum-Connes conjecture for a group bundle over a totally disconnected space is related to the Baum-Connes conjecture of the fibres. This uses a version of the so-called Going-Down principle for ample groupoids, which I will illustrate by means of the above example.
Topology SeminarSpeaker: Thomas Wasserman (Oxford)
Title: A Reduced Tensor Product of Braided Fusion Categories containing a Symmetric Fusion Category
Date: 1/6/2018
Time: 12:00
Web: http://mat.uab.cat/~topalg
Abstract: In this talk I will construct a reduced tensor product of braided fusion categories containing a symmetric fusion category $\mathcal{A}$. This tensor product takes into account the relative braiding with respect to objects of $\mathcal{A}$ in these braided fusion categories. The resulting category is again a braided fusion category containing $\mathcal{A}$. This tensor product is inspired by the tensor product of $G$-equivariant once-extended three-dimensional quantum field theories, for a finite group $G$.
Topology SeminarSpeaker: Jérôme Los
Title: Sequences in the mapping class group: some convergence/ divergence questions
Date: 23/5/2018
Time: 16:00
Web: http://mat.uab.cat/~topalg
Abstract:
Topology SeminarSpeaker: Jérôme Los (CNRS-Univ. Marseille)
Title: Sequences in the mapping class group: some convergence/ divergence questions
Date: 23/5/2018
Time: 16:00
Web: http://mat.uab.cat/~topalg
Abstract:
Topology SeminarSpeaker: Mark Weber
Title: Feynman categories as operads
Date: 23/5/2018
Time: 15:00
Web: http://mat.uab.cat/~topalg
Abstract: In various papers of Kaufmann and Ward, the notion of "Feynman
category" is introduced as a generalisation of "coloured
symmetric operad", and then developed further. In this talk it
will be explained that in fact Feynman categories and coloured
symmetric operads are the same things, in that one can set up a
biequivalence between 2-categories whose objects are these
structures. Moreover, this biequivalence induces equivalences
between the corresponding categories of algebras. Thus Feynman
categories are not really "new", but rather are an interesting
alternative point of view on coloured symmetric operads.
(Joint work with Michael Batanin and Joachim Kock)
Topology SeminarSpeaker: Mark Weber
Title: Feynman categories as operads
Date: 23/5/2018
Time: 15:00
Web: http://mat.uab.cat/~topalg
Abstract: In various papers of Kaufmann and Ward, the notion of "Feynman category" is introduced as a generalisation of "coloured symmetric operad", and then developed further. In this talk it will be explained that in fact Feynman categories and coloured symmetric operads are the same things, in that one can set up a biequivalence between 2-categories whose objects are these structures. Moreover, this biequivalence induces equivalences between the corresponding categories of algebras. Thus Feynman categories are not really "new", but rather are an interesting alternative point of view on coloured symmetric operads.
Geometry SeminarSpeaker: José Andrés Rodríguez Migueles (Université de Rennes I)
Title: Geodésicas en superficies hiperbólicas y complementos de nudos
Date: 8/5/2018
Time: 12:00
Abstract: Toda geodésica cerrada en una superficie hiperbólica tiene un levantamiento canónico en su haz tangente unitario, y lo podemos ver como un nudo dentro de una variedad de dimensión 3. Por ejemplo, Ghys demostró que los nudos de Lorentz son precisamente esos que se encuentran de dicha manera si usamos la superficie modular. Genéricamente el complemento de los nudos así construidos admiten una estructura hiperbólica, única por el teorema de rigidez de Mostow. En esta plática voy a relacionar algunas propiedades de la geodésica
inicial con la geometría de la 3-variedad.
Ring Theory SeminarSpeaker: Javier Sanchez (University of São Paulo)
Title: Embedding group algebras of torsion-free one-relator products of locally indicable groups in division rings
Date: 7/5/2018
Time: 15:00
Abstract:
Let $A$ and $B$ be locally indicable groups and $w$ a word in the free product $A\coprod B$ which is not
conjugate to an element of $A$ nor of $B$ and such that the one-relator product
$G=\frac{A\coprod B}{<| w |>}$ is torsion free.
Let $K$ be a division ring and $K*G$ a crossed product group ring.
We show that $K*G$ can be embedded in a division ring under certain conditions.
