Michael Lieberman: Induced stable independence, with applications

25th February, 2021

A stable independence relation on a category (a generalization of the model-theoretic notion of nonforking independence!) consists of a very special family of commutative squares, whose members have almost all the desirable properties of pushouts—this is exceedingly useful in categories in which pushouts do not exist. We describe conditions under which a stable independence notion can be
transferred from a subcategory to a category as a whole, and derive the existence of stable independence notions on a host of categories of groups and modules. We thereby extend results of Mazari-Armida, who has shown that the categories under consideration are stable in the sense of Galois types. Time permitting, we will also show that, provided the underlying category is locally finitely presentable, the existence of a stable independence relation immediately yields stable independence relations in every finite dimension.

Giles Gardam: Kaplansky’s conjectures

22nd February, 2021

Kaplansky made various related conjectures about group rings, especially for torsion-free groups. For example, the zero divisors conjecture predicts that if K is a field and G is a torsion-free group, then the group ring K[G] has no zero divisors. I will survey what is known about the conjectures, including their relationships to each other and to other group properties such as orderability, and present some recent progress.

Paul-Henry Leemann: Cayley graphs with few automorphisms

22nd February, 2021

Let G be a group and S a generating set. Then the group G naturally acts on the Cayley graph Cay(G,S) by left multiplications. The group G is said to be rigid if there exists an S such that the only automorphisms of Cay(G,S) are the ones coming from the action of G. While the classification of finite rigid groups was achieved in 1981, few results were known about infinite groups. In a recent work, with M. de la Salle we gave a complete classification of infinite finitely generated rigid groups. As a consequence, we also obtain that every finitely generated group admits a Cayley graph with countable automorphism group.

Lucas Mason-Brown: What is a Unipotent Representation?

22nd February, 2021

The concept of a unipotent representation has its origins in the representation theory of finite Chevalley groups. Let G(𝔽q) be the group of 𝔽q-rational points of a connected reductive algebraic group G. In 1984, Lusztig completed the classification of irreducible representations of G(𝔽q). He showed:

   1.  All irreducible representations of G(𝔽q) can be constructed from a finite set of building blocks -- called 'unipotent representations.'
   2.  Unipotent representations can be classified by certain geometric parameters related to nilpotent orbits for a complex group associated to G(𝔽q).

Now, replace 𝔽q by ℂ, the field of complex numbers, and replace G(𝔽q) with G(ℂ). There is a striking analogy between the finite-dimensional representation theory of G(𝔽q) and the unitary representation theory of G(ℂ). This analogy suggests that all unitary representations of G(ℂ) can be constructed from a finite set of building blocks - called 'unipotent representations' - and that these building blocks are classified by geometric parameters related to nilpotent orbits. In this talk I will propose a definition of unipotent representations, generalizing the Barbasch-Vogan notion of 'special unipotent'. The definition I propose is geometric and case-free. After giving some examples, I will state a geometric classification of unipotent representations, generalizing the well-known result of Barbasch-Vogan for special unipotents.

Swastik Kopparty: Geometric rank of tensors and the subrank of matrix multiplication

22nd February, 2021

I will talk about a new notion of rank for tensors called geometric rank, and discuss some of its basic properties, as well as its relationship with other well-studied notions of rank like subrank, slice rank and analytic rank. As an application, we will see a proof of tightness of an old bound of Strassen on the subrank of the matrix multiplication tensor.

Daniel Pomerleano: Intrinsic mirror symmetry and categorical crepant resolutions

19th February, 2021

Gross and Siebert have recently proposed an "intrinsic" programme for studying mirror symmetry. In this talk, we will discuss a symplectic interpretation of some of their ideas in the setting of affine log Calabi-Yau varieties. Namely, we describe work in progress which shows that, under suitable assumptions, the wrapped Fukaya category of such a variety X gives an intrinsic "categorical crepant resolution" of Spec(SH0(X)). No background in mirror symmetry will be assumed for the talk.

Salim Tayou: Exceptional Jumps of Picard Rank of K3 Surfaces over Number Fields

18th February, 2021

Given a K3 surface X over a number field K, we prove that the set of primes of K where the geometric Picard rank jumps is infinite, assuming that X has everywhere potentially good reduction. This result is formulated in the general framework of GSpin Shimura varieties and I will explain other applications to abelian surfaces. I will also discuss applications to the existence of rational curves on K3 surfaces.

The results in this talk are joint work with Ananth Shankar, Arul Shankar and Yunqing Tang.

Charles Walker: Distributive laws, pseudodistributive laws and decagons

18th February, 2021

The notion of a distributive law of monads was introduced by Beck, and gives a concise description of the data required to compose monads. In the 2-dimensional case, Marmolejo defined pseudodistributive laws of pseudomonads (where the required diagrams only commute up to an invertible modification). However, this description requires a number of coherence conditions due to the extra data involved.

In this talk we give alternative definitions of distributive laws and pseudodistributive laws involving the decagonal coherence conditions which naturally arise when the involved monads and pseudomonads are presented in their extensive form. As an application, we show that of Marmolejo and Wood’s eight coherence axioms for pseudodistributive laws, three are redundant.

We will then go on to give (likely) minimal definitions of distributive laws and pseudodistributive laws, which further simply the coherence conditions involved in this extensive viewpoint.

Nick Davidson: Type P Webs and Howe Duality

16th February, 2021

Webs are combinatorially defined diagrams which encode homomorphisms between tensor products of certain representations of Lie (super)algebras. I will describe some recent work with Jon Kujawa and Rob Muth which defines webs for the type P Lie superalgebra, and then uses these webs to deduce an analogue of Howe duality for this Lie superalgebra.

