Seminar Algebra–Geometrie–Kombinatorik

English
Internationally renowned scientists report on their work. The seminar is organised by  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.

Pierre Touchard (TU Dresden)

On model theory of Mekler groups

A Mekler group is a 2-nilpotent group G of prime exponent p, which is freely generated over the vertices of a 'nice´ graph Γ and where the edge relation of the vertice describes commutation of the generative elements of the group.

Since Mekler's pioneering work, model theorists have shown that a Mekler group G and its associated graph  Γ share many model-theoretic properties, suggesting a strong connection between G and Γ.

In a joint work with Blaise Boissonneau and Aris Papadopoulos , we  explained this connection in a syntactic way: we expressed the definable sets of G using the definable sets of Γ (relative quantifier elimination).

In this talk, I will describe the structure of Mekler groups and give an insight into this syntactic result. 

As application, we will explore some interactions between model theory and structural Ramsey theory and see that a Mekler group 'codes' an omega-categoric homogeneous Ramsey structure if and only if the associated graph does.

Contact: Prof. Dr. Arno Fehm

Réamonn Ó Buachalla (Charles University)

Nested Pairs of Quantum Homogeneous Spaces

In recent work Brzezinski and Szymanski formulated a q-deformation of the classical $2$-sphere fibration of the full flag manifold of $SU_3$ over the complex projective plane. Explicitly, they described the full quantum flag manifold $\mathcal{O}_q(\mathrm{F}_3)$ as the total space of a quantum fibration over the quantum projective plane, with a Podle\'s sphere fibre $\OO_q(S^2)$. The authors put this non-principal quantum fibration forward as a motivating example for a proposed theory of noncommutative fibrations with quantum homogeneous fibers. In this talk I will describe a simple but effective framework for producing more examples of noncommutative fibrations with quantum homogeneous fibers. Following Brzezinski and Szymanski, variations on Takeuchi's equivalence and Schneider's descent theorem will be established. Quantum flag manifolds and their associated quantum Poisson homogeneous spaces are taken as motivating examples, and a Dynkin diagram recipe for constructing quantum fibrations is given.

Contact: Prof. Dr. Ulrich Krähmer

Sebastian Meyer (TU Dresden)                                                                     Finite Simple Group in the Primitive Positive Constructability Poset

Primitive positive constructions between structures are motivated by theoretical computer science as they give logspace reductions between the associated constraint satisfaction problems. They define a partial order on equivalence classes of finite structures. We show that below two trivial layers, this poset has an infinite third layer, given by the equivalence class of a tournament graph and of structures in one-to-one correspondence with all finite simple groups.

Contact: Prof. Dr. Manuel Bodirsky

Colin Jahel (TU Dresden)                                                                 Unitary representations of isometry groups of Urysohn spaces

Unitary representation of groups are an essential part of abstract harmonic analysis and are classically studied for locally compact groups. Recently, the study of unitary representations has been extended to non-Archimedian Polish groups, notably through the work of Olshanski, Evans-Tsankov and Ibarlucia. The work I will present is in the continuation of those recent works, extending the classification of unitary representations to non-oligomorphic groups (isometry groups of Urysohn spaces). I intend to give an exposition on all involved objects, explain the many consequences of our result and present the main elements of proof. This is joint work with Remi Barritault and Matthieu Joseph.

Contact: Prof. Dr. Manuel Bodirsky

Kelly Maggs (Lausanne)
Cohomology classes in the RNA transcriptome

Single-cell sequencing data consists of a point cloud where the points are cells, with coordinates RNA expression levels in each gene. Since the tissue is destroyed by the sequencing procedure, the dynamics of gene expression must be inferred from the structure and geometry of the point cloud. In this talk, we will build a biological interpretation of the one-dimensional cohomology classes in hallmark gene subsets as models for transient biological processes. Such processes include the cell-cycle, but more generally model homeostatic negative feedback loops. Our procedure uses persistent cohomology to identify features, and integration of differential forms to estimate the cascade of genes associated with the underlying dynamics of gene expression.

