Seminar algebra-geometry-combinatorics
Internationally renowned scientists report on their work. The seminar is organised by Prof. Ellen Henke, Prof. Manuel Bodirsky and Prof. Arno Fehm from the Institute of Algebra and by Prof. Uli Kraehmer, Jun.-Prof. Mario Kummer and Prof. Andreas Thom from the Institute of Geometry.
We cordially invite everyone to visit the seminar. If you want to get informations about upcoming talks by email then please add your contact to our list.
16.12.2021 (Thu) 1:30 pm Online |
Gil Goffer Weizmann Institute of Science Is invariable generation hereditary? We will discuss the notion of invariably generated groups, with various motivating examples. We will then see how hyperbolic groups and small cancellation theory are used in answering the question in the title, which was asked by Wiegold and by Kantor-Lubotzky-Shalev. This is a joint work with Nir Lazarovich. |
09.12.2021 (Thu) 1:30 pm Online |
Rainer Sinn University of Leipzig Polypols and their adjoints Polypols were introduced by Emil Wachspress as more general geometric elements than simplices and their adjoints play a key role for generalizing barycentric coordinates. We define these objects, discuss their relation to positive geometries from particle physics, and give an open problem about adjoints of polypols in the plane. |
25.11.2021 (Thu) 1:30 pm Online |
Damien Gaboriau ENS Lyon On homology torsion growth for SL_{d}(Z), Artin groups and mapping class groups. This is joint work with Miklos Abert, Nicolas Bergeron and Mikolaj Fraczyk. The growth of the sequence of Betti numbers is quite well understood when considering a suitable sequence of finite sheeted covers of a manifold or of finite index subgroups of a countable group. We are interested in other homological invariants, like the growth of the mod p Betti numbers and the growth of the torsion of the homology. We produce new vanishing results on the growth of torsion homologies in higher degrees for SL_{d}(Z), Artin groups and mapping class groups. As a central tool, we introduce a quantitative homotopical method that constructs small classifying spaces of finite index subgroups, at the same time controlling the complexity of the homotopy. Our method easily applies to free abelian groups and then extends recursively to a wide class of residually finite groups. |
18.11.2021 (Thu) 1:30 pm In presence (WIL/A317) Also on zoom |
Vadim Alekseev Institute of Geometry What is... a lattice in a Lie group? |
11.11.2021 (Thu) 1:30 pm In presence (WIL/A317) Also on zoom |
Jens Kaad |
04.11.2021 (Thu) 1:30 pm In presence (WIL/A317) Also on zoom |
Ulrich Krähmer Institute of Geometry What is... a quantum group? The theory of quantum groups/Hopf algebras can be motivated and approached in many ways. This talk is designed as a survey talk and I wil present the maybe most elementary one that focuses on actions on rings and avoids the use of tensor products. Towards the end I will sketch the results of a series of joint papers with my PhD students Angela Tabiri, Manuel Martins and Blessing Oni in which we study quantum groups acting on the commutative coordinate rings of singular plane curves. |
21.10.2021 (Thu) 1:30 pm In presence (WIL/A317) Also on zoom |
Daniel Max Hoffmann |
14.10.2021 (Thu) 4:20 pm Online |
Slawomir Solecki Cornell University Closed groups generated by generic measure preserving transformations The behavior of a measure preserving transformation, even a generic one, is highly nonuniform. In contrast to this observation, a very different picture of a uniform behavior of the closed group generated by a generic measure preserving transformation has emerged. This picture included substantial evidence that pointed to all these groups being topologically isomorphic to a single group, namely, L^{0} - the topological group of all Lebesgue measurable functions from [0,1] to T. We will describe the background touched on above (including all the relevant definitions) and outline a proof of the following theorem: for a generic measure preserving transformation T, the closed group generated by T is not topologically isomorphic to L^{0}. This result answers a question of Glasner and Weiss. The proof rests on an analysis of unitary representations of L^{0}. |
22.07.2021 (Thu) 1:30 pm Online |
Harald Helfgott |
15.07.2021 (Thu) 1:30 pm Online |
Manuel Bodirsky Institute of Algebra What do I teach in … Combinatorics? Andreas Thom Institute of Geometry What do I teach in … Groups and Geometry? |
8.07.2021 (Thu) |
Mehmet Haluk Sengün University of Sheffield Hecke operators for operator K-theory Cohomology of arithmetic groups admits the action of a commuting family of endomorphisms, called 'Hecke operators'. These cohomology groups, as 'Hecke modules', play a crucial role in the Langlands program, sitting in the intersection of the theory of automorphic forms, representation theory and differential geometry. In the world of noncommutative geometry, one finds several 'cohomology groups' associated to arithmetic groups; these arise as K-groups of certain C*-algebras. With Bram Mesland (Leiden), we showed that these cohomology groups admit an action of Hecke operators as well. Could these new 'Hecke modules' play a similar role to the ordinary cohomology in the Langlands program? |
1.07.2021 (Thu) 1:30 pm Online |
Steffen Kionke FernUniversität in Hagen Amenability and profinite completions of finitely generated groups |
24.06.2021 (Thu) 1:30 pm Online |
Holger Kammeyer Karlsruher Institut für Technologie On the profinite rigidity of higher rank lattices |
17.06.2021 (Thu) 1:30 pm Online |
Michael Wibmer Graz University of Technology Solving difference equations in sequences Systems of finite difference equations occur in various models of real-world phenomena. Informally, they can be described as discrete analogs of systems of differential equations. A very simple example of a finite difference equation is a recursion formula, such as the recursion formula defining the Fibonacci sequence. Clearly, a solution to a recursion formula can always be found by iterating the formula. However, when studying systems of general (implicit, nonlinear) finite difference equations, algorithms for deciding the existence of a solution are very hard to come by. In fact, this problem was shown to be decidable only recently by Ovchinnikov, Pogudin and Scanlon. In this talk we will discuss the (un)decidability of some natural computational questions associated with systems of finite difference equations. This is joint work with Thomas Scanlon and Gleb Pogudin. |
10.06.2021 (Thu) 1:30 pm Online |
Katrin Tent |
6.05.2021 (Thu) 1:30 pm online |
Christoph Schulze Institute of Geometry Cones of locally non-negative polynomials |
29.04.2021 (Thu) |
Myriam Mahaman |
22.04.2021 (Thu) 1:30 pm online |
Nicolas Garrel Institute of Algebra Ring structure on hermitian forms over central simple algebras with involution In the classical theory of quadratic forms over fields, one of the central objects is the Witt ring, which is a commutative ring consisting of equivalence classes of quadratic forms, where the sum is the direct orthogonal sum of spaces, and the product is the tensor product. It turns out that for many purposes, including prominently the study of algebraic groups, it is necessary to replace quadratic forms over fields by hermitian forms over central simple algebras with involution. In that case we still have an orthogonal sum, but there is no longer a natural product, since we can't take tensor products of modules over non-commutative rings. We will introduce hermitian Morita equivalences, and explain how to use them to still define a commutative ring in that situation, at the price of having to treat quadratic forms and hermitian forms simultaneously. |
15.04.2021 (Thu) 13:30 Urh online |
Tim Netzer Universität Innsbruck Quantum Magic Squares link The notion of a magic square is familiar to many people, the similar notion of a doubly stochastic matrix at least to many mathematicians. A doubly stochastic matrix can be defined as a square matrix that contains a probability measure in each row and column. The famous Theorem of Birkhoff-von Neumann states that these matrices are all convex combinations of permutation matrices. |
