Archiv

09.04.2019, 15:00, WIL A 120: Seminar Geometrie
Andy Baker, University of Glasgow: Galois theory for (derived) commutative rings

Classical Galois theory of fields (as met in an undergraduate course) was generalised to commutative rings around 1960 by Auslander, Goldman, Chase, Harrison and Rosenburg. Around the same time, Grothendieck developed the theory of etale coverings of schemes and gave a more geometric view. In the ensuing half century the general framework has been expanded to many more settings including derived algebra. This talk will discuss the basic ideas that are common to these.

I will review the step from the field theory version to commutative rings, pointing out some of the extra tools required. If there is time I will explain some features of a derived version (originally developed in a topological context by Rognes).

16.04.2019, 15:00, WIL A 120: Seminar Geometrie
Jakob Schneider, TU Dresden: Isomorphism questions for metric ultraproducts of finite quasisimple groups

We compute double centralizers of semisimple torsion elements in metric ultraproducts of finite quasisimple groups. This information is used to show that (up to the exception:
Sp_U(q)=GO_U(q)?) the Lie type and the field size, is an invariant which is determined by the group structure of such an ultraproduct.

21.05.2019, 15:00, WIL A 120: Seminar Geometrie
Françoise Point, Institut de Mathématiques de Jussieu:  On pseudo-finite groups

First I will introduce the notion of being pseudo-C for a class C of groups. For the class of finite groups, I will relate it to other notions of approximability.
Then I will review results on one hand on a question of G. Sabbagh: are there infinite finitely generated pseudo finite groups and on the other hand on analogs of Tits alternative for certain classes of pseudo-finite groups.

09.10.2018, 15:00, WIL A 120: Seminar Geometrie
Tom Drescher, Universität Innsbruck: On Projections of Free Semi-Algebraic Sets

In classical semi-algebraic geometry one is interested in the study of
special subsets of the euclidean space: the semi-algebraic sets. Roughly
speaking, these are sets that can be defined by polynomial equalities
and inequalities as well as boolean combinations. One of the most
important results about semi-algebraic sets is the Projection Theorem:
Every linear projection of a semi-algebraic set is again semi-algebraic.
 From the perspective of mathematical logic, this theorem implies that
the theory of real closed fields has quantifier elimination.
Triggered by applications in a variety of different areas a new branch
of the theory has emerged: the free semi-algebraic geometry. The basic
idea is to consider tuples of matrices instead of tuples of numbers.
Because the free theory generalizes the classical one, the question is
whether some important classical properties like the Projection Theorem
can be transferred to the free setup.
In the talk we will point out the limits of a free quantifier
elimination theorem and show an approach for a weak version of a free
projection theorem.

13.11.2018, 15:00, WIL A 120: Seminar Geometrie
Sebastian Halbig, TU Dresden: Hopf algebras with and without modular pairs in involution

A modular pair in involution of a Hopf algebra H is a pair consisting of a grouplike element g in H and a character f on H such that f(g) = 1 and the antipode square is given by the adjoint action of g and f. The ground field k of H can be viewed as a module and a comodule via the character and the grouplike, respectively. The module comodule k is then a stable anti-Yetter-Drinfeld module playing the role of coefficients in Hopf-cyclic cohomology introduced by Connes and Moscovici. When H is finite dimensional, the aforementioned antipode condition for (g,f) is a square root of the celebrated Radford formula for the 4th power of the antipode, and is thus related to the work of Radford and Kauffman concerning the existence of a ribbon element in the Drinfeld double of H.

Using Radford's biproduct (also known as Majid's bosonization) of certain Nichols algebras over cyclic groups, we construct finite-dimensional Hopf algebras that do not admit a modular pair in involution.

10.04.2018, 15:00, WIL A 120: Seminar Geometrie
Niels Kowalzig, Sapienza Università di Roma: Higher brackets on cyclic and negative cyclic (co)homology

In this talk, we will embed the string topology bracket developed by Chas-Sullivan and Menichi on negative cyclic cohomology groups as well as the dual bracket found by de Thanhoffer de Voelcsey-Van den Bergh on negative cyclic homology groups into the global picture of a noncommutative differential (or Cartan) calculus up to homotopy on the (co)cyclic bicomplex in general, in case a certain Poincare' duality is given. For negative cyclic cohomology, this in particular leads to a Batalin-Vilkovisky algebra structure on the underlying Hochschild cohomology. In the special case in which this BV bracket vanishes, one obtains an e₃-algebra structure on Hochschild cohomology. The results are given in the general and unifying setting of (opposite) cyclic modules over (cyclic) operads.