Contact: Prof. Dr. A. Thom

Dr. Vincenzo Galgano (MPI-CBG)
Secant varieties to some generalised Grassmannians

Secant varieties are among the main protagonists in tensor decomposition, whose study involves both pure and applied mathematical areas. Any generalised Grassmannian G/P is a set of rank-1 "tensors" in the corresponding (minimal) irreducible representation, like Grassmannians for skew-symmetric tensors. The geometry of secant varieties to G/P is in general not completely understood. In this talk we discuss the 2nd secant variety to a generalised Grassmannian, and we give results on the identifiability and singularity in the case of Grassmannians. Some results are from a joint work with Reynaldo Staffolani

Contact: Jun.-Prof. Dr. Mario Kummer

Dr. Tariq Syed
Motivic cohomology of cyclic coverings

The generalized Serre question asks whether algebraic vector bundles over topologically contractible smooth affine complex varieties are always trivial or not. Many examples of topologically contractible smooth affine complex varieties are given by so-called cyclic coverings. In this talk, we present new results on the motivic cohomology of such cyclic coverings which might indicate a positive answer to the generalized Serre question in low dimensions

Contact: Prof. Dr. A. Thom

Prof. Dr. Andreas Thom (TU Dresden)
On non-isomorphic universal sofic groups.

We show that there are 2^{ℵ_0} non-isomorphic universal sofic groups. This proves a conjecture of Simon Thomas. This is joint work with Vadim Alekseev.

Contact: Prof. Dr. A. Thom

Dr. Aljaž Zalar (University of Ljubljana)
A gap between positive polynomials and sums of squares in various settings

In the talk we will present quantitative estimates for the gap between positive polynomials and sums of squares polynomials in three different vector spaces: 1) the space of biquadratic biforms, 2) the space of biquadratic biforms modulo the ideal of all orthonormal 2-frames, 3) the space of even quartic forms.The motivation to study the gap in 1) comes from quantum information theory, in 2) from financial mathematics, and in 3) from matrix theory. In 1) the gap corresponds to the gap between positive maps and completely positive maps between matrix algebras, in 2) to the gap between cross-positive maps and completely cross-positive maps between matrix algebras, and in 3) to the gap between copositive matrices and matrices that are a sum of a positive semidefinite matrix and entry-wise nonnegative one. This is joint work with Igor Klep, Scott McCullough, Klemen Šivic and Tea Štrekelj.

Contact: Jun.-Prof. Dr. Mario Kummer

Prof. Dr. Amador Martin-Pizarro (Albert-Ludwigs-Universität Freiburg)
Corners, Squares and Stability

Given an abelian group G, a corner is a a subset of pairs of the form (x,y),(x+d,y),(x,y+d) with d≠0 non trivial. Ajtai and Szemerédi proved that, asymptotically, every dense subset S of Z/NZ×Z/NZ contains a corner. Shkredov gave a quantitative lower bound on the density of the subset S for particular finite abelian groups. In this talk, we will explain how model-theoretic conditions on the subset S, such as stability, imply the existence of corners and of other configurations for (pseudo)-finite abelian groups. No prior knowledge of mathematical logic is necessary for this talk.

Contact: Prof. Dr. A. Fehm

Dr. Fabian Ruehle (Northeastern University)
Machine Learning for Knot Theory
While extremely powerful, the main problem in applying tools from machine learning to problems in mathematics is that the techniques are stochastic in nature and notoriously difficult to interpret. We discuss how problems from discrete mathematics can be tackled with reinforcement learning (RL), a subfield of machine learning. We illustrate this technique by applying RL to detect whether a given knot is the unknot, and whether a given knot is ribbon. The latter was used to rule out many proposed potential counterexamples to the smooth Poincare Conjecture in 4D. The talk is based on work with Gukov, Halverson, Manolescu, and Sulkowski.

Contact: Prof. Dr. A. Thom

Dr. Philip Ditmann (TU Dresden)
Definable sets in fields
I will give an introduction to some fundamental questions and results in the study of definable sets (in the sense of mathematical logic) in field theory. Such study is motivated classically by connections to geometry in various tame settings (such as over algebraically closed fields, the reals and p-adics), and on the other hand by undecidability results (in particular surrounding Hilbert's 10th Problem on the solvability of diophantine equations) with connection to Gödelian phenomena. In both tame and wild settings, I will hopefully be able to give a glimpse both of classical results and new achievements of the last decade. No background from mathematical logic will be required.

Contact: Prof. Dr. M. Bodirsky

Dr. Türkü Özlüm Celik (Koc University)
Theta Functions, Algebraic Curves and Water Waves
Theta functions play a central role in various branches of mathematics. In this talk, we will showcase their significance in algebraic geometry, particularly in studying complex algebraic curves with applications in the soliton equations which model shallow water waves. We will survey how deformations of the underlying objects lead to lots of beautiful combinatorics. We will explore how these applications can address the Schottky problem, a classical geometry problem, thereby spotting a mutually enriching relationship. An emphasis will be on exploiting current trends in computational algebraic geometry to advance our understanding of these functions, encompassing both symbolic and numerical computational approaches.

Contact: Prof. Dr. A. Thom

Dr. Matthieu Joseph (Laboratoire Mathématique d’Orsay)
From exchangeability theory to stabilizer rigidity of ergodic actions for Polish non-archimedean groups
Exchangeability theory is a branch of probability that studies stochastic processes subject to various symmetries. I will explain how the crucial notion of dissociation in exchangeability theory can be used in a very different context: that of stabilizer rigidity for probability measure-preserving ergodic actions of Polish non-archimedean groups. The main result that we obtain is that for a large class of oligomorphic groups, any probability measure-preserving ergodic action is either free or transitive (up to null sets). This is a joint work with Colin Jahel.

Contact: Prof. Dr. M. Bodirsky

Colin Jahel
Dynamics of non-Archimedian (oligomorphic) groups
The study of group dynamics, i.e. the study of their continuous or measured actions, is classically focused on locally compact groups, such as free groups or special groups SL_n(R). My goal in this talk is to present a class of non-locally compact groups with interesting dynamical properties: non-Archimedian groups, i.e. closed subgroups of the permutation group of N. I will explore topics of amenability, rigidity of pmp actions and invariant random subgroups, in order to display the interesting behaviors we encounter in the realm of non-Archimedian groups.

Contact: Prof. Dr. M. Bodirsky

Thursday,
25.01.2024
WIL A 124

Andy Zucker (University of Waterloo)
Ultracoproducts and weak containment for flows of topological groups
We develop the theory of ultracoproducts and weak containment for flows of arbitrary topological groups. This provides a nice complement to corresponding theories for p.m.p. actions and unitary representations of locally compact groups. For the class of locally Roelcke precompact groups, the theory is especially rich, allowing us to define for certain families of G-flows a suitable compact space of weak types. When G is locally compact, all G-flows belong to one such family, yielding a single compact space describing all weak types of G-flows.
Contact: Prof. Dr. M. Bodirsky

Thursday,
18.01.2024    13.30 - 14.30  WIL A 124

Winfried Hochstättler (FernUniversität in Hagen)
A generalization of Hadwiger's Conjecture to Regular
and Orientable Matroids
Let G=(V,E) be a graph with a proper k-coloring c : V -> {1,...,k}. Choosing an abitrary orientation D of G and interpreting c as a potential yields a tension with values in {-k+1,-k+2,...,-1,1,2,...,k-1} which we call a nowhere-zero-k-tension. Up to a circular permutation of the colors these are in bijection with proper k-colorings.
Dualizing this approach Tutte introduced nowhere-zero-k-flows. Taking Hadwiger's conjecture - that any graph G without a K_k-minor is k-1-colorable- as a model, he conjectured that any bridgeless graph without a Petersen-minor has a nowhere-zero-4-flow and any arbitrary bridgeless graph has a nowhere zero-5-flow.
The natural class of matroids which is closed under duality and contains graphs are the regular matroids, those matroids which are representable by a totally unimodular matrix. With Goddyn we proved, that a generalization of Hadwiger's conjecture to regular matroids is equivalent to

Tutte's 4-flow conjecture if k=5
Tutte's 5 flow conjecture if k=6
Hadwiger's Conjecture if k>=7.

Contact: Prof. Dr. M. Bodirsky

Lukasz Grabowski (Leipzig)
Analogues of expanders and hyperfinite graph sequences for
directed graphs
This is a report on a joint work with Endre Csoka (available on arxiv, and in "Combinatorics, Probability and Computing"), where we construct "extender" graphs - sparse directed graphs with the property that it is impossible to remove a small amount of edges and obtain a graph which does not have long directed paths (long is |V|^delta, where delta>0 is an absolute constant). Extenders are a directed analogue of expander graphs, which I will explain in the talks, Similarly, we study some basic properties of hypershallow graph sequences  which are a directed analogue of hyperfinite graph sequence, I'll discuss some conjectures in circuit complexity  which motivated us to introduce extenders and hypershallow graphs.