4.3.2021 (Thu) 13:30 Uhr online |
Colin Jahel U Lyon Some progress on the unique ergodicity problem link In 2005, Kechris, Pestov and Todorcevic exhibited a correspondence between combinatorial properties of structures and dynamical properties of their automorphism groups. In 2012, Angel, Kechris and Lyons used this correspondence to show the unique ergodicity of all the actions of some subgroups of $S_\infty$. In this talk, I will give an overview of the aforementioned results and discuss recent work generalizing results of Angel, Kechris and Lyons. |
4.2.2021 (Thu) 13:30 Uhr online |
Jitendra Bajpai Istitute of Geometry Arithmeticity and Thinness of hypergeometric groups link Password (usually not needed): *U#C2L The monodromy groups of hypergeometric differential equations of type nFn-1 are often called hypergeometric groups. These are subgroups of GL_n . Recently, Arithmeticity and Thinness of these groups have caught a lot of attention. In the talk, a gentle introduction and recent progress to the theory of hypergeometric groups will be presented. |
28.1.2021 (Thu) 13:30 Uhr online |
Julian Kaspczyk Institute of Algebra A characterization of the groups PSL(n,q) and PSU(n,q) by their 2-fusion systems, q odd link Password (usually not needed): *U#C2L The classification of finite simple groups (CFSG) is one of the greatest achievements of modern mathematics. Its proof required thousands of pages in various journals and was developed by more than hundred mathematicians from all over the world. A shorter and more accessible proof of the CFSG would be very valuable. A programme working towards this goal was set up by Aschbacher. The goal of Aschbacher's programme is to obtain a new proof of the CFSG by using fusion systems. Roughly, Aschbacher's programme consists of two steps. The first step is the classification of simple fusion systems over finite $2$-groups. In the second step, the first step is used in order to obtain a new proof of the CFSG. A crucial part of Step 2 is to identify finite simple groups from their $2$-fusion systems. In this talk, I am going to present some recent progress on Step 2. In particular, I will present the main result of my PhD thesis, which characterizes the finite groups $PSL(n,q)$ and $PSU(n,q)$ by their $2$-fusion systems, where $q$ is an odd prime power and $n \ge 2$. |
21.1.2021 (Thu) 13:30 Uhr online |
Martin Schneider TU Freiberg Skew-amenability of topological groups link with ZIH-login link without ZIH-login The talk will revolve around an amenability-like property of topological groups, its connections with representation theory and its manifestations in the realm of topological permutation groups. After briefly reviewing some aspects of (classical) amenability for topological groups, I will introduce the concept of skew-amenability, which originated in a recent work by Pestov about topological groups of finite-energy paths and loops. I will give an overview of closure properties of the class of skew-amenable topological groups, explain representation-theoretic and dynamical consequences of skew-amenability, and discuss some examples. The latter will include topological permutation groups built from Thompson's group F or Monod's group of piecewise projective homeomorphisms of the real line. |
14.1.2021 (Thu) 13:30 Uhr online |
Philip Dittmann Institut für Algebra Axiomatisability and definability in finitely generated structures link Password (usually not needed): *U#C2L Given a finitely generated group or ring or field, can it be pinned down (i.e. characterised up to isomorphism) by adding a single axiom (i.e. first-order sentence) to the group/ring/field axioms? This is a property known as quasi-finite axiomatisability. Although it appears unexpected from the perspective of model theory for reasons to be discussed, it has recently been shown to be very common (work of Nies, Aschenbrenner-Khélif-Naziazeno-Scanlon, D.-Pop). Secondly, which sets in a given finitely generated group/ring/field can "explicitly described", i.e. are definable (by a first-order formula)? This problem turns out to be closely related to the first one, and indeed one can frequently show that "every conceivable" subset is definable in a sense to be made precise. We will have a look at the underpinnings of these results both from mathematical logic (going back to Gödel) and arithmetic (connected to results on Hilbert's 10th Problem). |
7.1.2021 (Thu) 13:30 Uhr online |
Arno Fehm Institute of Algebra What is ... a diophantine set? link Password (usually not needed): *U#C2L I will motivate the study of diophantine sets in fields and will discuss various ways to look at them. Then I will survey some results and will try to give an idea of different proof techniques. |
17.12.2020 (Thu) 13:30 Uhr online |
Vadim Alekseev Institute of Geometry What is ... an arithmetic group? link Password (usually not needed): *U#C2L Arithmetic groups are fascinating objects which connect the worlds of countable groups, number theory, Lie groups, geometry and even ergodic theory. In this talk I will give a survey of some classical results about arithmetic groups, illustrating some of these connections, and - if time permits - explain how a question from a seemingly faraway field of operator algebras can get one to study arithmetic groups more profoundly. |
10.12.2020 (Thu) 13:30 Uhr online |
Manuel Bodirsky Institute of Algebra Ramsey Expansions and Extreme Amenability link This talk is about classes of finite structures that are closed under substructures, have the joint embedding property, and have the so-called Ramsey property, which is a strong combinatorial property generalizing Ramsey's theorem. I will present the connection between Ramsey classes and extreme amenability of the automorphism group of the respective Fraisse-limits. An important open problem in the field is the *Ramsey expansion conjecture*: does every homogeneous structure with a finite relational signature have a homogeneous finite-signature expansion whose age is Ramsey? I will also present a recent transfer result for the existence of Ramsey expansions by Mottet and Pinsker. |
3.12.2020 (Thu) 13:30 Uhr online |
Andreas Thom Institute of Geometry Questions about word maps on groups link |
19.11.2020 (Thu) 13:30 Uhr online |
Ellen Henke Institute of Algebra Topics and Methods in Finite Group Theory link Password (usually not needed): M@?22m In the talk I will introduce some important methods from finite group theory, which are used in the proof of the classification of finite simple groups but play also a role in other contexts. I will moreover briefly summarize how some aspects of finite group theory generalize to the theory of certain other algebraic structures called fusion systems and how one can hope to achieve a simplified proof of the classification of finite simple groups through this theory. |
12.11.2020 (Thu) 13:30 Uhr online |
Mario Kummer Instiitute of Geometry Some results and problems in real algebraic geometry link A widespread principle in real algebraic geometry is to find and use algebraic certificates for geometric statements. This covers for example writing a globally nonnegative polynomial as a sum of squares or expressing a polynomial with only real zeros as the minimal polynomial of a symmetric matrix. In the first part of the talk I will survey some classical results and open problems in this direction. Then I will present a quite general result from a joint work with Christoph Hanselka that implies several of the aforementioned results. |
5.11.2020 (Thu) 13:30 Uhr online |
Uli Krähmer Institute of Geometry What is an operad? link |
30.1.2020 (Thu) 13:30 Uhr WIL/B321 |
Dirk Kreimer HU Berlin Feynman Graphs: From Combinatorics to Algebra and Geometry Quantum field theory is studied with the help of Feynman graphs. We introduce the Hopf algebra structure underlying them and discuss how it raises interesting problems in combinatorics, algebra and geometry. |