17.04.2018, 15:00, WIL A 120: Seminar Geometrie
Tom Kaiser, Université de Neuchâtel: Sofic approximations and property A

Consider a residually finite group, then given a filtration we can construct the box space of this group with respect to the chosen filtration. It is a known result that the box space having property A is equivalent to the group being amenable. The same is true for hyperfiniteness. Firstly we will optimise this result by replacing filtrations by Farber sequences  and secondly determine which arrows still hold when we replace box spaces by sofic approximations.
Knowledge of the above concepts is not required since these will be introduced during the talk.

24.04.2018, 15:00, WIL A 120: Seminar Geometrie
Francesca Arici, MPI for Mathematics in the Sciences Circle and sphere bundles in noncommutative geometry

In this talk I will recall how Pimsner algebras of self Morita equivalences can be thought of as total spaces of quantum circle bundles, and the associated six term exact sequence in K-theory can be interpreted as an operator algebraic version of the classical Gysin sequence for circle bundles. After reviewing some results in this direction, I will  report on work in progress concerning the construction of higher dimensional quantum sphere bundles in terms of Cuntz-Pimsner algebras of sub-product systems. Based on (ongoing) joint work with G. Landi and J. Kaad.

08.05.2018, 15:00, WIL A 120: Seminar Geometrie
Ulrich Krähmer, TU Dresden: Mixed vs cyclic homology

In this talk I will (try to) explain the main result of the paper I have written with Dylan Madden. Therein, we consider so-called mixed complexes, that is, graded vector spaces C together with linear maps b and d on them such that (C,b) is a chain complex and (C,d) is a cochain complex. One can then consider the homology of the former and the cohomology of the latter, but there is also a way to define a mixed homology that involves both b and d at the same time, and to develop the general theory of such mixed homology group is the aim of the game. Two important examples are given by the differential forms on an oriented Riemannian manifold with the de Rham differential and codifferential as d and b, and the Hochschild chains of an associative algebra.

15.05.2018, 15:00, WIL A 120: Seminar Geometrie
Michael Whittaker, University of Glasgow: New directions in self-similar group theory

A self-similar group (G,X) consists of a group G acting faithfully on a homogeneous rooted tree such that the action satisfies a self-similarity condition. In this talk I will generalise the above definition to faithful groupoid actions on the path space of more general graphs. This new definition allows us to work out the structure of the KMS state space of associated Toeplitz and Cuntz-Pimsner algebras. This is joint work with Marcelo Laca, Iain Raeburn, and Jacqui Ramagge.

29.05.2018, 15:00, WIL A 120: Seminar Geometrie
Manuel Martins, TU Dresden: The nodal cubic is a quantum homogeneous space

Starting from the classical point of view, the notion of quantum homogeneous space will be introduced.  The coordinate ring of the nodal cubic in the plane yields such an example, whose corresponding coalgebra Galois extension is expressed in terms of quantum groups at roots of unity and is shown to be cleft.

05.06.2018, 15:00, WIL A 120: Seminar Geometrie
David Kyed, Syddansk Universitet: Measure equivalence for locally compact groups

In the talk I will discuss the notion of measure equivalence within the framework of locally compact second countable groups, with particular emphasis on the unimodular case, and explain how classical results, such as the Ornstein-Weiss theorem, generalize to this setting. If time permits, I will also discuss the notion of uniform measure equivalence and its relation to coarse equivalence. The talk is based on joint work with Juhani Koivisto and Sven Raum.

12.06.2018, 15:00, WIL A 120: Graduate Lectures
Maxime Gheysens, TU Dresden: On, around, and beyond Ryll-Nardzewski's theorem I

After recalling some facts about group actions and functional analysis, we will state and prove Ryll-Nardzewski's theorem.

19.06.2018, 15:00, WIL A 120: Graduate Lectures
Maxime Gheysens, TU Dresden: On, around, and beyond Ryll-Nardzewski's theorem II: Applications

The second lecture will focus on applications of Ryll-Nardzewski's theorem. We will cover in details the existence of Haar measures for compact groups and the unbounded vs. fixed-point dichotomy for isometric actions on reflexive Banach spaces. Other consequences (for instance, for von Neumann algebras or for compactifications of groups) will be mentioned, according to the interest of the audience.

02.07.2018, 15:00, WIL A 317: Geometry Seminar
Alcides Buss, Federal University of Santa Catarina, Florianópolis:
Exotic crossed products of group actions

Motivated by exactness obstructions to the Baum-Connes conjecture with coefficients in C*-algebras, Baum, Guentner and Willett recently introduced a new formulation of the conjecture that replaces the reduced crossed product by the smallest exact crossed product functor. In this talk I will present a more concrete description of this functor that shows, among other things, that the reformulated conjecture coincides with the original one for trivial coefficients. I also intend to present a “dual" construction of the largest injective crossed product functor and explain some interesting connections to exactness and local lifting properties of groups.
 

03.07.2018, 15:00, WIL A 120: Graduate Lectures
Maxime Gheysens, TU Dresden: On, around, and beyond Ryll-Nardzewski's theorem III: Sibling theorems

We will explain three interesting related theorems: a fixed-point theorem for L¹-spaces (an unexpected application of Ryll-Nardzewski's theorem), a fixed-point theorem for actions on convex cones (a generalisation thereof), and the Day–Rickert characterization of amenability (which shows that Ryll-Nardzewski's theorem is somehow optimal).

10.07.2018, 15:00, WIL A 120: Geometry Seminar
Henry Bradford, Georg-August-Universität Göttingen: On LEF Growth in Groups

In this talk I will propose a quantitative version of the LEF (local embeddability into finite groups) property. I will estimate this new invariant in some examples, and compare it with existing invariants which quantify residual finiteness and soficity. I will give a construction of a family of finitely generated residually finite groups which nevertheless admit many more local embeddings to finite groups than they do finite quotients. It would be of great interest to identify more groups satisfying this last property since, as I shall show, such groups are never finitely presentable.


17.07.2018, 15:00, WIL A 120: Geometry Seminar
Madeleine Jotz Lean, Georg-August-Universität Göttingen: Ideals in Lie algebroids and the Atiyah class

I will describe a notion of ideals in Lie algebroids, named
'infinitesimal ideal systems' and motivate from several point of views
why this is in my opinion the 'right' notion of symmetry in the Lie
algebroid setting: I will discuss quotients by infinitesimal ideal
systems, their equivalence to multiplicative foliations on Lie
groupoids, and how they define sub-representations of Lie algebroid adjoint
representations (up to homotopy).
Then I will talk about the Atiyah class of an infinitesimal ideal system and give an
obstruction to the existence of an ideal system structure on a Lie pair.
This is partly based on joint work with Ortiz and Drummond.

24.10.2017, 15:00, WIL A 120: Seminar Geometrie
Vladimir Finkelshtein, Universität Göttingen: Diophantine approximation problems for groups of toral automorphisms

I will present sharp rates for a shrinking target problem for the action of arbitrary subgroups of SL(2,Z) on the 2-torus. This can also be viewed as a noncommutative Diophantine approximation problem. The methods require construction of spectrally optimal random walks on groups acting properly cocompactly on Gromov hyperbolic spaces. Additionally, I will explain how similar estimates for this problem in higher dimension can be obtained using harmonic analysis.

07.11.2017, 15:00, WIL A 120: Seminar Geometrie
Mario Kummer, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig: Interlacing Ehrhart Polynomials of Reflexive Polytopes

There are constructions that associate to a graph a lattice polytope. Furthermore, for each lattice polytope there is the Ehrhart polynomial that computes the number of lattice points in dilations of the polytope. Recently, the roots of Ehrhart polynomials became an object of intensive studies. For many graphs these roots have a remarkable property: They all have real part –1/2.  Still, proving that this property holds for a family of graphs is often very hard. One of the first positive results was the case of the complete (1,n)-bipartite graphs proved independently by Kirschenhofer, Pethö and Tichy and by Bump, Choi, Kurlberg and Vaaler. The latter studied this family of polynomials in the context of the local Riemann hypothesis. We prove several conjectures confirming that certain families of polynomials have real part –1/2. The main ingredient is the theory of interlacing polynomials.

14.11.2017, 15:00, WIL A 120: Seminar Geometrie
Maria Gerasimova, TU Dresden: Geometric aspects of unitarisability of discrete groups

A discrete group Γ called unitarisable if for every uniformly bounded representation π: Γ → B(H) of Γ on some Hilbert space H there exists an operator S ∈ B(H) such that S^-1π(g)S is unitary for all g ∈ Γ. It is well-known that amenable groups are unitarisable, but it is still unknown if unitarisable groups are amenable.

We will define some analytic characteristic of a group Γ which on one hand is related to unitarisability and on the other hand – to some geometry of Γ. More precisely, it is related to the behaviour of Cayley graphs of Γ when one increases the generating sets. We will also discuss some corollaries of this connection, for example, about the number of colours one need to colour the Cayley graphs of some groups (e.g. for Burnside groups B(m,n)).


21.11.2017, 15:00, WIL A 120: Seminar Geometrie
Marco Matassa, Université Blaise Pascal – Clermont-Ferrand II: Regularity of twisted spectral triples and pseudodifferential calculi

We investigate the regularity condition for twisted spectral triples. This condition is equivalent to the existence of an appropriate pseudodifferential calculus compatible with the spectral triple. A natural approach to obtain such a calculus is to start with a
twisted algebra of abstract differential operators, in the spirit of Higson. Under an appropriate algebraic condition on the twisting, we obtain a pseudodifferential calculus which admits an asymptotic expansion, similarly to the untwisted case. We present some examples coming from the theory of quantum groups. Finally we discuss zeta functions and the residue (twisted) traces on differential operators.

28.11.2017, 15:00, WIL A 120: Seminar Geometrie
Markus Steenbock, ENS Paris: Product Set Growth in Hyperbolic Geometry

Improving a theorem of Razborov (Ann. of Math. 2014), Safin (Mat. Sb. 2011) proved that there is a constant α > 0 such that the cardinality of any finite subset U of a free group that does not generate a cyclic subgroup satisfies |U³| ⩾ α |U²|. In addition, the growth of the cardinalities of the sets Uⁿ satisfies Uⁿ ⩾ αⁿ |U|^{⌊(n+1)/2⌋} for all natural numbers n.

We will discuss the extension of this result to hyperbolic groups. This includes cocompact Fuchsian groups, fundamental groups of compact Riemannian manifolds of negative sectional curvature, and finitely presented small cancellation groups.

05.12.2017, 15:00, WIL A 120: Seminar Geometrie
Igor Burban, Universität zu Köln: Algebraic geometry of the classical Yang-Baxter equation

In my talk, based on a joint work with Lennart Galinat,  I am going to give a short introduction into the theory the classical Yang-Baxter equation. Then I shall explain how solutions of this equation can be canonically obtained from appropriate  sheaves   of Lie algebras on possibly singular Weierstraß cubics.

12.12.2017, 15:00, WIL A 120: Seminar Geometrie
Dominik Gruber, ETH Zürich: Hyperbolicity in graphical small cancellation monsters

Infinitely presented small cancellation groups are a major source of examples and counterexamples in the theory of infinite groups.
Gromov's graphical small cancellation theory is a generalization of classical small cancellation theory that has provided the only known groups with expander graphs embedded in their Cayley graphs, the so-called Gromov's monsters. In my talk, I will discuss advances in understanding features of negative curvature of infinitely presented graphical small cancellation groups, such as admitting interesting actions on Gromov hyperbolic spaces. As an application, I will present a whole new small cancellation theory in varieties of n-periodic groups.
This is based on joint works with multiple co-authors.

09.01.2018, 15:00, WIL A 120: Seminar Geometrie
Tim de Laat, Universität Münster: Superexpanders from group actions on compact manifolds

Expanders are sequences of finite highly connected sparse graphs with an increasing (and unbounded) number of vertices. It is well known that expanders do not coarsely embed into Hilbert spaces. It is a deep open problem whether every expander is a superexpander, i.e. whether all expanders do not coarsely embed into any superreflexive Banach space. In this talk, I will explain a couple of constructions of superexpanders, focusing on the recent constructions coming from certain group actions on compact manifolds.

16.01.2018, 15:00, WIL A 120: Seminar Geometrie
Chiara Esposito, Universität Würzburg: Formality and Universal deformation formulas

In this talk we present an application of formality of Lie algebroids to the case where we have an infinitesimal Poisson action on a manifold M to obtain a generalization of the Drinfeld twists approach to the problem of quantization of symmetries. In particular we obtain a generalized notion of quantum action that yields universal deformation formulas. This is a joint work with N. de Kleijn.

23.01.2018, 15:00, WIL A 120: Seminar Geometrie
Masato Mimura, École polytechnique fédérale de Lausanne: Upgrading fixed points without bounded generation

A natural question on group actions asks the following. "Suppose that an isometric group action on a metric space admits fixed points for certain (small) subgroups. Under which conditions, can we 'upgrade' them to the existence of fixed points for a bigger subgroup, ultimately, for the whole group?"
This question for actions on Hilbert spaces has strong connection to proving Kazhdan's property (T) from relative properties. In 1999 and 2006, Shalom answered it based on (BG) ("Bounded Generation"). I will present the first solution to a natural question from Shalom's work that asks whether we can upgrade fixed points (in an intrinsic way) without employing (BG).
 

30.01.2018, 15:00, WIL A 120: Seminar Geometrie
Christian Voigt, University of Glasgow: A deformation of the Baum–Connes assembly map

(joint with A. Monk)
A central theme in the study of K-theory of operator algebras is the Baum-Connes conjecture, which predicts the K-theory of group C*-algebras and crossed products. In the case of complex semisimple Lie groups like SL(n,ℂ), the conjecture can be reformulated as saying that a certain deformation induces an isomorphism in K-theory.

In this talk I’ll first review these constructions and results, and then explain a similar picture for complex semisimple quantum groups. These quantum groups can be viewed as deformations of classical complex semisimple groups; on a technical level they are obtained using the Drinfeld double construction. In particular, I’ll show how to obtain an analogue of the Baum-Connes assembly map in this setting. The quantum assembly map is an isomorphism, and it contains the classical Baum-Connes assembly map as a direct summand. I’ll also discuss how the various deformations involved fit together in a conceptual way.

13.02.2018, 15:00, WIL A 120: Seminar Geometrie
Anton Claußnitzer, TU Dresden: Aspects of p-adic operator algebras

In my talk, I define possible p-adic analogues of certain well-known functional analytic concepts such as Hilbert spaces, Banach algebras and group von Neumann algebras. As an example, I will talk about several p-adic Lück type approximation lemmas for residually finite groups.

04.04.2017, 15:00, WIL A 120: Seminar Geometrie
Marcus de Chiffre, TU Dresden: Uniformity norms, inverse theorems and ε-representations for amenable groups

The Gowers uniformity norms ‖ ⋅ ‖_U^k were originally defined for functions φ : G → ℂ from a finite cyclic (or abelian) group G as part of Gowers' work on arithmetic progressions. Recently Gowers and Hatami extended the definition of the norm ‖ ⋅ ‖_U² to maps φ : G → 𝕄_n from a general finite group G taking values in the n × n -matrices 𝕄_n . Using this definition, they stated and proved a so-called inverse theorem for the U² -norm and, as a corollary, proved a stability result for ε-representations. In this talk, we present an operator algebraic approach to the proof of these theorems, which enables us to extend the definition and results to maps φ : G → ℳ from an amenable group G into a von Neumann algebra ℳ .

18.04.2017, 15:00, WIL A 120: Seminar Geometrie
Jordan Venters, University of Warwick: Combinatorial Nullstellensatz

Many problems in combinatorics may be rephrased as questions about polynomials and in particular the zeroes of these polynomials, one powerful tool in this direction is the combinatorial Nullstellensatz developed by Noga Alon which gives us conditions under which a polynomial restricted to some set is not the zero polynomial.

I will state the combinatorial Nullstellensatz and give several quick applications of it to additive number theory and graph theory.

