John Bourke: Accessible ∞-cosmoi

10th December, 2020

Riehl and Verity introduced ∞-cosmoi - certain simplicially enriched categories - as a framework in which to give a model-independent approach to ∞-categories. For instance, there is an infinity cosmos of ∞-categories with finite limits or colimits, or of cartesian fibrations. In this talk, I will introduce the notion
of an accessible ∞-cosmos and explain that most, if not all, ∞-cosmoi arising in practice are accessible. Applying results of earlier work, it follows that accessible ∞-cosmoi have homotopy weighted colimits and admit a broadly applicable homotopical adjoint functor theorem.

Colin Reid: Abelian chief factors of locally compact groups

9th December, 2020

Recent work in the theory of locally compact second-countable (l.c.s.c.) groups has highlighted the importance of chief factors, meaning pairs of closed normal subgroups K/L such that no closed normal subgroups lie strictly between K and L. In particular, the group K/L is then topologically characteristically simple, meaning it has no proper nontrivial closed subgroup that is preserved by all automorphisms. I will present a classification of the abelian l.c.s.c. topologically characteristically simple groups: these all occur as chief factors of soluble groups, and naturally fall into five families with a few parameters. Each family has a straightforward characterization within the class of abelian l.c.s.c. groups, without directly invoking the property of being topologically characteristically simple.

Stephan Tornier: Think globally, act locally

9th December, 2020

Let G be a group acting on a regular tree. The 'local' actions that vertex stabilisers in G induce on balls around the fixed vertex are innately connected to the 'global' structure of G. I demonstrate this relationship and define a particularly accessible class of groups acting on (locally finite) regular trees by 'prescribing' said local actions, following Burger-Mozes. Being defined solely in terms of finite permutation groups, these groups allow us to introduce computational methods to the world of locally compact groups: I will outline the capabilities of a recently developed GAP package that provides methods to create, analyse and find suitable local actions.

Nicolas Jacon: Cores of Ariki-Koike algebras

8th December, 2020

We study a natural generalization of the notion of cores for l-partitions: the (e,s)-cores. We relate this notion with the notion of weight as defined by Fayers and use it to describe the blocks of Ariki-Koike algebras.

Laura Rider: Modular Perverse Sheaves on the Affine Flag Variety

7th December, 2020

There are two categorical realizations of the affine Hecke algebra: constructible sheaves on the affine flag variety and coherent sheaves on the Langlands dual Steinberg variety. A fundamental problem in geometric representation theory is to relate these two categories by a category equivalence. This was achieved by Bezrukavnikov in characteristic 0 about a decade ago. In this talk, I will discuss a first step toward solving this problem in the modular case joint with R. Bezrukavnikov and S. Riche.

Daniel Nakano: A new Lie theory for classical Lie superalgebras

4th December, 2020

In 1977, Kac classified simple Lie superalgebras over ℂ and showed they play an analogous role to simple Lie algebras over the complex numbers. For simple algebraic groups and their Lie algebras, the notions of a maximal torus, Borel subgroups and the Weyl groups provide a uniform method to treat the structure and representation theory for these groups and Lie algebras. Historically, much of the work for simple Lie superalgebras has involved dealing with these objects using a case by case analysis. Fifteen years ago, Boe, Kujawa and the speaker introduced the concept of detecting subalgebras for classical Lie superalgebras. These algebras were constructed by using ideas from geometric invariant theory. More recently, D. Grantcharov, N. Grantcharov, Wu and the speaker introduced the concept of a BBW parabolic subalgebra.

Given a Lie superalgebra 𝔤, one has a triangular decomposition 𝔤=𝔫– ⨁ 𝔣 ⨁ 𝔫+ with 𝔟 = 𝔣 ⨁ 𝔫– where 𝔣 is a detecting subalgebra and 𝔟 is a BBW parabolic subalgebra. This holds for all classical ‘simple’ Lie superalgebras, and one can view 𝔣 as an analogue of the maximal torus, and 𝔟 like a Borel subalgebra. This setting also provide a useful method to define semisimple elements and nilpotent elements, and to compute various sheaf cohomology groups R ∙ indBG (-). The goal of my talk is to provide a survey of the main ideas of this new theory and to give indications of the interconnections within the various parts of this topic. I will also indicate how this treatment can further unify the study of the representation theory of classical Lie superalgebras.