9.1.2020 (Thu) 13:30 Uhr WIL/B321 |
Maja Pech U Novi Sad All that homogeneity One of the most fascinating phenomena in model theory is homogeneity. In this talk I will present and discuss different notions of homogeneity with special emphasis to the youngest one - polymorphism homogeneity. |
19.12.2019 (Thu) 13:30 Uhr WIL/B321 |
Maxim Gheysens Institut für Geometrie A Group from Beyond Infinity The order topology on the first uncountable ordinal yields a fascinating topological space, well known for featuring "pathological" phenomena, such as sequential compactness without compactness, or a Stone--Cech compactification reduced to Alexandrov's one-point compactification. In this talk, we introduce its homeomorphism group, which also comes with its own niceties. For instance, it is an amenable topological group which is strongly allergic to any idea of distance: all its continuous isometric actions have bounded orbits and all its continuous morphisms to metrisable groups are trivial. |
5.12.2019 (Do) 13:30 Uhr WIL/B321 |
Andrew Zucker U Paris Weak amalgamation versus amalgamation Fraïssé classes are those classes F of finite structures satisfying the Hereditary Property (HP), the Joint Embedding Property (JEP), and the Amalgamation Property (AP). The JEP allows one to build some countably infinite structure K whose age (the collection of finite structures which embed into K) is the class F that we started with. If AP also holds, then there is a canonical such K which we denote K_F, the Fraïssé limit of the class. One can interpret this fact dynamically by considering the action of the group S_\infty on the space X_F of countable structures whose age is contained in F. One can give X_F a natural Polish topology, and the theorem of Fraisse is precisely the fact that this action has a comeager orbit. However, Ivanov, and later Kechris-Rosendal, isolated a strictly weaker property, the Weak Amalgamation Property (WAP), which also guarantees that the space X_F has a comeager orbit. This gives rise to a natural question: can we detect the difference between AP and WAP from the dynamical point of view? This talk will discuss an affirmative answer, linking these amalgamation properties to the dynamical notion of a highly proximal extension. |
28.11.2019 (Do) 13:30 Uhr WIL/B321 |
Pablo Cubides Kovacsics Inst. f. Algebra Berkovich analytification of algebraic curves and definability Let k be a complete rank 1 algebraically closed non-trivially valued field. After recalling the construction of the analytification X^an of an algebraic variety X over k in the sense of Berkovich, we will show how, in the case of curves, one can have a definable version of this construction. More precisely, when X is a curve, we will show how to functorially associate to X^an a definable set X^S in a certain fist order language. If time permits, we will show how this definable version allows us to recover a result of Temkin about the "radiality" of the set of points of a given fixed multiplicity with respect to the analytification of a morphism between curves. This is a joint work with Jérôme Poineau. |
21.11.2019 (Do) 13:30 Uhr WIL/B321 |
Aleksandra Kwiatkowska U Wroclaw Simplicity of the automorphism groups of order and tournament expansions of structures We prove that the automorphism groups of order and tournament expansions of Fraïssé limits of free, transitive and nontrivial amalgamation classes, in particular, the automorphism group of the ordered random graph, are simple. We obtain similar results for the bounded countable Urysohn metric space. This is joint work with Filippo Calderoni and Katrin Tent. |
14.11.2019 (Do) 13:30 Uhr WIL/B321 |
Oleg Bogopolski U Düsseldorf Equations in acylindrically hyperbolic groups. Algebraic, verbal and existential closedness of subgroups of groups. |
24.10.2019 (Do) 13:30 Uhr WIL/B321 |
Ezra Waxman TU Dresden Hecke Characters and Angles of Gaussian Primes A Gaussian prime is a prime element in the ring of Gaussian integers Z[i]. As the Gaussian integers lie on the plane, interesting questions about their geometric properties can be asked which have no classical analogue among the ordinary primes. Hecke proved that the Gaussian primes are equidistributed across sectors of the complex plane by making use of Hecke characters and their associated L-functions. In this talk I will present a refined conjecture for the variance of Gaussian primes across sectors, and introduce a function field analogue to the question of angular distribution across sectors. |
17.10.2019 (Do) 13:30 Uhr WIL/B321 |
Arkady Leiderman Ben Gurion University of the Negev Embeddings of free (abelian) topological groups |
11.7.2019 (Do) 13:30 Uhr WIL/B321 |
Gregor Schaumann U Würzburg Module categories and oriented 3d TFTs Tensor categories have a far-reaching interpretation as quantum symmetries of 3-dimensional topological field theory (3d TFT). This leads to a correspondence between algebraic structures related to tensor categories and TFT. In this talk I will discuss instances of this correspondence for pivotal structures on tensor categories, module categories, and defects in oriented TFTs. |
27.6.2019 (Do) 13:30 Uhr WIL/B321 |
Lars Louder London Coherence for one relator groups with torsion |
20.6.2019 (Do) 13:30 Uhr WIL/B321 |
Istvan Heckenberger Uni Marburg From PBW deformations to matrix algebras |
23.05.2019 (Do) 13:30 Uhr WIL/B321 |
Michal Marcinkowski Universität Wroclaw Bounded cohomology of transformation groups Let M be a finite-volume Riemannian manifold and let \mu be the measure induced by the volume form. Denote by G_M the group of all \mu-preserving homeomorphisms of M isotopic to the identity. We show how to construct classes in the bounded cohomology of G_M. As an application we show that, under certain conditions on \pi_1(M), the third bounded cohomology of G_M (and some of its subgroups) is highly non-trivial. This is a joint work with Michael Brandenbursky. |
9.5.2018 (Do) 13:30 Uhr WIL/B321 |
Ulrich Krähmer Institut für Geometrie What is ... noncommutative geometry? |
02.05.2019 (Do) 13:30 Uhr WIL/B321 |
Jakub Gismatullin Uni Wroclaw Simplicity and amenability of metric ultraproducts of groups |
25.4.2019 (Do) 13:30 Uhr WIL/B321 |
Manuel Bodirsky Institut für Algebra What is ... an oligomorphic permutation group? |
04.04.2019 (Do) 13:30 Uhr WIL/B321 |
Adam Skalski Uni Warschau On C*-completions of discrete (quantum) group rings To any discrete group one can associate its complex group ring, which is a unital *-algebra. We will investigate when this algebra admits a unique C*-completion (in other words, all its embeddings into a *-algebra of bounded operators on a Hilbert space yield the same norm). Grigorchuk, Musat and Rordam observed this happens when the group in question is locally finite and asked if this is the only case. We will discuss what is known about their question and show that in the quantum world, i.e. when we look at C*-completions of Hopf*-algebras associated to compact quantum groups, the answer is negative. Based on the joint work with Martijn Caspers. |
24.01.2019 (Do) 13:30 Uhr WIL/B321 |
Andreas Thom TU Dresden, Institut für Geometrie What is ... representation theory? |
29.11.2018 (Do) 13:30 Uhr WIL/B321 |
Martin Schneider TU Dresden, Institut für Algebra Concentration and dissipation in topological groups |
22.11.2028 (Do) 13:30 Uhr WIL/B321 |
Christoph Schweigert Uni Hamburg Hochschild cohomology and the modular group |
8.11.2018 (Do) 13:15 Uhr WIL/B321 |
Franziska Jahnke WWU Münster Henselian fields in Shelah's classification theory In contemporary model theory, first-order theories are classified by the infinite combinatorial patterns they encode. A fundamental theme is whether these purely combinatorial `dividing lines' correspond to natural algebraic properties when applied to fields. One particularly interesting combinatorial pattern is when a theory encodes a bipartite random graph; if not, the theory is called NIP. A conjecture by Shelah states that every infinite NIP field is either algebraically closed, real closed, or `like the p-adic numbers'. We will discuss this conjecture and its consequences, and present some evidence for it. The talk contains joint work with Sylvy Anscombe, Yatir Halevi and Assaf Hasson. |
