We give a gentle introduction to the theory of self-similar sets and self-similar measures. Connections of this topic to Diophantine approximation on Lie groups as well as to additive combinatorics will be exposed. In particular, we will discuss recent progress on Bernoulli convolutions. If time permits, we mention recent joint work with Samuel Kittle on absolutely continuous self-similar measures. 
 

Heegner points are a powerful tool for understanding the structure of the group of rational points on elliptic curves. In this talk, I will describe these points and the ideas surrounding their generalisation to other situations.

We explore the weak coupling limit for stochastic Burgers type equation in critical dimension, and show that it is given by a Gaussian stochastic heat equation, with renormalised coefficient depending only on the second order Hermite polynomial of the nonlinearity. We use the approach of Cannizzaro, Gubinelli and Toninelli (2024), who treat the case of quadratic nonlinearities, and we extend it to polynomial nonlinearities. In that sense, we extend the weak universality of the KPZ equation shown by Hairer and Quastel (2018) to the two dimensional generalized stochastic Burgers equation. A key new ingredient is the graph notation for the generator. This enables us to obtain uniform estimates for the generator. This is joint work with Nicolas Perkowski.

In 1975, Bollobás, Erdős, and Szemerédi asked the following Zarankiewicz-type problem. What is the smallest $\tau$ such that an $n \times n \times n$ tripartite graph with minimum degree $n + \tau$ must contain $K_{t, t, t}$? They further conjectured that $\tau = O(n^{1/2})$ when $t = 2$.

I will discuss our proof that $\tau = O(n^{1 - 1/t})$ (confirming their conjecture) and an infinite family of extremal examples. The bound $O(n^{1 - 1/t})$ is best possible whenever the Kővári-Sós-Turán bound $\operatorname{ex}(n, K_{t, t}) = O(n^{2 - 1/t})$ is (which is widely-conjectured to be the case).

This is joint work with Francesco Di Braccio (LSE).

A zero-one matrix $M$ is said to contain another zero-one matrix $A$ if we can delete some rows and columns of $M$ and replace some 1-entries with 0-entries such that the resulting matrix is $A$. The extremal number of $A$, denoted $\operatorname{ex}(n,A)$, is the maximum number of 1-entries that an $n\times n$ zero-one matrix can have without containing $A$. The systematic study of this function for various patterns $A$ goes back to the work of Furedi and Hajnal from 1992, and the field has many connections to other areas of mathematics and theoretical computer science. The problem has been particularly extensively studied for so-called acyclic matrices, but very little is known about the general case (that is, the case where $A$ is not necessarily acyclic). We prove the first asymptotically tight general result by showing that if $A$ has at most $t$ 1-entries in every row, then $\operatorname{ex}(n,A)\leq n^{2-1/t+o(1)}$. This verifies a conjecture of Methuku and Tomon.

Our result also provides the first tight general bound for the extremal number of vertex-ordered graphs with interval chromatic number two, generalizing a celebrated result of Furedi, and Alon, Krivelevich and Sudakov about the (unordered) extremal number of bipartite graphs with maximum degree $t$ in one of the vertex classes.

Joint work with Barnabas Janzer, Van Magnan and Abhishek Methuku.

In a graph $H$ whose edges are coloured (not necessarily properly) a rainbow copy of a graph $G$ is a (not necessarily induced) subgraph of $H$ that is isomorphic to $G$ and whose edges are all coloured differently. In this talk I will explain why the problem of finding such rainbow copies is interesting, survey what we know, concentrating mainly on the case where $G$ is a Hamilton cycle, and then tell you a bit about a new result about finding rainbow Hamilton cycles resiliently in random graphs (which is joint work with Peter Allen and Liana Yepremyan).