Speaker: 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?
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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).
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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 $\mathcal{M}_{H& gt0}(p,q)$ 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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).
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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).
Speaker: 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.
Speaker: 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"
Speaker: 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).
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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:
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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)
Speaker: 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.
Speaker: 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).
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: Agustí Reventós (UAB)
Title: Algunes aplicacions geomètriques de les sèries de Fourier
Date: 29/1/2018
Time: 11:00
Abstract:
Speaker: 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 $X$.
Speaker: 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: