Speaker: Eduard Vilalta
Title: Pure C*-algebras and *-homomorphisms
Date: 19/3/2024
Time: 14:30
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: Dolors Herbera
Title: Finite free resolutions (part 2)
Date: 5/2/2024
Time: 16:00
Speaker: 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.
Speaker: Dolors Herbera (UAB)
Title: Finite free resolutions
Date: 22/1/2024
Time: 16:00
Speaker: 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.
Speaker: Michal Hrbek
Title: Some new results about the Telescope Conjecture in D (X)
Date: 20/11/2023
Time: 16:00
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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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}
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: Guillem Quingles (UAB)
Title: Finiteness properties of local cohomology modules
Date: 27/3/2023
Time: 16:00
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.
Speaker: 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).
Speaker: 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.
Speaker: 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).
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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
Speaker: 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 $W(k)$.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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 $X-gt{0,1,...,\infty }$ 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]
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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
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.
Speaker: 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.
Speaker: 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 -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.
Speaker: 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.
Speaker: 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.
Speaker: 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).
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: Ferran Cedó (Universitat Autònoma de Barcelona)
Title: "Braided algebras of Gelfand-Kirillov dimension one"
Date: 11/4/2019
Time: 11:30
Speaker: 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.
Speaker: Pere Ara (Universitat Autònoma de Barcelona)
Title: The Realization Problem
Date: 17/1/2019
Time: 09:00
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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
Speaker: University of Muenster
Title: Bivariant K-theory - Categorical Algebra Approach
Date: 11/10 /2018
Time: 10:00
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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.
Speaker: 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
Speaker: 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
Speaker: 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.
Speaker: 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.