25.10.2018 (Do) 13:15 Uhr WIL/B321 |
Pablo Cubides Kovacsics Universite de Caen Polynomially bounded valued fields A theorem of Chris Miller states that given an o-minimal expansion R of the real field R, either R is polynomially bounded or the exponential function is definable in R. The aim in this talk is to show that over non-archimedean fields we have a very different behavior: all definable functions in "tame" expansions of Qp and Cp are polynomially bounded. A brief introduction to both o-minimality and their non-archimedean analogues will be given. No prior knowledge of model theory will be needed. This is a joint work with Francoise Delon. |
18.10.2018 (Do) 13:15 Uhr WIL/B321 |
Bruno Duchesne Université de Lorraine Groups acting on dendrites Roughly speaking, a dendrite is a compact topological tree. More precisely, a dendrite is a connected compact metrizable space that is locally connected and such that any two points are joined by a unique arc. We will be interested in groups acting on dendrites by homeomorphisms. We will see obstructions to have interesting such actions. On the other side, we will see dendrites with a very interesting homeomorphism group. These groups also appear to be the automorphism groups of some countable structures. |
19.7.2018 (Do) 13:15 Uhr WIL/B321 |
Philip Dittmann U Oxford Diophantine Predicates over the Rational Numbers A set of tuples of rational numbers is called diophantine if it is the projection of the zero set of a polynomial in some number of variables. Such sets are of interest in algebraic geometry and logic, where they are exactly the sets definable by existential first-order formulae. In this talk I will present some recent diophantine definitions; notably, I will sketch a proof that irreducibility of a polynomial over the rationals is a diophantine condition on its coefficients, and give a geometric application. It follows for instance that the set of non-squares is unexpectedly diophantine. The proof uses class field theory and central simple algebras, with some group-theoretic input. |
12.7.2018 (Do) 13:15 Uhr WIL/B321 |
Ehud Meir Hamburg symmetric monoidal categories, invariant theory and Hopf algrebas. In this talk I will explain a general approach for the study of algebraic structures using symmetric monoidal categories and geometric invariant theory. I will explain a construction of a symmetric monoidal category which is related to generic central simple algebras on the one hand, and to results of Deligne on the other hand. I will then concentrate especially on algebraic structures such as Hopf algebras and Hopf-Galois objects, and explain what new results can be proven for these structures using the symmetric monoidal categories approach. |
28.6.2018 (Do) 13:15 Uhr WIL/B321 |
Clara Löh U Regensburg Simplicial volume of one-relator groups The size of singular homology classes can be measured in terms of the $\ell^1$-semi-norm. One-relator groups have a canonical homology class in degree 2; in analogy with the case of manifolds, we define the simplicial volume of one-relator groups as the $\ell^1$-semi-norm of this class. In this talk, we will discuss the relation between simplicial volume of one-relator groups and stable commutator length of the relator. Moreover, we will give an application of these calculations to simplicial volume of manifolds. This is joint work with Nicolaus Heuer (Oxford). |
21.6.2018 (Do) 13:15 Uhr WIL/B321 |
Sylvy Anscombe U of Central Lancashire (Preston) Generalised measurable homogeneous structures The notion of a `measurable structure' was introduced by Macpherson and Steinhorn, motivated by the work on pseudofinite fields by Chatzidakis, van den Dries, and Macintyre. In joint work between the speaker, Macpherson, Steinhorn, and Wolf we propose the broader framework of `generalised measurable structures'. Both the new definition and the original definition abstract the idea of assigning a `measure' and a `dimension' to sets definable in a given structure. Originally, measures were positive real numbers and dimensions were natural numbers; but in the newer framework we combine measure and dimension into one element of a more general algebraic object: a measuring semiring. Consequently, there is a much richer zoo of examples and we have more exotic model theory. In this talk I will give a fresh introduction to these notions and discuss a range of key motivating examples, including pseudofinite fields, the random graph, and vector spaces with bilinear forms. I will also discuss more recent (and ongoing) work with the generic triangle-free graph and Urysohn space, which gets into more interesting combinatorial territory. |
14.6.2018 (Do) 13:15 Uhr WIL/B321 |
Jakob Schneider TU Dresden, Institut für Geometrie Word images are dense |
7.6.2018 (Do) 13:15 Uhr WIL/B321 |
Rémi Coulon CNRS - Université de Rennes 1 Growth gap in hyperbolic groups and amenability (joint work with Françoise Dal'Bo and Andrea Sambusetti) Given a finitely generated group G acting properly on a metric space X, the exponential growth rate of G with respect to X measures "how big" the orbits of G are. If H is a subgroup of G, its exponential growth rate is bounded above by the one of G. In this work we are interested in the following question: what can we say if H and G have the same exponential growth rate? This problem has both a combinatorial and a geometric origin. For the combinatorial part, Grigorchuck and Cohen proved in the 80's that a group Q = F/N (written as a quotient of the free group) is amenable if and only if N and F have the same exponential growth rate (with respect to the word length). About the same time, Brooks gave a geometric interpretation of Kesten's amenability criterion in terms of the bottom of the spectrum of the Laplace operator. He obtained in this way a statement analogue to the one of Grigorchuck and Cohen for the deck automorphism group of the cover of certain compact hyperbolic manifolds. These works initiated many fruitful developments in geometry, dynamics and group theory. We focus here one the class of Gromov hyperbolic groups and propose a framework that encompasses both the combinatorial and the geometric point of view. More precisely we prove that if G is a hyperbolic group acting properly co-compactly on a metric space X which is either a Cayley graph of G or a CAT(-1) space, then the growth rate of H and G coincide if and only if H is co-amenable in G. In addition if G has Kazhdan property (T) we prove that there is a gap between the growth rate of G and the one of its infinite index subgroups. |
31.5.2018 (Do) 13:15 Uhr WIL/B321 |
Martin Widmer Royal Holloway, University of London Heights - a survey: from dynamics to (un)decidability A height is a real-valued function that measures the arithmetic complexity of an algebraically defined object such as an algebraic number. The concept was introduced and developed by Weil, Siegel and Northcott in the first half of the last century, and since then it has played a major role in number theory. In this talk I will explain Northcott's original motivation, describe the construction and basic properties of the so-called Weil height, and discuss more recent applications of heights, including to (un)decidability questions about certain subfields of the algebraic numbers. This talk is aimed at a general audience of mathematicians. |
17.5.2018 (Do) 13:15 Uhr WIL/B321 |
Uli Krähmer TU Dresden Clones form comonoids The endomorphism operad of a cocommutative comonoid in an arbitrary monoidal category is shown to be a clone. An example associated to coquasitriangular Hopf algebras is discussed. |
3.5.2018 (Do) 13:15 Uhr WIL/C133 |
Lucia Rotheray TU Dresden Trees, quivers, bigraphs: combinatorial bialgebras from monoidal Möbius categories |
26.4.2018 (Do) 13:15 Uhr WIL/B321 |