18.04.2017, 16:30, WIL B 122: Seminar Geometrie
Peter Patzt, Freie Universität Berlin: Representation stability for filtrations of Torelli groups

The notion of representation stability for the symmetric groups, the general linear groups and the symplectic groups was introduced by Church-Farb. In this talk, we give a criterion for a sequence of algebraic representations of the general linear groups to be representation stable. We use this to prove that the factors of the lower central series of the Torelli subgroups of the automorphism groups of free groups are representation stable.

02.05.2017, 15:00, WIL A 120: Seminar Geometrie
Friedrich Wagemann, Université de Nantes: Shelves in algebra and topology

This will be a survey talk about shelves in algebra and topology. A shelf is a set X with a product "<" such that for all x,y,z in X: (x<y)<z = (x<z)<(y<z) (selfdistributivity). This is an instance of the third Reidemeister move, so there are natural applications to knot invariants. To mention just one other application in Lie theory, in case X is a pointed manifold and "<" is smooth, it makes sense to derive (x,y) -> x<y with respect to x and y at the point 1. This gives a Leibniz bracket on the tangent space at 1, and one may ask conversely whether all Leibniz algebras integrate to such shelves. In joint work with Martin Bordemann, we have constructed an integration process which gives back the conjugation shelf underlying a Lie group in case the Leibniz algebra was a Lie algebra.

09.05.2017, 15:00, WIL A 120: Seminar Geometrie
Anastasia Khukhro, Université de Neuchâtel: Geometry of finite quotients of groups

Geometric properties of a collection of finite quotients of a group can provide information about the group if the set of finite quotients is sufficiently rich. Using a metric space constructed using Cayley graphs of these finite quotients, one can exploit the connections between the world of group theory and graph theory to give examples of metric spaces with interesting and often surprising properties. In this talk, we will describe some results in this direction, and then give recent results concerning geometric rigidity of finite quotients of a group (joint work with Thiebout Delabie).

16.05, 23.05, 30.05.2017, 15:00, WIL A 120: Graduate Lectures
Jan Hladky, TU Dresden: Szemerédi regularity lemma and its applications

In 1978, Szemerédi published a three page paper "Regular partitions of graphs". The result of that paper, now known as the Regularity lemma, became one of the most important tools in graph theory, with deep connections to other parts of combinatorics, number theory, and to theoretical computer science.
The first lecture will cover the concept of graph quasirandomness and the Regularity lemma. We shall also derive the Removal lemma. In the second lecture, we will give applications to extremal graph theory, property testing, and number theory. In the third lecture, we shall use the Regularity lemma to construct graphons which are the key objects of the theory of limits of dense graphs.

13.06.2017, 15:00, WIL A 120: Seminar Geometrie
Stefan Colb, University of Newcastle: Braided module categories via quantum symmetric pairs

Abstract: In the early 90s V. Turaev and N. Reshetikhin formulated a
graphical calculus for invariants of knots and links. This calculus is
based on the notion of a ribbon category, which is a braided monoidal
category with some additional structure (twist and duality). Categories of
representations of quantum groups provide an important example for this
theory.

The Reshetikhin-Turaev calculus can be extended to braided module
categories over ribbon categories. In this talk I will explain the notion
of a braided module category and discuss its incarnation via
quasitriangular coideal subalgebras. I will then outline how quantum group
analogs of symmetric spaces provide examples of braided module categories.

21.06.2017, 17:00: Seminar Geometrie
Martín Miglioli, Instituto Argentino de Matemática: Geometric aspects of similarity problems

Abstract: Some similarity problems study conditions under which a unital homomorphism between operator algebras is conjugated to a self-adjoint homomorphism. We study aspects of these problems through the isometric action of the invertible elements of a C*-algebra on the positive invertible elements given by g.a=gag*. Minimality properties of projections to closed convex sets in the non-positively curved cone of positive invertible operators are used to study minimality properties of canonical orthogonalizers of some unital algebra homomorphisms. The convexity of the distance along geodesics is used to prove interpolation results of norms of conjugated representations. We also address the question of existence of unitarizers of groups of invertible operators when these groups act on manifolds of positive invertible operators endowed with a metric derived from a trace. Here the Bruhat–Tits fixed point theorem is used to show that the square root of the circumcenter of orbits are unitarizers of the groups.