Shukun Wu: On the largest sum-free subsets of integers

15th February, 2021

An old conjecture in additive combinatorics asks: what is the largest sum-free subset of any set of N positive integers? Here the word "largest" should be understood in terms of cardinality. In this talk, I will discuss some recent progress on this conjecture, and the analogous conjecture on (k,l)-sum-free sets. The main method we used is Fourier analysis.

Cheuk Yu Mak: Non-displaceable Lagrangian links in four-manifolds

12th February, 2021

One of the earliest fundamental applications of Lagrangian Floer theory is detecting the non-displaceablity of a Lagrangian submanifold. Many progress and generalisations have been made since then but little is known when the Lagrangian submanifold is disconnected. In this talk, we describe a new idea to address this problem. Subsequently, we explain how to use Fukaya-Oh-Ohta-Ono and Cho-Poddar theory to show that for every S2 × S2 with a non-monotone product symplectic form, there is a continuum of disconnected, non-displaceable Lagrangian submanifolds such that each connected component is displaceable.

Alina Vdovina: Buildings, quaternions and Drinfeld-Manin solutions of Yang-Baxter equations

11th February, 2021

We will give a brief introduction to the theory of buildings and present their geometric, algebraic and arithmetic aspects. In particular, we present explicit constructions of infinite families of quaternionic cube complexes, covered by buildings. We will introduce new connections of geometric group theory and theoretical physics by using quaternionic lattices to find new infinite families of Drinfeld-Manin solutions of Yang-Baxter equations.

Paolo Perrone: Kan extensions are partial colimits

11th February, 2021

One way of interpreting a left Kan extension is as taking a kind of 'partial colimit', where one replaces parts of a diagram by their colimits. We make this intuition precise by means of the 'partial evaluations' sitting in the so-called bar construction of monads. The (pseudo)monads of interest for forming colimits are the monad of diagrams and the monad of small presheaves, both on the category CAT of locally small categories. We also define a morphism of monads between them, which we call 'image', and which takes the 'free colimit' of a diagram. This morphism allows us in particular to generalize the idea of 'cofinal functors', i.e. of functors which leave colimits invariant in an absolute way. This generalization includes the concept of absolute colimit as a special case. The main result of this work says that a pointwise left Kan extension of a diagram corresponds precisely to a partial evaluation of its colimit. This categorical result is analogous to what happens in the case of probability monads, where a conditional expectation of a random variable corresponds to a partial evaluation of its centre of mass.

George Shakan: Effective Khovanskii Theorems

8th February, 2021

Let A be a subset of the d dimensional integer lattice and NA be the N-fold sumset. In 1992, Khovanskii proved that |NA| can be written as a polynomial in |A| of degree at most d, provided N is sufficiently large. We provide an effective bound for "sufficiently large", and discuss some related results.

Yusuf Barış Kartal: Algebraic torus actions on Fukaya categories

5th February, 2021

The purpose of this talk is to explore how Lagrangian Floer homology groups change under (non-Hamiltonian) symplectic isotopies on a (negatively) monotone symplectic manifold (M,ω) satisfying a strong non-degeneracy condition. More precisely, given two Lagrangian branes L,L′, consider family of Floer homology groups HF(ϕv(L),L′), where v ∈ H1(M,ℝ) and ϕv is the time-1 map of a symplectic isotopy with flux v. We show how to fit this collection into an algebraic sheaf over the algebraic torus H1(M,𝔾m). The main tool is the construction of an "algebraic action" of H1(M,𝔾m) on the Fukaya category. As an application, we deduce the change in Floer homology groups satisfy various tameness properties, for instance, the dimension is constant outside an algebraic subset of H1(M,𝔾m). Similarly, given closed 1-form α, which generates a symplectic isotopy denoted by ϕtα, the Floer homology groups HF(ϕtα(L),L′) have rank that is constant in t, with finitely many possible exceptions.

Vladimir Shpilrain: What, if anything, can be done in sublinear time?

4th February, 2021

In his talk on September 10, 2020, Yuri Gurevich discussed some algorithms that run in linear time (in the "length" of an input). We are going to take it up a notch and discuss what can be done in sublinear time; in particular, without reading the whole input but only a small part thereof. One well-known example is deciding divisibility of a decimal integer by 2, 5, or 10: this is done by reading just the last digit. We will discuss some less obvious examples from (semi)group theory.

Jiří Rosický: Metric monads

4th February, 2021

We develop universal algebra over an enriched category and relate it to finitary enriched monads. Using it, we deduce recent results about ordered universal algebra where inequations are used instead of equations. Then we apply it to metric universal algebra where quantitative equations are used instead of equations. This contributes to understanding of finitary monads on the category of metric spaces.

Alistair Savage: Affinization of monoidal categories

2nd February, 2021

We define the affinization of an arbitrary monoidal category, corresponding to the category of string diagrams on the cylinder. We also give an alternative characterization in terms of adjoining dot generators to the category. The affinization formalizes and unifies many constructions appearing in the literature. We describe a large number of examples coming from Hecke-type algebras, braids, tangles, and knot invariants.

Oliver Edtmair: 3D convex contact forms and the Ruelle invariant

29th January, 2021

Is every dynamically convex contact form on the three sphere convex? In this talk I will explain why the answer to this question is no. The strategy is to derive a lower bound on the Ruelle invariant of convex contact forms and construct dynamically convex contact forms violating this lower bound.