Peter P. Palfy Renyi Institute, Budapest Groups of finite abelian width There are various ways to measure how abelian a group is. A few examples: the class of a nilpotent group; the probability that two random elements commute; the order of the commutator subgroup; etc. Our new concept is the abelian width, that is the smallest number of factors in a decomposition of the group into a product of abelian subgroups. If no such decomposition exists, then we say that the group has infinite abelian width. For example, the group of finitary permutations of an infinite set has infinite abelian width. However, quite surprisingly, the full symmetric group on an infinite set has finite abelian width. With the exception of the alternating groups, all finite simple groups have bounded abelian width. This is a joint project with M. Abért, A. Maróti, L. Pyber, and B. Szegedy. |
19.4.2018 (Do) 13:15 Uhr WIL/B321 |
Manuel Bodirsky TU Dresden Semi-linear relations preserved by the median operation |
12.4.2018 (Do) 13:15 Uhr WIL/B321 |
Andreas Thom TU Dresden Stability and Invariant Random Subgroups |
1.2.2018 (Do) 13:15 Uhr WIL/C133 |
François Legrand TU Dresden, Inst. f. Algebra On a Weak version of the Inverse Galois Problem The Inverse Galois Problem over a field k asks whether every finite group occurs as the Galois group of a finite Galois extension of k. The original problem of Hilbert and Noether is the case k=Q and, despite many efforts in the last decades, remains widely open. In 1980, Fried proved by elementary techniques that every finite group occurs as the automorphism group of a finite extension of Q (not necessarily Galois). This motivates the study of a Weak version of the Inverse Galois Problem where the notion of "Galois group" is replaced by that of "automorphism group". In this talk, we will discuss this variant over more general fields and, using (inverse) Galois theoretic and group theoretic tools, reveal the gap between it and the usual Inverse Galois Problem. This is based on a joint work with Elad Paran and a work in progress with Bruno Deschamps. |
18.01.2018 (Do) 13:15 Uhr WIL/C 133 |
Franz-Viktor Kuhlmann University of Szczecin Ball spaces - generic fixed point theorems for contracting functions In Fixed Point Theory there are many well known theorems that work with functions that are (explicitly or implicitly) contracting. Just to mention a few areas of mathematics where fixed point theorems play a role: metric spaces (Banach Fixed Point Theorem), ultrametric spaces, topological spaces, partially ordered sets, lattices (Knaster-Tarski Fixed Point Theorem), logic programming. The question arises: what is the common denominator of the proofs of such theorems, and of the conditions that have to be met by the functions and the spaces on which they live? I will report on an approach that reduces the structure on the spaces to the bare minimum that is necessary to formulate the conditions on the spaces. In analogy to the ultrametric world, we call this condition "spherical completeness": every chain of distinguished subsets, called "balls", must have a nonempty intersection. I will show how this translates into, e.g., completeness for metric and compactness for topological spaces. This approach allows us to prove generic fixed point theorems (with proofs based on Zorn's Lemma) which then can be specialized to the various areas of application. Moreover, it allows us to transfer ideas and notions from one area to another. For instance, the Knaster-Tarski Theorem can be transferred to topological and ultrametric spaces, and there is also a generic form of the Tychonoff Theorem, which in turn can be specialized e.g. to ultrametric spaces. The metric spaces used to be the misfits in the big picture, because in contrast to ultrametric balls the metric balls do not behave well and one is still better off working with Cauchy sequences instead. However, recently we found that new insight into the famous Caristi-Kirk Theorem changes the picture completely. There are alternatives to the metric balls which fit perfectly into the framework of our ball spaces. This is joint work with Katarzyna Kuhlmann. |
14.12.2017 (Do) 13:15 Uhr WIL/C 133 |
Andreas Thom TU Dresden, Institut für Geometrie Stability of asymptotic representations, cohomology vanishing, and non-approximable group |
07.12.2017 (Do) 13:15 Uhr WIL/C133 |
Tobias Fritz MPI-MIS, Leipzig On the decidability of noncommutative sums of squares, with applications to Connes' embedding problem and quantum logic Due to Positivstellensätze, sums of squares play a fundamental role in real algebraic geometry. While this remains true in the noncommutative case, the noncommutativity makes the problem generally undecidable: there is no algorithm to decide whether a given noncommutative polynomial can be approximated by sums of squares. I will explain why this is the case, how it implies the undecidability of quantum logic, and how it relates to Connes' embedding problem. Based on arXiv:1207.0975 and arXiv:1607.05870. |
30.11.2017 (Do) 13:15 Uhr WIL/C133 |
Gabriella Böhm Wigner Research Centre for Physics of the Hungarian Academy of Sciences, Budapest A unified treatment of quantum symmetries Symmetries in various situations are described classically by groups. In the last decades, however, many indications lead to the appearance of more general symmetry structures, and many (interrelated) competing notions have been proposed. In the talk we will survey some of these so-called `quantum’ symmetries. As a highlight, we will illustrate how these apparently different algebraic structures can be seen as instances of the unifying notion of Hopf monad in a monoidal bicategory. |
23.11.2017 (Do) 13:15 Uhr WIL/C133 |
Lukas Pottmeyer U Duisburg-Essen Small points and free abelian groups The height of an algebraic number can be seen as a measure for the arithmetic complexity of the algebraic number. For any algebraic extension $F$ of the rationals, the height induces a norm on the group $F^*/Tors(F^*)$ - the torsion points in $F^*$ are precisely the roots of unity in $F$. Using some group theory it follows that $F^*/Tors(F^*)$ is free abelian if there are no points in $F$ of arbitrarily small positive height. In this talk we prove that the converse is not true; i.e. there are fields $F$ such that $F^*/Tors(F^*)$ is free abelian, but there are points of arbitrarily small positive height in $F$. We conclude by giving some (weak) group theoretical support for a widely open arithmetical conjecture due to G. Rémond. Most of this is joint work with R. Grizzard and P. Habegger. |
16.11.2017 (Do) 13:15 Uhr WIL/C133 |
Andrei Krokhin U Durham The complexity of valued constraint satisfaction problems The Valued Constraint Satisfaction Problem (VCSP) is a well-known combinatorial problem. An instance of VCSP is given by a finite set of variables, a finite domain of labels for the variables, and a sum of functions, each function depending on a subset of the variables. Each function can take finite rational values specifying costs of assignments of labels to its variables or the infinite value, which indicates an infeasible assignment. The goal is to find an assignment of labels to the variables that minimizes the sum. The case when all functions takeonly values 0 and infinity corresponds to the standard CSP. We study (assuming that P\neq NP) how the computational complexity of VCSP depends on the set of functions allowed in the instances, the so-called constraint language. Helped greatly by algebra, massive progress has been made in the last three years on this complexity classification question, and our work gives, in a way, the final answer to it, modulo the complexity of CSPs. This is joint work with Vladimir Kolmogorov and Michal Rolinek (both from IST Austria). |
2.11.2017 (Do) 13:15 Uhr WIL/C133 |
Martin Hils U Münster Imaginärenklassifikation in separabel abgeschlossenen bewerteten Körpern |
26.10.2017 (Do) 13:15 Uhr WIL/C133 |