Contact: Prof. Dr. A. Thom

Thursday,
04.01.2024
13.30 – 14.30  WIL A 124

Matěj Konečný
Extending partial automorphisms
The infinite symmetric group Sym(N) can be viewed as a subspace of N^N, thereby becoming a topological group with the pointwise-convergence topology. It turns out that G is a closed subgroup of Sym(N) if and only if it is an automorphism group of a countable relational structure. In fact, one can pick the structure to be homogeneous (i.e. every isomorphism between finite substructures extends to an automorphism of the whole structure).

In turn, Fraisse theory tells us that homogeneous structures correspond to classes of finite structures with the so-called amalgamation property. It is thus not that surprising that various dynamical properties of closed subgroups of Sym(N) are equivalent to combinatorial properties of the corresponding classes of finite structures.

In this talk, I will outline an example of this: A class C of finite structures has the extension property for partial automorphisms (EPPA, also called the Hrushovski property) for every A in the class there is B in the class containing A as a substructure such that every partial automorphisms of A extends to an automorphism of B. This has, for example, been a key ingredient in the proof of the small index property for the automorphism group of the random graph by Hodges, Hodkinson, Lascar, and Shelah.

Contact: Prof. Dr. M. Bodirsky

Thursday,
14.12.2023
13.30 – 14.30 WIL A 124

Jan Hubicka (Charles University)
Recent progress on big Ramsey degrees
We say that a countably-infinite relational structure M has finite big Ramsey degrees if for every finite substructure A of M there exists a finite integer TM(A) (the big Ramsey degree of A in M) such that for every finite colouring of copies of A in M, there exists a copy M’ of M in M such that all copies of A in M’ have at most TM(A) many different colours.  This is an infinitary extension of the concept of Ramsey classes.  In this talk, we introduce the main notions and discuss recent progress on big Ramsey degrees of structures with forbidden substructures.

Contact: Prof. Dr. M. Bodirsky

Thursday,
07.12.2023
13.30 – 14.30 WIL A 124

Édouard Bonnet (ÉNS Lyon)
Introduction to twin-width
Twin-width is a new graph parameter whose boundedness generalizes classes of bounded clique-width, classes excluding a fixed minor, and proper permutation classes.
We will survey some results and open questions on twin-width related to algorithms, finite model theory, permutations, and groups.
Contact: Prof. Dr. M. Bodirsky

Hidden symmetries in Hecke algebra and stochastic applications
Many interacting particle systems which are of interest in mathematical physics can be viewed as random walks on Hecke algebras. It was recently realised that certain hidden algebraic symmetries are crucial for the asymptotic analysis of these systems. In this talk I will give an introduction into this circle of ideas, with an emphasis on one particular example.
Contact: Prof. Dr. A. Thom

Thursday,
16.11.2023
13.30 – 14.30
WIL A 124
/ link

In this talk we introduce the audience to the basics of  deformation quantization theory of Operator Algebras as developed  originally by Mark Rieffel in the 90’s in the setting of C*-algebras  carrying an action of R^n. For locally compact abelian groups this was  later generalized by Kasprzak using Landstad duality for crossed  products. For non-abelian groups one can replace actions by coactions of  groups and develop a similar procedure for deforming C*-algebras via  2-cocycles on the group. Some difficulties appear if the cocycle is not  continuous. We propose a deformation process that works for every Borel  2-cocycle and extends previous work by Bhowmick, Neshveyev, and Sangha.

This is based on joint work with Siegfried Echterhoff.

Daniel Plaumann
Hyperbolicity Cones and Duality
Hyperbolic polynomials generalize determinants of pencils of real symmetric matrices. We may think of their study as "eigenvalue theory without matrices". The hyperbolicity cone of such a polynomial is the analogue of the cone of positive definite matrices (in a pencil) and has been studied in particular in real algebraic geometry and convex optimization. Many good properties and constructions carry over from the matrix case. However, a big exception is convex duality, which is not well understood in the polynomial setting. In this talk, I will explain the context of this problem and present some partial results and examples.
Contact: Jun.-Prof. Dr. M. Kummer