This is a joint work with Dolors Herbera.
Geometry SeminarSpeaker: Antonin Guilloux (Univ. Pierre et Marie Curie)
Title: Volume function and Mahler measure of polynomials
Date: 17/4/2018
Time: 12:00
Abstract: The Mahler measure of a polynomial is a kind of size of this polynomial, introduced around 1930 for questions related to prime numbers. It has since proven useful and intriguing in a variety of contexts of number theory. It is very hard to compute in general.
I will present this Mahler measure, and its computation for a class of 2-variables polynomials through elementary considerations about a function called volume function. Indeed, for this class, strong and surprising links exist between Mahler measure and volume of hyperbolic 3-manifolds.
Geometry SeminarSpeaker: Philippe Castillon (Université de Montpellier)
Title: Prescribing the Gauss curvature of hyperbolic convex bodies
Date: 10/4/2018
Time: 12:00
Abstract: The Gauss curvature of a convex body can be seen as a measure on the unit sphere (with some properties). For such a measure $\mu$, Alexandrov problem consists in proving the existence of a convex body whose curvature measure is $\mu$. In the Euclidean space, this problem is equivalent to an optimal transport problem on the sphere.
In this talk I will consider Alexandrov problem for convex bodies of the hyperbolic space. After defining the curvature measure, I will explain how to relate this problem to a non linear Kantorovich problem on the sphere and how to solve it.
Joint work with Jérôme Bertrand.
Ring Theory SeminarSpeaker: SIMONE VIRILI (Universidad de Múrcia)
Title: When is the heart of a t-structure a Grothendieck category?
Date: 9/4/2018
Time: 15:00
Abstract: Let D be a triangulated category endowed with a t-structure t = (U, ΣV) and denote by H := U ∩ ΣV its heart. In this seminar I will report on some recent results, obtained in collaboration with Manuel Saorı́n and Jan Šťovı́ček, partially answering the following well-known question:
Under what conditions on D and t can we say that H is a Grothendieck category?
We will concentrate on the case when D is the base of a stable derivator. In this generality we will see that, under very natural hypotheses on t, direct limits in H are exact. If, moreover, D is a well-generated algebraic or topological triangulated category, then the heart of any accessibly embedded (e.g., compactly generated) t-structure has a generator. As a consequence, it follows that the heart of any compactly generated t-structure of a well-generated algebraic or topological triangulated category is a Grothendieck category. Joint work with MANUEL SAORÍN AND JAN ŠŤOVÍČEK
Geometry SeminarSpeaker: Laurent Meersseman (Université d'Angers)
Title: Local Structure of the Teichmüller Stack
Date: 6/3/2018
Time: 11:00
Abstract: In this talk, I will discuss the local structure of the Teichmüller stack of a fixed smooth compact manifold M, which encodes the set of complex structures on M up to biholomorphisms isotopic to the identity. I will focus on the differences between Kähler components (that is components corresponding to Kähler structures) and non-Kähler ones.
Topology SeminarSpeaker: Alex Cebrian (UAB)
Title: A simplicial groupoid for plethystic substitution
Date: 2/3/2018
Time: 11:00
Web: http://mat.uab.cat/~topalg
Abstract: We give a simple combinatorial model for plethystic substitution: precisely, the plethystic bialgebra is realised as the homotopy cardinality of the incidence bialgebra of a simplicial groupoid, obtained from surjections by a construction reminiscent of Waldhausen S and Quillen Q-construction.
Topology SeminarSpeaker: Bob Oliver (Université Paris 13)
Title: Recent constructions and theorems on fusion systems due to Michael Aschbacher
Date: 23/2/2018
Time: 11:00
Web: http://mat.uab.cat/~topalg
Abstract: Fix a prime $p$. The fusion system of a finite group $G$ with respect to a Sylow subgroup $S \in\mathop {\rm Syl}_p(G)$ is the category $\mathcal{F}_S(G)$ whose objects are the subgroups of $S$, and whose morphisms are the homomorphisms induced by conjugation in $G$. More generally, an abstract fusion system over a $p$-group $S$ is a category whose objects are the subgroups of $S$ and whose morphisms are injective homomorphisms between the subgroups that satisfy certain axioms formulated by Lluis Puig and motivated by the Sylow theorems for finite groups.