Jan Hladky TU Dresden Relating the cut-distance and weak* convergence for graphons Around 2004, Borgs, Chayes, Lovasz, Sos, Szegedy, Vesztergombi came up with limit objects, so-called graphons, that can be associated to sequences of graphs. This has subsequently led to a number of breakthrough applications in extremal graph theory and random graph theory. All these results rely on the key fact that the space of graphons equipped with the so-called cut-distance is compact. In the talk I will present a new view on the cut-distance, which in particular gives an alternative proof of compactness. The talk will be self-contained. This is based on joint work with Dolezal [arXiv:1705.09160] and further extensions with Dolezal, Grebik, Noel, Piguet, Rocha, Rozhon, Saumell. |
19.10.2017 (Do) 13:15 Uhr WIL/C133 |
Jakub Oprsal TU Dresden Infinite algebras with few subalgebras of powers For a finite algebra, there can be up to doubly exponentially many in $n$ subalgebras of the $n$-th power. It is known that if this is not the case, the algebra has few subpowers, i.e., there is a polynomial $p$ such that there are at most $2^{p(n)}$ subalgebras of the $n$-power. All finite algebras having this property have been characterized in 2009 by Berman, Idziak, Markovic, McKenzie, Valeriote, and Willard. In the talk, we will focus on infinite algebras with few subpowers; in particular we require that we have a finite number of subalgebras of powers. These algebras seems to be much rarer than their finite counterparts. We will give a few examples, and describe how the clasification of the finite algebras generalize to the infinite, and in particular to algebras that are obtained by taking polymorphisms of countable structures. |
12.10.2017 (Do) 13:15 Uhr WIL/C133 |
Arno Fehm TU Dresden Elementary equivalence of profinite groups I will discuss some results and questions concerning elementary equivalence and isomorphisms of profinite groups. |
29.6.2017 (Do) 13:15 Uhr WIL/C133 |
Sylvy Anscombe U of Central Lancashire, Preston, UK Multidimensional asymptotic classes and generalised measurable structures Macpherson and Steinhorn introduced `1-dimensional asymptotic classes' of finite structures, which are those classes that satisfy an asymptotic counting condition on definable sets, which is modelled on Chatzidakis, van-den-Dries, and Macintyre's work on finite fields. This notion gives insights into the model theory of such classes: for example their infinite ultraproducts are supersimple of finite rank. In this talk I will introduce `multidimensional asymptotic classes', which broadens the original notion to allow more exotic examples, such as finite-dimensional vectors spaces over finite fields, equipped with a bilinear form. I will also define the notion of `generalised measurable structures', which are infinite structures satisfying an analogous condition on definable sets. I will show that free homogeneous structures are generalised measurable; and that certain homogeneous structures are not elementarily equivalent to ultraproducts of classes satisfying an *exact* version of the asymptotic condition. This is joint work with Dugald Macpherson, Charles Steinhorn, and Daniel Wolf. |
22.6.2017 (Do) 13:15 Uhr WIL/C133 |
Oleg Verbitsky HU Berlin The many facets of logic with two variables First-order logic with two variables has surprising connections to seemingly distant concepts of discrete mathematics. For example, two graphs are indistinguishable by sentences with two first-order variables and counting quantifiers if and only if they are fractionally isomorphic and if and only if their universal covers are isomorphic. Due to these connections, 2-variable logic provides a convenient framework for analysis of various problems in theory of computing. We will survey these applications with a focus on the graph isomorphism problem. In particular, we will discuss the following result and its algorithmic consequences: If a graph is definable in 2-variable logic with counting quantifiers, then the polytope of the fractional automorphisms of this graph is integral, i.e., all extreme points of this polytope have integer coordinates (or, in other terms, every definable graph is compact in the sense of Tinhofer). The talk is based on joint work with A.Krebs, V.Arvind, J.Köbler, and G.Rattan. |
15.6.2017 (Do) 13:15 Uhr WIL/C133 |
Alejandra Garrido Uni Düsseldorf Primitive groups of intermediate word growth |
1.6.2017 (Do) 13:15 Uhr WIL/C133 |
Ulrich Krähmer TU Dresden Cyclic homology arising from adjunctions This will be a talk on joint work with Niels Kowalzig and Paul Slevin in which I'll explain how distributive laws between (co)monads lead to cyclic homology theories. |
18.5.2017 (Do) 13:15 Uhr WIL/C133 |
Martin Schneider TU Dresden Topological groups of measurable maps |
11.5.2017 (Do) 13:15 Uhr WIL/C133 |
Andres Aranda U Calgary Ramsey properties of metrically homogeneous graphs Homogeneous structures are countably infinite structures that satisfy the strong symmetry condition of having every isomorphism between finite substructures induced by an automorphism. Gregory Cherlin has produced a classification of metrically homogeneous graphs, conjectured to be complete. It is known that the class of finite structures embeddable in a homogeneous structure can often be extended to a class with the Ramsey property; we present a recent approach to proving the Ramsey and other related properties (EPPA, existence of a stationary independence relation) that can be applied to Cherlin's catalogue and has some potential for more general application. |
4.5.2017 (Do) 13:15 Uhr WIL/C133 |
Itay Kaplan Hebräische Universität Jerusalem On dense subgroups of permutation groups I will present a criterion that ensures that Aut(M) has a 2-generated dense subgroup when M is a countable structure (which holds in many natural examples), and then show how expanding an omega categorical structure might help in getting a finitely generated dense subgroup. This is joint work with Pierre Simon. |
27.4.2017 (Do) 13:15 Uhr WIL/C133 |
Jakob Schneider TU Dresden Metric approximation of groups by finite groups Sofic groups are precisely the groups which can be approximated by finite symmetric groups with normalized Hamming metric. It is an open question whether non-sofic groups exist. In the first part of this talk we define the concepts of C-approximable abstract and topological groups for a class C of finite groups, as a generalization of sofic and weakly sofic groups, and present examples of such groups. In the second part we give examples of groups which cannot be approximated by certain classes of finite groups, varying the above question. At first we discuss the question raised by Glebsky whether all abstract groups are approximable by finite solvable groups. Finally, referring to questions of Doucha and Zilber, we turn to the approximability of Lie groups by finite groups, considering two different kinds of approximation. All results we present will be consequences of theorems on commutators and generators in finite groups by Nikolov and Segal. This is joint work with A. Thom and N. Nikolov. |
20.4.2017 (Do) 13:15 Uhr WIL/C133 |
Tim Netzer U Innsbruck Operator Systems and Spectrahedra Spectrahedra are the feasible sets of semidefinite programming, and important objects in convex algebraic geometry. Many of its properties can only be fully understood by considering their free versions, i.e. adding matrix levels to the usual setup. In this way one obtains a special class of operator systems, namely those that admit a finite-dimensional realization. We thus examine abstract operator systems, and their concrete realizations. We give a characterization of finite-dimensional realizability, and apply it to systems constructed from convex cones at scalar level. In this way we obtain interesting results about classical spectrahedra. For example, we can completely classify the cases where an approach of Ben-Tal and Nemirovski for detecting inclusion is exact. We can also deduce results about the number of different matrix inequalities defining the same spectrahedron. The talk is based on joint results with T. Fritz and A. Thom. |
06.04.2017 (Do) 13:15 Uhr WIL C133 |