Kıvanç Ersoy
Centralizer depth in locally finite groups
Let $G$ be an infinite group. Let $r$ be the smallest number of generators of a finitely generated subgroup $F$ of $G$ such that the centralizer of $F$ is also finitely generated. If there is no such finitely generated subgroup $F$ of $G$ with finitely generated centralizer, say $r$ is infinite.  We call this number  $r$ the centralizer depth of $G$. There are two classical open problems related to simple locally finite groups, asked by Brian Hartley.
Problem 1. Does there exist a non-linear simple locally finite group with a linear centralizer?
Problem 2. Does there exist a non-linear simple locally finite group with a finite subgroup whose centralizer is finite.
In this talk, we first talk about simple locally finite groups and their structure. Then we will investigate how this concept of centralizer depth transforms these classical problems, suggesting a number measuring linearity. In particular, we will prove some results suggesting the following conjecture could be true.
Conjecture. A simple locally finite group is linear if and only if the centralizer depth is $2$. A simple locally finite group is non-linear if and only if the centralizer depth is infinite.

Simon Telen
Chebyshev varieties
Chebyshev varieties are algebraic varieties parametrized by Chebyshev polynomials. They arise naturally when solving polynomial equations expressed in the Chebyshev basis. More precisely, when passing from monomials to Chebyshev polynomials, Chebyshev varieties replace toric varieties. I will introduce these objects, discuss their de􏰂fining equations and present key properties. Via examples, I will motivate their use in practical computations. This is joint work with Zaïneb Bel-Afia and Chiara Meroni.
Contact: Jun.-Prof. Dr. M. Kummer

Cyclic objects from string links and surfaces
A (co)cyclic object in a category is, roughly speaking, a (co)simplicial object equipped with compatible actions of cyclic groups. In the 80's, Connes defined this notion while introducing cyclic cohomology for algebras. In this talk, we illustrate two constructions of cyclic objects which arise in low-dimensional topology and discuss how these relate to 3-dimensional TQFTs. More precisely, we first equip the set of string links (which contains pure braids) with a structure of a (co)cyclic set. Next, we equip the family of closed oriented surfaces, starting with torus, with a structure of a (co)cyclic object in the category of 3-dimensional cobordisms. As a corollary, every 3-dimensional TQFT gives rise to a (co)cyclic object in vector spaces, which we compute algebraically for the Reshetikhin–Turaev TQFT. In the course of the talk, I will give some historical motivations and examples. No prerequisite in category theory or quantum topology is necessary.

Santiago Guzmán Pro (TU Dresden)
Structural Graph Theory, finite bounds, and some CSPs.
A common technique to characterize hereditary graph classes is to exhibit their minimal obstructions. Sometimes, the set of minimal obstructions might be infinite, or too complicated to describe. For instance, the minimal obstructions for chordal graphs are all cycles of length at least 4, and for any k ≥ 3, the set of minimal obstructions of the class of k-colourable graphs is yet unknown. Nonetheless, the Roy-Gallai-Hasse-Vitaver Theorem asserts that a graph G is k-colourable if and only if it admits and orientation with no directed walk on k + 1 vertices. Similarly, it is well-known that a graph G is a chordal graph if and only if its vertex set admits a linear ordering ≤ such that (G, ≤) avoids ({1 ≤ 2 ≤ 3}, {12, 13}) as an induced structure. In this talk we explore characterizations of graph classes that follow the previous template, i.e., expressions of graphs classes in terms of reducts of finitely bounded classes. The driving meta-problem of this project is attempting to understand which graph classes can be expressed by such expression systems. We will provide several examples of these characterizations, and present some limitations of these expression systems. We conclude by noticing that some open problems in this subject correspond to complexity classification problems for certain infinite CSPs. Moreover, these turn out to be particular open instances of the Tractability Conjecture.

Thursday,
22.06.2023
13.30 – 14.30
Z21/242
link

Sebastian Mentemeier (Universität Hildesheim)
Random Walks on Matrix (Semi-)-Groups

Contact: Prof. Dr. A. Behme

Thursday,
06.04.2023
13.30 – 14.30
Z21/250
link

Inna Capdebosq (University of Warwick)
Few things about Kac-Moody groups
In this talk we will gently introduce Kac-Moody groups over finite fields (both incomplete and topological ones) and talk about some
properties of the groups.