This video was produced by the Universidade de São Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.

Ualbai Umirbaev: Automorphism groups of free algebras

4th December, 2020

There are many interesting results on the structure of the automorphism group Aut(Fn) and the outer automorphism group Out(Fn) of the free group Fn of rank n. Unfortunately, the theory of automorphism groups of free algebras over a field is not very rich and many problems are still open. I will describe some results and recall some open questions on the structures of the automorphism groups of:

   1.  the polynomial algebra K[x1,x2,…,xn] of rank n over a field K;
   2.  the free associative algebra K<x1,x2,...,xn> of rank n over K; and
   3.  the free Lie algebra Lie<x1,x2,...,xn> of rank n over K.

Eugene Plotkin: On logical rigidity of groups

4th December, 2020

We will survey a series of recent developments in the area of first-order descriptions of linear groups. The goal is to illuminate the known results and to pose the new problems relevant to logical characterizations of Chevalley groups and Kac-Moody groups. We also dwell on the principal problem of isotipicity of finitely generated groups.

Henry Wilton: Negative immersions and one-relator groups

4th December, 2020

One-relator groups G=F/≪w≫ pose a challenge to geometric group theorists. On the one hand, they satisfy strong algebraic constraints (eg Magnus's theorem that the word problem is soluble). On the other hand, they are not susceptible to geometric techniques, since some of them (such as Baumslag-Solitar groups) exhibit extremely pathological behaviour.

I will relate the subgroup structure of one-relator groups to a measure of complexity for the relator w introduced by Puder - the primitivity rank π(w), the smallest rank of a subgroup of F containing w as an imprimitive element. A sample application is that every subgroup of G of rank less than π(w) is free. These results in turn provoke geometric conjectures that suggest a beginning of a geometric theory of one-relator groups.

Agatha Atkarskaya: Small cancellation rings

4th December, 2020

The theory of small cancellation groups is well known. In this paper we introduce the notion of Group-like Small Cancellation Ring. This is the main result of the paper. We define this ring axiomatically, by generators and defining relations. The relations must satisfy three types of axioms. The major one among them is called the Small Cancellation Axiom. We show that the obtained ring is non-trivial. Moreover, we show that this ring enjoys a global filtration that agrees with relations, find a basis of the ring as a vector space and establish the corresponding structure theorems. It turns out that the defined ring possesses a kind of Gröbner basis and a greedy algorithm. Finally, this ring can be used as a first step towards the iterated small cancellation theory which hopefully plays a similar role in constructing examples of rings with exotic properties as small cancellation groups do in group theory. Joint results with A. Kanel-Belov, E. Plotkin, E. Rips.

Anna Erschler: Ordering Ratio Function and Travelling Salesman Breakpoint for Groups and Metric Spaces

4th December, 2020

(Abstract taken from arXiv paper.) We study asymptotic invariants of metric spaces and infinite groups related to the universal Travelling Salesman Problem (TSP). We prove that spaces with doubling property (in particular virtually nilpotent groups) admit Gap for Ordering Ratio functions which holds for all orders on these spaces. We describe Travelling Salesman Breakpoint for finite graphs. We characterize groups with Travelling Salesman Breakpoint ≤ 3 as virtually free ones. We show that Ordering Ratio function is bounded (which is the best possible situation) for all uniformly discrete δ-hyperbolic spaces of bounded geometry, in particular for all hyperbolic groups. We prove that any metric space, containing weakly a sequence of arbitrarily large cubes, has infinite Travelling Salesman Breakpoint; this means that any order on such spaces satisfies OR(s)=s for all s. This is the worst possible case for Ordering Ratio functions. For a sequence of finite graphs, we provide a sufficient spectral condition for OR(s)=s. This condition is in particular satisfied for any sequence of expander graphs. Under this stronger assumption of being a family of expander graphs, we prove a stronger claim about snakes of bounded width. We show that any metric space of finite Assouad-Nagata dimension admits an order satisfying OR(s) ≤ Const ln s, and discuss general Gap Problems for Ordering Ratio functions.