Barnaby Martin Durham University Generating sets for powers of finite algebras and the complexity of Quantified Constraints. Let A be a finite-domain algebra. We can associate a function f_A:N->N, giving the cardinality of the minimal generating sets of the sequence A, A^2, A^3, ... as f(1), f(2), f(3), ..., respectively. We may say A has the g-GP if f(m) < g(m) for all m. The question then arises as to the growth rate of f and specifically regarding the behaviours constant, logarithmic, linear, polynomial and exponential. Wiegold proved that if A is a finite semigroup then f_A is either linear or exponential, with the former prevailing precisely when A is a monoid. This dichotomy classification may be seen as a gap theorem because no growth rates intermediate between linear and exponential may occur. We say A enjoys the polynomially generated powers property (PGP) if there exists a polynomial p so that f_A=O(p) and the exponentially generated powers property (EGP) if there exists a constant b so that f_A=Omega(g) where g(i)=b^i. A new result by Dmitriy Zhuk states that this gap between PGP and EGP holds for all finite-domain algebras. Furthermore, the PGP cases obey a certain property called switchability. This "switchability" was introduced in the context of the Quantified Constraint Satisfaction Problem (QCSP) as something more general than "collapsibility", also introduced in the context of the QCSP. We study the relationship between switchability and collapsibility and use the PGP-EGP Gap Theorem of Zhuk to give a full dichotomy classification for the computational complexity of QCSPs, where the PGP cases correspond to QCSPs in NP while the EGP cases correspond to QCSPs that are co-NP-hard. |
15.12.2016 (Do) 13:15 Uhr WIL C133 |
Dr. Vadim Alekseev TU Dresden Operator algebras, discrete groups and finite approximations |
17.11.2016 (Do) 13:15 Uhr WIL C133 |
Prof. Dr. Eamonn O'Brien University of Auckland, New Zealand Effective algorithms for matrix groups |
10.11.2016 (Do) 13:15 Uhr WIL C 133 |
Dr. Wied Pakusa RWTH Aachen The search for a logic capturing polynomial time (Abstract) |
03.11.2016 (Do) 13:15 Uhr WIL C 133 |
Gabor Kun MTA Alfred Renyi Institute of Mathematics, Budapest, Ungarn Sofic groups A group is called sofic if any of its labeled Cayley graphs admits a local approximation by finite labeled graphs. Sofic groups were introduced by Gromov (the word sofic was coined by Weiss). Many classical conjectures are proved for sofic groups: Gottschalk’s conjecture, Kaplansky’s direct finiteness conjecture and Connes’ embedding conjecture. It is not known if every group is sofic. Sofic groups and related notions have attracted a great attention in many fields from algebra to logic and graph theory. I will explain the basics. |
27.10.2016 (Do) 13:15 Uhr WIL C 133 |
Szymon Torunczyk U Warschau Entropy bounds for conjunctive queries I will talk about a problem emerging from database theory, concerning bounding the worst-case size of a result of a conjunctive query. In the basic form, this problem has been solved by Atserias, Grohe, Marx in '13 using entropy. I will present yet another proof of their result using Shannon's information inequalities, and discuss the relationship with several inequalities in geometry, e.g., the Loomis-Whitney inequality and Bollobas-Thomason inequality and Friedgut's inequality. The result can be also stated in terms of bounds concerning finite groups, using a connection between entropy and groups first discovered by Zhang and Yeung in '98. I will also briefly discuss an open problem concerning a generalization obtained by allowing functional dependencies. |
20.10.2016 (Do) 13:15 Uhr WIL C 133 |
Julius Jonusas U St. Andrews, UK Universal words and sequences Given a group, I am particularly interested in finding (preferably minimal) generating set of the group. However, every generating set of an uncountable group must be of the same cardinality as the group itself, in which case generating sets proved very little new information about the structure of the group. In this talk, I will introduce notions of universal words and universal sequences for groups and semigroups, and then show how universal sequences can be thought as an extension of the notion of generation. |
Sommersemester 2016
16.6.2016 (Do) 13:15 Uhr WIL C 133 |
Dr. Henry Bradford U Göttingen Expansion, Random Walks and Sieving in SL2(Fp [t]) |
09.6.2016 (Do) 13:15 Uhr WIL C 133 |
Dr. Martin Finn-Sell U Wien Semigroups in coarse geometry and operator algebras |
02.6.2016 (Do) 13:15 Uhr WIL C 133 |
Tamás Waldhauser Bolyai Institute, Szeged, Hungary Polycyclic monoids and Cuntz algebras via strange number systems |
12.5.2016 (Do) 13:15 Uhr WIL C 133 |
Dr. Alessandro Carderi TU Dresden An exotic group as limit of finite special linear groups |
28.4.2016 (Do) 13:15 Uhr WIL C 133 |
Dr. Benedikt Stufler Mathematisches Institut der LMU München Limits of random tree-like combinatorial structures |
21.4.2016 (Do) 13:15 Uhr WIL C 133 |
Dr. Johann Thapper Université Paris-Est, France Fractional polymorphisms and LP relaxations of VCSPs |
14.4.2016 (Do) 13:15 Uhr WIL C 133 |
Prof. Dr. Miklós Abért MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary Entropy and graph convergence |
Wintersemester 2015/16
21.1.2016 (Do) 13:15 Uhr WIL C 133 |
Manuel Bodirsky TU Dresden Tree-like Homogeneous Structures, their reducts, automorphism groups, and polymorphism clones |
7.1.2016 (Do) 13:15 Uhr WIL C 133 |
Marcello Mamino TU Dresden Max-closed semilinear constraints |
17.12.2015 (Do) 13:15 Uhr WIL C 133 |
Jérémie Brieussel Université de Montpellier Speed of random walks, isoperimetry and compression on finitely generated groups |
10.12.2015 (Do) 13:15 Uhr WIL C 133 |
Michael Pinsker Czech Academy of Sciences, Prague Uniform Birkhoff |
3.12.2015 (Do) 13:15 Uhr WIL C 133 |
Katrin Tent U Münster Kurze Beschreibungen endlicher Gruppen |
26.11.2015 (Do) 13:15 Uhr WIL C 133 |
Sean Eberhard MMath (Cantab) Oxford University Invariable generation of the symmetric group |
19.11.2015 (Do) 13:15 Uhr WIL C 133 |
Andreas Thom TU Dresden Quasi-isometry and topological orbit-equivalence |
5.11.2015 (Do) 13:15 Uhr WIL C 133 |
Hanno Lefmann TU Chemnitz Infinite partition regular linear systems of equations |
29.10.2015 (Do) 13:15 Uhr WIL C 133 |
Damain Osajda U Wroclawski and IMPAN Embedding expanders in groups and applications |
15.10.2015 (Do) 13:15 Uhr WIL C 133 |
Agelos Georgakopoulos U Warwick Group Walk Random Graphs I will discuss a construction of finite 'geometric' random graphs motivated by the study of random walks on infinite groups. This construction has connections to other topics, including the Poisson boundary and Sznitman's random interlacements (which I will try to introduce in a gentle way). |
Sommersemester 2015
16.7.2015 (Do) 13:15 Uhr WIL C 133 |
Phillip Wesolek UC Louvain Elementary amenable groups and the space of marked groups The space of marked groups is a compact totally disconnected space that parameterizes all countable groups. This space allows for tools from descriptive set theory to be applied to study group-theoretic questions. The class of elementary amenable groups is the smallest class that contains the abelian groups and the finite groups and that is closed under group extension, taking subgroups, taking quotients, and taking countable directed unions. In this talk, we first give a characterization of elementary amenable groups in terms of a chain condition. We then show the set of elementary amenable marked groups is not in the Borel sigma algebra of the space of marked groups. This gives a new proof of a theorem of Grigorchuk: There are finitely generated amenable non-elementary amenable groups. We conclude by discussing further questions and possible generalizations of the techniques. |
11.6.2015 (Do) 13:15 Uhr WIL C 133 |
