L6

TBC

A subset $A$ of $A_n$ is $k$-product-free if for all $a_1,a_2,\dots,a_k\in A$, $a_1a_2\dots a_k$ $\notin A$.
We determine the largest $3$-product-free and $4$-product-free subsets of $A_n$ for sufficiently large $n$. We also obtain strong stability results and results on multiple sets with forbidden cross products. The principal technical ingredient in our approach is the theory of hypercontractivity in $S_n$. Joint work with Peter Keevash.

Dirichlet polynomials are useful in the study of the Riemann zeta function & Dirichlet L functions, serving as approximations to them via the approximate functional equation. Understanding how often they can be large gives bounds on the number of zeroes of these functions in vertical strips - known as zero density estimates - which are relevant to the distribution of primes in short intervals. Based on Guth-Maynard, we study large values of Dirichlet polynomials with characters, relevant to Dirichlet L functions. Joint work with Yung Chi Li. 

Given a mod $p$ Galois representation, one often wonders whether it arises by reducing a $p$-adic one, and whether these lifts are suitably 'well-behaved'. In this talk, we discuss how ideas from homotopy theory aid the study of Galois deformations, reviewing work of Galatius-Venkatesh.

The arithmetic regularity lemma says that any dense set A in F_p^n can be cut along cosets of some small codimension subspace H <= F_p^n such that on almost all cosets of H, A is either random or structured (in a precise quantitative manner). A standard example shows that one cannot hope to improve "almost all" to "all", nor to have a good quantitative dependency between the constants involved. Adding a further combinatorial assumption on A to the arithmetic regularity lemma makes its conclusion so strong that one can essentially classify such sets A. In this talk, I will use use the analogous problem with F_p^n replaced with R^n as a way the motivate the funny title.

I will construct a fully-faithful functor from the category of co-admissible D-cap modules of Ardakov—Wadsley, to the category of quasi-coherent sheaves on the "analytic de Rham space”, at least in the case when the rigid variety is affinoid and étale over a polydisk. 

Is it possible to tell the isomorphism type of an infinite group from its collection of finite quotients? This question, known as profinite rigidity, has deep roots in various areas of mathematics, ranging from arithmetic geometry to group theory. In this talk, I will introduce the question, its history and context. I will explain how profinite rigidity is studied using the machinery of profinite completions, including elementary proofs and counterexamples. Then I will outline some of the key results in the field, ranging from 1970 to the present day. Time permitting, I will elaborate on recent results of myself on the profinite rigidity of certain classes of solvable groups. 

In the definition of the skein lasagna module of a $4$-manifold $X$, it is essential that the input TQFT be fully functorial for link cobordisms in $S^3 \times [0, 1]$. I will describe an approach to resolve existing sign ambiguities in Kronheimer and Mrowka's spectral sequence from Khovanov homology to singular instanton link homology. The goal is to obtain a theory that is fully functorial for link cobordisms in $S^3 \times [0,1]$, and where the $E_2$ page carries a canonical isomorphism to Khovanov-Rozansky $\mathfrak{gl}_2$ link homology. Possible applications include non-vanishing theorems for $4$-manifold Khovanov skein lasagna modules à la Ren-Willis.

In this talk, we give an accessible introduction to the theory of coarse cohomology of metric spaces in the sense of Margolis, which we present in direct analogy with group cohomology for discrete groups. We explain how this yields the robust notion of coarse cohomological dimension (due to Margolis), which is a genuine quasi-isometry invariant of metric spaces generalising the cohomological dimension of groups when the latter is finite. We then give applications to geometric properties of quasimorphisms and motivate how such considerations might be useful in the setting of non-positively curved groups. This is joint reading/work with Paula Heim.