Jingwei Xiao: A Unitary Analogue of Friedberg-Jacquet Periods and Central Values of Standard L Functions on GL(2n)

3rd December, 2020

Let G be a reductive group over a number field F and H a subgroup. Automorphic periods study the integrals of cuspidal automorphic forms on G over H(F)\H(AF). They are often related to special values of certain L-functions. One of the most notable cases is when (G,H)=(U(n+1)☓U(n), U(n)), and these periods are related to central values of Rankin-Selberg L-functions on GL(n+1)☓GL(n). In this talk, I will explain my work in progress with Wei Zhang that studies central values of standard L-functions on GL(2n) using (G,H)=(U(2n), U(n)☓U(n)) and some variants. I shall explain the conjecture and a relative trace formula approach to study it. We prove the required fundamental lemma using a limit of the Jacquet-Rallis fundamental lemma and Hironaka’s characterization of spherical functions on the space of non-degenerate Hermitian matrices. Also, the question admits an arithmetic analogue.

Asilata Bapat: A Thurston compactification of Bridgeland stability space

30th November, 2020

The space of Bridgeland stability conditions on a triangulated category is a complex manifold. We propose a compactification of the stability space via a continuous map to an infinite projective space. Under suitable conditions, we conjecture that the compactification is a real manifold with boundary, on which the action of the autoequivalence group of the category extends continuously. We focus on 2-Calabi-Yau categories associated to quivers, and prove our conjectures in the A2 and affine A1 cases.

Brandon Hanson: Higher convexity and iterated sumsets

30th November, 2020

Erdős and Szemerédi made the (still open) conjecture that for a finite set of natural numbers, A, either the sumset A+A, or else the productset AA, must be nearly as large as possible. A slightly different interpretation is that either A+A is large or log(A)+log(A) is large, where log(A) is the image of A under the (convex) logarithm function. This phenomenon is in fact more general, and extends to arbitrary convex functions f: if f has non-vanishing second derivative, then either A+A or else f(A)+f(A) is large. In recent work with Roche-Newton and Rudnev, we show that this growth persists when f has further non-vanishing derivatives.

Uriel Sinichkin: Enumeration of algebraic and tropical singular hypersurfaces

29th November, 2020

It is classically known that there exist (n+1)(d-1)n singular hypersurfaces of degree d in complex projective n-space passing through a prescribed set of points (of the correct size). In this talk we will deal with the analogous problem over the real numbers and construct, using tropical geometry, Ω(dn) real singular hypersurfaces through a collection of points in ℝℙn. We will also consider the enumeration of hypersurfaces with more than one singular point. No prior knowledge of tropical geometry will be assumed.

Eli Aljadeff: Generic Azumaya G-graded algebras

26th November, 2020

The algebra of generic nxn-matrices and its localizations (e.g. the generic division algebra) has attracted much attention among researchers in different areas as PI theory, Brauer theory and algebraic geometry. We construct the corresponding generic objects for an arbitrary finite dimensional G-graded simple algebra where G is a finite group. In particular we construct a generic G-graded Azumaya algebra which represents all forms in the sense of descent theory of a finite dimensional G-graded simple algebra.

Martin Bidlingmaier: Model categories of lcc categories and the gros model of dependent type theory.

26th November, 2020

In this talk we discuss various model categories of locally cartesian closed (lcc) categories and their relevance to coherence problems, in particular the coherence problem of categorical semantics of dependent type theory. We begin with Lcc, the model category of locally cartesian closed (lcc) sketches. Its fibrant objects are precisely the lcc categories, though without assigned choices of universal objects. We then obtain a Quillen equivalent model category sLcc of strict lcc categories as the category of algebraically fibrant objects of Lcc. Strict lcc categories are categories with assigned choice of lcc structure, and their morphisms preserve these choices on the nose. Conjecturally, sLcc is precisely Lack’s model category of algebras for a 2-monad T , where T is instantiated with the free lcc category functor on Cat. We then discuss the category of algebraically cofibrant objects of sLcc and show how it can serve as a "gros" model of dependent type theory.