Anusch Taraz TU Hamburg-Harburg Embedding and packing large graphs into dense and sparse graphs Extremal combinatorics is often concerned with the forced appearance of highly organized structures. In this talk, we explain two major research avenues that deal with such situations. On the one hand, density results assert that these substructures must be present in any sufficiently dense host configuration. On the other hand, partition theorems guarantee that, no matter how we partition a sufficiently large object, at least one of the partition classes must contain the desired substructure. We first survey old and new results of both types, to give a flavour of the field and its methods, and then focus on a sequence of results that generalize the existence of paths and cycles to graphs of sublinear bandwidth. |
21.5.2015 (Do) 13:15 Uhr WIL C 133 |
Alessandro Carderi ENS Lyon Space of actions, ultraproducts and sofic entropy |
7.5.2015 (Do) 13:15 Uhr WIL C 133 |
Viola Meszaros TU Berlin Voronoi game on graphs The discrete Voronoi game is played on a graph G by two players A and B for a fixed number t of rounds. Player A starts and they alternatingly claim vertices of G in each round. No vertex can be claimed more than once. At the end of the game also the remaining vertices are divided among the players. Each player receives the vertices that are closer to his/her claimed vertices. If a vertex is equidistant to both players then we split it among them. In general it is hard to determine who gets more and therefore wins the game. First Demaine, Teramoto and Uehara dealt with this question. They proved that it is NP-complete to determime the winner of the Voronoi game on a general graph G. The game on trees was studied further by Kiyomi, Saitoh and Uehara. They showed that on a path it always ends in a draw unless the number of vertices is odd and t=1 when A wins by one. We were interested in knowing when either player could control a large portion of the graph by the end of the game. We proved that there are graphs for which player B gets almost all vertices. The proof is inspired by a geometric construction. On a tree, player A can get at least one quarter of the vertices. If the game lasts for two rounds on a tree, then A can even get one third of all vertices. But A cannot get much more. A construction shows that this result cannot be improved when t is least two. In one round A can always claim half of the vertices of a tree. Inspired by the outcome of the game on a star, we investigated how much B can ensure on graphs with bounded degree. We also made some observations relating the result with many rounds to the one-round game. It is a joint work with D. Gerbner, D. Palvolgyi, A. Pokrovskiy and G. Rote. |
30.4.2015 (Do) 13:15 Uhr WIL C 133 |
Christian Pech TU Dresden Reconstructing the topology of polymorphism clones Every clone of functions comes naturally equipped with a topology - the topology of pointwise convergence. A clone C is said to have automatic homeomorphicity with respect to a class K of clones, if every clone-isomorphism of C to a member of K is already a homeomorphism (with respect to the topology of pointwise convergence). I am going to talk about automatic homeomorphicity-properties for polymorphism clones of countable homogeneous relational structures. The results base on (and extend) previous results by Bodirsky, Pinsker, and Pongrácz. |
23.4.2015 (Do) 13:15 Uhr WIL C 133 |
Andreas Thom TU Dresden Group laws for finite simple groups |
Wintersemester 2014/15
5.2.2015 (Do) 13:15 Uhr WIL C 133 |
Friedrich Martin Schneider TU Dresden Invariants of group actions and their connection to amenability |
29.1.2015 (Do) 13:15 Uhr WIL C 133 |
Anton Klyachko Moscow Surprising divisibilities in group theory I shall talk about a general theorem which implies some amusing facts. For instance, in any group, the number of elements whose 2015th powers belong to a given subgroup is always a multiple of the order of this subgroup. The talk is based on joint results with Anna Mkrtchyan. |
15.1.2015 (Do) 13:15 Uhr WIL C 133 |
Marcello Mamino TU Dresden Groups definable in two orthogonal sorts We will investigate the following question. Let (Z,R) be a first order structure with two sorts Z and R, which is simply the disjoint union of a structure Z and a structure R -- i.e. there is no function and no relation in the language connecting the two structures. Can groups definable in (Z,R) be analyzed in terms of groups definable in Z and groups definable in R separately? |
8.1.2015 (Do) 13:15 Uhr WIL C 133 |
Jan Hladky Czech Academy of Sciences, Prague The Loebl-Komlos-Sos conjecture Many problems in extremal graph theory fit in the following framework: Does a certain density condition imposed on a host graph guarantee the existence of a given subgraph? Perhaps the most famous example in this direction is Mantel's Theorem from 1907: If a graph on n vertices contains at more than n^2/4 edges, then it must contain a triangle. I will give further examples, explain the basic concepts of extremal graphs and stability, and show the role of the celebrated Szemeredi Regularity Lemma in proving similar results. I will then report on joint progress with Janos Komlos, Diana Piguet, Miklos Simonovits, Maya Stein, Endre Szemeredi on the Loebl-Komlos-Sos conjecture, an extremal problem about containment of trees, which has been open for two decades. |
11.12.2014 (Do) 13:15 Uhr WIL C 133 |
Isolde Adler U Frankfurt PAC learning of FO definable concepts & nowhere dense graph classes In machine learning, the problem of "concept learning" is to identify an unknown set from a given concept class (i.e. a collection of sets). In the model of "probably approximately correct" (PAC) learning, the learner receives a number of samples and must be able to identify the unknown set approximately, in a probabilistic sense. It is well-known the the sample size for PAC learning is characterised by the Vapnik-Cervonenkis (VC) dimension of the concept class. We are interested in the VC dimension of concept classes that are definable in some logic on graph classes. In 2004, Grohe and Turán showed that for any subgraph closed class C, monadic second-order definable concept classes have bounded VC dimension on C if and only if C has bounded tree-width. We show that for any subgraph closed class C, first-order definable concept classes have bounded VC dimension on C if and only if C is nowhere dense. |
20.11.2014 (Do) 13:15 Uhr WIL C 133 |
Jan-Christoph Schlage-Puchta U Rostock Der Product Replacement Algorithmus auf der symmetrischen Gruppe Der Product Replacement Algorithmus auf der symmetrischen Gruppe erzeugenden k-Tupel von G eine Irrfahrt gestartet, und am Ende ein Element dieses Tupels ausgegeben. In der Praxis hat sich dieser Algorithmus gut bewährt, eine theoretische Analyse ist in den meisten Fällen jedoch nicht möglich. In diesem Vortrag werde ich für die symmetrische Gruppe S_n zeigen, dass der Graph auf dem die Irrfahrt abläuft einerseits für kleinere k zusammenhängend ist, als man bisher wusste, andererseits der Zusammenhang numerisch schwerer nachweisbar ist, als man zunächst dachte. |
13.11.2014 (Do) 13:15 Uhr WIL C 133 |
John Wilson U Oxford Metric ultraproducts of finite simple groups Metric ultraproducts of structures have arisen in a variety of contexts. The study of the case when the structures are finite groups is recent and motivated partly by the connection with sofic groups. We report on current joint work with Andreas Thom on the topological and algebraic properties of metric ultraproducts of finite simple groups. |
6.11.2014 (Do) 13:15 Uhr WIL C 133 |
Tim Netzer Tu Dresden Convex sets with semi-definite representations |
30.10.2014 (Do) 13:15 Uhr WIL C 133 |
Wieslav Kubis U Prag Banach-Mazur games on categories We shall discuss a natural game played on a category with amalgamations, in the spirit of the classical Banach-Mazur game. Among applications, we obtain new characterizations of various generic objects, including Fraisse limits, as well as a new tool for detecting generic structures in continuous model theory. |
23.10.2014 (Do) 13:15 Uhr WIL C 133 |
Andreas Thom Tu Dresden Sofic groups and applications |
16.10.2014 (Do) 13:15 Uhr WIL C 133 |
Manuel Bodirsky Tu Dresden Ramsey Theory and Topological Dynamics |