Charlotte Hoffmann: Short words of high imprimitivity rank yield hyperbolic one-relator groups

8th March, 2021

It is a long-standing question whether a group of type F that does not contain Baumslag–Solitar subgroups is necessarily hyperbolic. One-relator groups are of type F and Louder and Wilton showed that if the defining relator has imprimitivity rank greater than 2, they do not contain Baumslag-Solitar subgroups, so they conjecture that such groups are hyperbolic. Cashen and I verified the conjecture computationally for relators of length at most 17. In this talk I'll introduce hyperbolic groups and the imprimitivity rank of elements in a free group. I'll also discuss how to verify hyperbolicity using versions of combinatorial curvature on van Kampen diagrams.

Jieru Zhu: Transitioning between the polytabloid and web bases for the Specht modules

8th March, 2021

SLn webs first emerged in invariant theory and have a recent reformulation by Cautis-Kamnitzer-Morrison (2014). A collection of these webs form a basis of the Specht modules for the symmetric groups. On the other hand, classical construction of the Specht modules uses the polytabloids basis parameterized by standard Young tableaux. Russell-Tymoczko (2020) showed that the transitioning matrix from the polytabloid basis to the web basis is unitriangular. We further proved their conjecture that the upper-triangular entries are positive. The talk will be mostly focused on SL2 webs, with some preliminary results on SL3 webs.

Aled Walker: Extremal problems for GCDs

8th March, 2021

In this talk we will discuss how ideas from Koukoulopoulos and Maynard's proof of an old conjecture in diophantine approximation led to some surprisingly subtle questions in combinatorial number theory. These questions ask for a bound on the maximal size of two finite sets of natural numbers A and B with the property that 1% of the pairs (a,b) have a large greatest common divisor. We will also describe our recent joint work with Ben Green, in which we proved close-to-optimal bounds on these problems.

Sobhan Seyfaddini: Periodic Floer homology and the large-scale geometry of Hofer’s metric on the sphere

5th March, 2021

The group of Hamiltonian diffeomorphisms of a symplectic manifold admits a remarkable bi-invariant metric, called Hofer's metric. My talk will be about a recent joint work with Dan Cristofaro-Gardiner and Vincent Humilière resolving the following two open-questions related to the large-scale geometry of this metric. The first, due to Kapovich and Polterovich, asks whether the two-sphere, equipped with Hofer's metric, is quasi-isometric to the real line; we show that it is not. The second, due to Fathi, asks whether the group of area and orientation preserving homeomorphisms of the two-sphere is a simple group; we show that it is not. Key to our proofs is a new sequence of spectral invariants defined via Hutchings's Periodic Floer Homology.

Harald Helfgott: Expansion, divisibility and parity

5th March, 2021

We will discuss a graph that encodes the divisibility properties of integers by primes. We show that this graph is shown to have a strong local expander property almost everywhere. We then obtain several consequences in number theory, beyond the traditional parity barrier. For instance: for λ the Liouville function (that is, the completely multiplicative function with λ(p) = -1 for every prime), (1/log x) ∑n ≤ x λ(n) λ(n+1)/n = O(1/√(log log x)), which is stronger than a well-known result by Tao. We also manage to prove, for example, that λ(n+1) averages to 0 at almost all scales when n restricted to have a specific number of prime divisors Ω(n)=k, for any "popular" value of k (that is, k = log log N + O(√(log log N)) for n ≤ N. We will give a quick overview of the combinatorial ideas behind the proof.

Vladimir Dotsenko: Diamond Lemma and the Maurer-Cartan equation

5th March, 2021

I shall outline a new approach to the Composition-Diamond Lemma for rewriting systems/Gröbner-Shirshov bases; more specifically, I shall explain how the Maurer-Cartan equation in the tangent complex of a monomial algebra leads to many different versions of the Composition-Diamond Lemma, one for each representative of the tangent complex arising from a multigraded resolution of such algebra.

Gal Dor: Monoidal Structures on GL2-Modules and Abstractly Automorphic Representations

4th March, 2021

Consider the function field F of a smooth curve over 𝔽q, with q>2. L-functions of automorphic representations of GL2 over F are important objects for studying the arithmetic properties of the field F. Unfortunately, they can be defined in two different ways: one by Godement-Jacquet, and one by Jacquet-Langlands. Classically, one shows that the resulting L-functions coincide using a complicated computation. Each of these L-functions is the GCD of a family of zeta integrals associated to test data. I will categorify the question, by showing that there is a correspondence between the two families of zeta integrals, instead of just their L-functions. The resulting comparison of test data will induce an exotic symmetric monoidal structure on the category of representations of GL2. It turns out that an appropriate space of automorphic functions is a commutative algebra with respect to this symmetric monoidal structure. I will outline this construction, and show how it can be used to construct a category of automorphic representations.

Tamar Bar-On: Profinite completion of free profinite groups

4th March, 2021

The profinite completion of a free profinite group on infinite set of generators is a profinite group of greater rank. However, it is still unknown whether it is a free profinite group too. I am going to present some partial results regarding to this question, which is equivalent to ask: what abstract embedding problems can a free profinite group solve.

Chaitanya Leena Subramaniam: Dependently typed algebraic theories

4th March, 2021

For S a set, S-sorted algebraic (or 'Lawvere') theories are, equivalently, finite-product categories whose objects are freely generated by S, finitary monads on Set/S, or monoids in a category of 'S-coloured cartesian collections'.

When S is a suitable direct category, I will describe equivalences of categories between finitary monads on [Sop, Set], monoids in a category of 'S-coloured cartesian collections', and a certain category of contextual categories (in the sense of Cartmell) under Sop.

