University of Oxford

In this talk, I will firstly give asymptotic formulas for the moments of the n-th derivative of the characteristic polynomials from the CUE. Secondly, I will talk about the connections between them and a solution of certain Painleve differential equation. This is joint work with Jonathan P. Keating.
 

Let $p_c$ and $q_c$ be the threshold and the expectation threshold, respectively, of an increasing family $F$ of subsets of a finite set $X$. Recently, Park and Pham proved Kahn–Kalai conjecture stating that a not-too-large multiple of $q_c$ is an upper bound on $p_c$. In the talk, I will present a slight improvement to the Park–Pham theorem, which is obtained from transferring the threshold result from the small $p$ regime to general $p$. Based on joint work with Oliver Riordan.

It is known that there exists certain randomness in the values of the Moebius function. It is widely believed that this randomness predicts significant cancellations in the summation of the Moebius function times any 'reasonable' sequence. This rather vague principle is known as an instance of the 'Moebius randomness principle'. Sarnak made this principle precise by identifying the notion 'reasonable' as deterministic. More precisely, Sarnak's Moebius Disjointness Conjecture predicts the disjointness of the Moebius function from any arithmetic functions realized in any topological dynamical systems of zero topological entropy. In this talk, I will firstly introduce some background and progress on this conjecture. Secondly, I will talk about some of my work on this. Thirdly, I will talk some related problems to this conjecture.

Let $p$ be a prime, let $1 \le t < d < p$ be integers, and let $S$ be a non-empty subset of $\mathbb{F}_p$ (which may be thought of as being $\{0,1\}$). We will establish that if a polynomial $P:\mathbb{F}_p^n \to \mathbb{F}_p$ with degree $d$ is such that the image $P(S^n)$ does not contain the full image $A(\mathbb{F}_p)$ of any non-constant polynomial $A: \mathbb{F}_p \to \mathbb{F}_p$ with degree at most $t$, then $P$ coincides on $S^n$ with a polynomial $Q$ that in particular has bounded degree-$\lfloor d/(t+1) \rfloor$-rank in the sense of Green and Tao, and has degree at most $d$. Likewise, we will prove that if the assumption holds even for $t=d$ then $P$ coincides on $S^n$ with a polynomial determined by a bounded number of coordinates and with degree at most $d$.

Many problems in number theory are equivalent to determining all of the rational points on some curve or family of curves. In general, finding all the rational points on any given curve is a challenging (even unsolved!) problem. 

The focus of this talk is rational points on so-called Erdős-Selfridge curves. A deep conjecture of Sander, still unproven in many cases, predicts all of the rational points on these curves. 

I will describe work-in-progress proving new cases of Sander's conjecture, and sketch some ideas in the proof. The core of the proof is a `mass increment argument,' which is loosely inspired by various increment arguments in additive combinatorics. The main ingredients are a mixture of combinatorial ideas and quantitative estimates in Diophantine geometry.

The closest thing we have to a general method for finding primes in sets is to use sieve methods to turn the problem into some other (hopefully easier) arithmetic questions about the set.

Unfortunately this process is still poorly understood - we don’t know ‘how much’ arithmetic information is sufficient to guarantee the existence of primes, and how much is not sufficient. Often arguments are rather ad-hoc.

I’ll talk about work-in-progress with Kevin Ford which shows that many of our common techniques are not optimal and can be refined, and in many cases these new refinements are provably optimal.

The computation of matrix function is an important task appearing in many areas of scientific computing. We propose two algorithms, one for banded matrices and the other for general sparse matrices that approximate f(A) using matrix-vector products only. Our algorithms are inspired by the decay bound for the entries of f(A) when A is banded or sparse. We show its exponential convergence when A is banded or sufficiently sparse and we demonstrate its performance using numerical experiments.

Asymptotic dimension was introduced by Gromov as an invariant of finitely generated groups. It can be shown that if two metric spaces are quasi-isometric then they have the same asymptotic dimension. In 1998, the asymptotic dimension achieved particular prominence in geometric group theory after a paper of Guoliang Yu, which proved the Novikov conjecture for groups with finite asymptotic dimension. Unfortunately, not all finitely generated groups have finite asymptotic dimension. 

In this talk, we will introduce some basic tools to compute the asymptotic dimension of groups. We will also find upper bounds for the asymptotic dimension of a few well-known classes of finitely generated groups, such as hyperbolic groups, and if time permits, we will see why one-relator groups have asymptotic dimension at most two.

A common pastime of geometric group theorists is to try and derive algebraic information about a group from the geometric properties of its Cayley graphs. One of the most classical demonstrations of this can be seen in the work of Maschke (1896) in characterising those finite groups with planar Cayley graphs. Since then, much work has been done on this topic. In this talk, I will attempt to survey some results in this area, and show that the class group with planar Cayley graphs is QI-rigid.