Thursday,
15.12.2022
13.30 - 14.30
WIL A 124
link

Paul Wedrich (Universität Hamburg)
A Kirby color for Khovanov homology
The Jones polynomial is a famous knot invariant that has been categorified by Khovanov to a knot homology theory. Jones polynomials of knots can be computed relatively straightforwardly using the Temperley-Lieb algebras, which admit a diagrammatic presentation. Surprisingly, closely related diagrammatic algebras, the dotted Temperley-Lieb algebras (a.k.a. nil-blob algebras) appear when extending Khovanov homology to an invariant of smooth 4-dimensional manifolds. I will define these algebras, assemble them into a monoidal category, and then construct a special ind-object that we call the Kirby color for Khovanov homology. This object, discovered in joint work with Hogancamp and Rose, allows a concrete description of a certain 2-handle gluing rule for the invariants of smooth 4-manifold constructed in joint work with Morrison and Walker.

Thursday,
10.11.2022
13.30 - 14.30
WIL A 124
link

Pascal Schweitzer (TU Darmstadt)
Symmetry in discrete structures: graphs, groups, and algorithms
The concept of symmetry is ubiquitous in the study of discrete combinatorial objects. The ensemble of symmetries of an object G is its automorphism group Aut(G). It provides us with structural information of the object. The group also captures how its symmetries relate to one another. While symmetries are interesting to study in their own right, symmetries also have direct applications in diverse areas that make use of combinatorial objects.
In this talk, I will describe various facets of research surrounding the structure, detection and application of symmetries. For example I will describe how and when groups can be represented as automorphisms of graphs. I will also discuss asymmetry and aspects of the graph isomorphism problem, which lies at the heart of the algorithmic task to find symmetries.

Thursday,
13.10.2022
13.30 - 14.30
WIL A 124
link

Claus Scheiderer
Representing spectrahedral shadows using small matrix sizes
A spectrahedron is the solution set of a linear matrix inequality (LMI), or equivalently, an affine-linear section of the cone of positive semidefinite matrices of some size. Semidefinite programming allows to efficiently optimize linear functions over spectrahedra, or more generally over spectrahedral shadows, which by definition are the linear images of spectrahedra. Large families of convex semialgebraic sets are known to be spectrahedral shadows, but there exist prominent examples that fail to be. Given a spectrahedral shadow $K$, one would like to present $K$ using LMIs of matrix size as small as possible, for reasons of computational performance. We plan to present some recent results on what is possible in this respect.

Mario Kummer
Spectrahedral Shadows and Completely Positive Maps on Real Closed Fields
Nemirovski asked in his plenary talk at ICM 2006 whether every convex semialgebraic set is a spectrahedral shadow. Recently, Scheiderer answered this question to the negative by exhibiting first counter examples. In this talk I will present a joint work with Manuel Bodirsky and Andreas Thom in which we answer this question for certain sets that resisted the efforts of Scheiderer. Our approach relies on a simple model theoretic observation and differs from Scheiderer's methods substantially.

Rafael Oliveira
Radical Sylvester-Gallai theorem for cubics - and beyond In 1893, Sylvester asked a basic question in combinatorial geometry: given a finite set of distinct points $v_1, \ldots, v_m \in \R^N$ such that the line defined by any pair of distinct points $v_i, v_j$ contains a third point $v_k$ in the set, must all points in the set be collinear? Generalizations of Sylvester's problem, which are known as Sylvester-Gallai type problems, have found applications in algebraic complexity theory (in Polynomial Identity Testing - PIT) and coding theory (Locally Correctable Codes). The underlying theme in all these types of questions is the following: Are Sylvester-Gallai type configurations always low-dimensional? In 2014, Gupta, motivated by such applications in algebraic complexity theory, proposed wide-ranging non-linear generalizations of Sylvester's question, with applications on the PIT problem. In this talk, we will discuss these non-linear generalizations of Sylvester's conjecture, their intrinsic relation to algebraic computation, and a recent theorem proving that radical Sylvester-Gallai configurations for cubic polynomials must have small dimension. Joint work with Akash Kumar Sengupta

Thomas Nikolaus
K-Theory and arithmetic cohomology theories
Algebraic K-theory is an invariant of rings. It has several relations to questions in geometric topology, homotopy theory, algebraic geometry and number theory. Recently there have been advances how to understand and compute algebraic K-theory using arithmetic cohomology theories, specifically using the newly developed prismatic cohomology of Bhatt--Scholze. We will describe sample results, how the relation philosophically works and how it fostered important advances in p-adic cohomology (notably prismatic cohomology).

Thursday, 19.05.2022
13.30 - 14.30
WIL A 124 https://tu-dresden.zoom.us/my/mariokummer

Alexander Mang
Quantizing groups through combinatorics
A major source of examples of compact quantum groups is provided by a particular kind of categorial combinatorics. Partitions of finite sets, possibly equipped with some coloring, or finite graphs have been used to "liberate" the orthogonal, unitary or symmetric groups. Following an introduction to Woronowicz's quantum groups, I will demonstrate how all the aforementioned examples can be explained as special cases of subcategories of non-standard binary relations. I shall then focus on the particular case of colored partitions, report on the progress of their classification program, and illustrate how their combinatorics can be utilized to establish algebraic and analytic properties of the discovered quantum groups. In part, this is joint work with Johannes Flake and Moritz Weber.

Thursday, 12.05.2022
13.30 - 14.30
WIL A 124 Link: https://tu-dresden.zoom.us/my/mariokummer

Kevin Wolf
Representation theory of deformed Nichols algebras We study some methods to do representation theory on deformed Nichols algebras. We will do so on the examples of Nichols algebras known as the Fomin-Kirillov algebras. We will point out, that these algebras and their deformations all have multiple subalgebras isomorphic to Hecke algebras. We then induce representations of these Hecke algebras to representations of larger subalgebras. We find that with the exception of a few special parameter choices, these subalgebras, and therefore their module categories, are semisimple. We will also discuss the open problems of using these techniques.

Thursday, 21.04.2022
13.30 - 14.30
WIL A 124

Emily Roff
Magnitude homology, length functions, and holes in the plane
Magnitude homology is a homology theory for enriched categories, first defined in 2015 by Hepworth and Willerton and later extended substantially by Leinster and Shulman. Specialized to ordinary categories, magnitude homology recovers the ordinary homology of the classifying space; specialized to metric spaces, it yields something new: an (N x R)-graded abelian group whose first few R-graded pieces carry subtle information about convexity.

This talk will introduce the theory and discuss recent developments, without assuming detailed knowledge of category theory or homological algebra. Leinster and Shulman’s construction suggests a means to build a ‘magnitude nerve’ for any group G equipped with a conjugation-invariant length function L, and this will be our central example. The magnitude homology of (G,L) - the homotopy of its magnitude nerve - captures information about the spectrum of L as well as the traditional group homology of G. To illuminate the picture, we’ll construct a variant on the fundamental group of a subset of the plane and analyse its magnitude homology.

Ulrich Krähmer (TU Dresden)
Cyclic duality for slice and orbit 2-categories
Assuming that everyone has at least heard of the simplicial category and is aware of its relevance in algebra and topology, I will explain how it can be extended to the duplicial and the closely related paracyclic category and why I and a few others care about these. The main topic will then be one possible answer to the question why these more exotic index categories are self-dual while the simplicial category is not.

What is... a lattice in a Lie group?
Lattices in Lie groups are prominent and very important examples of discrete groups whose understanding reveals surprising connections between geometric group theory, differential geometry, number theory, functional analysis and ergodic theory. In this talk I will try to explain how these interactions work and what open questions around lattices in Lie groups connecting all these are still pertaining.

Cones of locally non-negative polynomials
An important motivation from optimization for the study of non-negative polynomials is the obvious fact that global minima of a real polynomial f take the maximal value c such that f-c is non-negative. Similarly, a local minimum x_0 of f induces the polynomial f-f(x_0) which takes value 0 and is locally non-negative at x_0.
I will present results of my PhD thesis on the convex cone of locally non-negative polynomials. We will see geometric interpretations and examples of faces of this cone, some general theory of cones in infinite-dimensional vector spaces and classifications of faces using tools from singularity theory. I will also give a short outlook on an application to sums of squares in real formal power series rings.

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.

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.
Now in quantum theory, the notion of a probability measure is replaced by a so-called positive operator valued measure (POVM), a set of positive semidefinite matrices that sums to the identity matrix. So a quantum magic square is a matrix that contains a POVM in each row and column. There also exists the well-studied notion of a quantum permutation matrix, which is a quantum magic square whose entries are all idempotent (i.e. orthogonal projections). So a quantum version of the Birkhoff-von Neumann Theorem would ask whether each quantum magic square is a convex combination of quantum permutation matrices.
Taking classical convex combinations does not suffice here, but there exists a much more suitable notion, that of a free convex hull. We prove that the result is true when restricted to special quantum magic squares, which we call semi-classical. But we also prove that the general version of the Birkhoff-von Neumann Theorem fails even under the very general notion of free convexity.