Examples of such S are the categories of semi-simplices, globes and opetopes. Opetopes will be a running example, and we will see that there are three idempotent finitary monads on the category of opetopic sets, whose algebras are, respectively, small categories, coloured planar Set-operads, and planar coloured combinads (in the sense of Loday).

Sarah Olson: Sperm Navigation: from hydrodynamic interactions to parameter estimation

3rd March, 2021

Microorganisms can swim in a variety of environments, interacting with chemicals and other proteins in the fluid. In this talk, we will highlight recent computational methods and results for swimming efficiency and hydrodynamic interactions of swimmers in different fluid environments. Sperm are modeled via a centerline representation where forces are solved for using elastic rod theory. The method of regularized Stokeslets is used to solve the fluid-structure interaction where emergent swimming speeds can be compared to asymptotic analysis. In the case of fluids with extra proteins or cells that may act as friction, swimming speeds may be enhanced, and attraction may not occur. We will also highlight how parameter estimation techniques can be utilized to infer fluid and/or swimmer properties.

Aaron Yi Rui Low: Adjustment matrices

2nd March, 2021

James's Conjecture predicts that the adjustment matrix for weight w blocks of the Iwahori-Hecke algebras ℋn and the q-Schur algebras 𝒮n is the identity matrix when w < char(F). Fayers has proved James's Conjecture for blocks of ℋn of weights 3 and 4. We shall discuss some results on adjustment matrices that have been used to prove James's Conjecture for blocks of 𝒮n of weights 3 and 4 in an upcoming paper. If time permits, we will look at a proof of the weight 3 case.

David Galban: First and Second Cohomology Groups for BBW Parabolics for Lie Superalgebras

1st March, 2021

For semisimple Lie algebras, a well-known theorem of Kostant computes the cohomology groups of parabolic subalgebras, but it is unknown whether an analogue of Kostant’s theorem exists for Lie superalgebras. Seeking to provide the first calculations in this direction, in this talk, I will describe the cohomology groups for the subalgebra 𝔫+ relative to the BBW parabolic subalgebras constructed by D. Grantcharov, N. Grantcharov, Nakano and Wu. These classical Lie superalgebras have a triangular decomposition 𝔤 = 𝔫- + 𝔣 + 𝔫+, where 𝔣 is a detecting subalgebra as introduced by Boe, Kujawa and Nakano. I will show that there exists a Hochschild-Serre spectral sequence that collapses for all infinite families of classical simple Lie superalgebras. Using this, I will provide examples of computation of the first and second cohomologies for various 𝔫+.

Jimmy He: Random walks on finite fields with deterministic jumps

1st March, 2021

Recently, Chatterjee and Diaconis showed that most bijections, if applied between steps of a Markov chain, cause the resulting chain to mix much faster. However, explicit examples of this speedup phenomenon are rare. I will discuss recent work studyingsuch walks on finite fields where the bijection is algebraically defined. This work gives a large collection of examples where this speedup phenomenon occurs. These walks can be seen as a non-linear analogue of the Chung-Diaconis-Graham process, where the bijectionis multiplication by a non-zero element of the finite field. This work is partially joint with Huy Pham and Max Xu.

Agata Smoktunowicz: Some questions related to nilpotent rings and braces

27th February, 2021

In around 2005, Wolfgang Rump introduced braces, a generalization of nilpotent rings to describe all involutive, non-degenerate set-theoretic solutions of the Yang-Baxter equation. This formulation then rapidly found application in other research areas. This talk will review these applications.

Definition. A set A with binary operations of addition +, and multiplication ∘ is a brace if (A, +) is an abelian group, (A, ∘) is a group and a ∘ (b+c)+a = a ∘ b+a ∘ c for every a, b, c ∈ A. It follows from this definition that every nilpotent ring with the usual addition and with multiplication a ∘ b = ab + a + b is a brace.

Braces have been shown to be equivalent to several concepts in group theory such as groups with bijective 1-cocycles and regular subgroups of the holomorph of abelian groups. In algebraic number theory there is a correspondence between braces and Hopf-Galois extensions of abelian type first observed by David Bachiller. There is also connection between R-braces and pre-Lie algebras discovered by Wolfgang Rump in 2014. One generator braces have been shown to describe indecomposable, involutive solutions of the Yang-Baxter equation.

On the other hand, Anastasia Doikou and Robert Weston have recently discovered some fascinating connections between braces and quantum integrable systems. In particular, to find solutions of the set-theoretic reflection equation it is needed to solve problems on some polynomial identities in nilpotent rings. Because previously the theory of polynomial identities was mainly developed for prime rings, and for the reflection equation we only consider nilpotent rings, there are no known methods for solving such problems. We will mention some open problems on polynomial identities in nilpotent rings which appear in this situation.

Sylvain Courte: Twisted generating functions and the nearby Lagrangian conjecture

26th February, 2021

I will explain the notion of twisted generating function and show that a closed exact Lagrangian submanifold L in the cotangent bundle of M admits such a thing. The type of function arising in our construction is related to Waldhausen's tube space from his manifold approach to algebraic K-theory of spaces. Using the rational equivalence of this space with BO, as proved by Bökstedt, we conclude that the stable Lagrangian Gauss map of L vanishes on all homotopy groups. In particular when M is a homotopy sphere, we obtain the triviality of the stable Lagrangian Gauss map and a genuine generating function for L.

Alexander Smith: Selmer groups and a Cassels-Tate pairing for finite Galois modules

25th February, 2021

I will discuss some new results on the structure of Selmer groups of finite Galois modules over global fields. Tate's definition of the Cassels-Tate pairing can be extended to a pairing on such Selmer groups with little adjustment, and many of the fundamental properties of the Cassels-Tate pairing can be reproved with new methods in this setting. I will also give a general definition of the theta/Mumford group and relate it to the structure of the Cassels-Tate pairing, generalizing work of Poonen and Stoll.