Starting 10–15 years ago, Michael Aschbacher and some other finite group theorists became interested in fusion systems, hoping that they can be used to help shorten some parts of the proof of the classification of finite simple groups. This has led to many new structures and results such as generalized Fitting subsystems of fusion systems, as well as intersections, central products, and centralizers of normal fusion subsystems. In many cases, these are analogs of basic, elementary structures or operations in finite groups, but are surprisingly difficult to define in the context of fusion systems.
Topology SeminarSpeaker: Sune Precht Reeh (UAB)
Title: Constructing a transporter infinity category for fusion systems
Date: 21/2/2018
Time: 11:00
Web: http://mat.uab.cat/~topalg
Abstract: In this research talk, I will give a tour of the progress I have made in the last two weeks on constructing an infinity category that is supposed to model the transporter category for a fusion system (when given a choice of locality/linking system).
I will explain the construction itself as a category enriched in Kan complexes. I will talk about the results obtained so far, with details as time permits, and I will explain the open problems that I am still working on, including how to adapt this transporter category into a working orbit category.
Topology SeminarSpeaker: Jesper M. Møller (University of Copenhagen)
Title: The Alperin weight conjecture, the Knörr-Robinson conjecture, and equivariant Euler characteristics
Date: 16/2/2018
Time: 11:00
Web: http://mat.uab.cat/~topalg
Abstract: A topologically biased amateur marvels at the Alperin weight conjecture from different angles without getting anywhere near a solution.
Ring Theory SeminarSpeaker: Eric Jespers (Vrije Universiteit Brussel, Brussels, Belgium)
Title: Groups, Rings, Braces and Set-theoretic Solutions of the Yang-Baxter Equation
Date: 12/2/2018
Time: 15:15
Abstract: Drinfeld in 1992 proposed to study the set-theoretical solutions of the Yang- Baxter equation. Recall that a set-theoretical solution is a pair (X, r), where X is a set and
r : X × X → X × X is a bijective map such that (r × id)(id × r)(r × id) = (id×r)(r×id)(id×r). For every x,y∈X,writer(x,y)=(\sigma x(y),\gamma_y(x)) where \sigma_x and \gamma_y aremaps X→X.
In order to describe all set-theoretic non-degenerate (i.e. each \sigma_x and \gamma_y is bijective) involutive (i.e. r^2 = id) solutions, Rump introduced a new algebraic structure called a brace. The aim of this talk is to survey some of the recent results on this topic and show that there are deep connections with several structures in group and non-commutative ring theory. Time permitting we also present some more general strucures.
Ring Theory SeminarSpeaker: Jan Okninski (Warsaw University)
Title: Hecke-Kiselman algebras: combinatorics and structure.
Date: 12/2/2018
Time: 14:00
Abstract: To every finite simple G graph with n vertices one can associate the so called Hecke-Kiselman monoid HK(G).
This is a finitely presented monoid with n generators and with defining relations that are either of the form xy=yx
or of the form of the braid relation xyx=yxy. This talk is motivated by the general problem concerning the interplay between the combinatorial and structural properties of the semigroup algebra K[HK(G)] (over a field K) of this monoid. In particular, we will focus on: the Gelfand-Kirllov dimension and the automaton property on one hand and on ring theoretical properties such as the PI-property and the Noetherian property, on the other hand.
The talk is based on a joint work with A.Mecel, M.Wiertel and L.Kubat.
Geometry SeminarSpeaker: Agustí Reventós (UAB)
Title: Algunes aplicacions geomètriques de les sèries de Fourier
Date: 29/1/2018
Time: 11:00
Abstract:
Geometry SeminarSpeaker: Jean Gutt (Univ. Köln)
Title: Knotted symplectic embeddings
Date: 22/1/2018
Time: 11:00
Abstract: I will discuss a joint result with Mike Usher, showing that many toric domains $X$ in the 4-dimensional euclidean space admit symplectic embeddings $f$ into dilates of themselves which are knotted in the strong sense that there is no symplectomorphism of the target that takes $f(X)$ to $X$.
Geometry SeminarSpeaker: Michael Heusener (Univ. Clermont Auvergne)
Title: Deformations of abelian representations of knot groups into $\mathrm{SL}(n,\mathbb C)$
Date: 8/1/2018
Time: 11:00
Abstract: