The neutron transport equation is a linear Boltzmann-type PDE that models radiative transfer processes, and fission nuclear reactions. The computation of the largest eigenvalue of this Boltzmann operator is crucial in nuclear safety studies but it has classically been formulated only at a discretized level, so the predictive capabilities of such computations are fairly limited. In this talk, I will give an overview of the modeling for this equation, as well as recent analysis that leads to an infinite dimensional formulation of the eigenvalue problem. We leverage this point of view to build a numerical scheme that comes with a rigorous, a posteriori estimation of the error between the exact, infinite-dimensional solution, and the computed one.

We prove that every finite colouring of the plane contains a monochromatic pair of points at an odd (integral) distance from each other. We will also discuss some further results with Rose McCarty and Michal Pilipczuk concerning prime and polynomial distances.

Tom Hutchcroft and I have been working to develop a general theory of percolation on arbitrary finite transitive graphs. This extends from percolation on local approximations to infinite graphs, such as a sequence of tori, to percolation on the complete graphs - the Erdős-Rényi model. I will summarise our progress on the basic questions: When is there a phase transition for the emergence of a giant cluster? When is the giant cluster unique? How does this relate to percolation on infinite graphs? I will then sketch our proof that for finite transitive graphs with uniformly bounded vertex degrees, the supercritical giant cluster is unique, verifying a conjecture of Benjamini from 2001.

In 1973, Erdős conjectured the existence of high girth $(n,3,2)$-Steiner systems. Recently, Glock, Kühn, Lo, and Osthus and independently Bohman and Warnke proved the approximate version of Erdős' conjecture. Just this year, Kwan, Sah, Sawhney, and Simkin proved Erdős' conjecture. As for Steiner systems with more general parameters, Glock, Kühn, Lo, and Osthus conjectured the existence of high girth $(n,q,r)$-Steiner systems. We prove the approximate version of their conjecture. This result follows from our general main results which concern finding perfect or almost perfect matchings in a hypergraph $G$ avoiding a given set of submatchings (which we view as a hypergraph $H$ where $V(H)=E(G)$). Our first main result is a common generalization of the classical theorems of Pippenger (for finding an almost perfect matching) and Ajtai, Komlós, Pintz, Spencer, and Szemerédi (for finding an independent set in girth five hypergraphs). More generally, we prove this for coloring and even list coloring, and also generalize this further to when $H$ is a hypergraph with small codegrees (for which high girth designs is a specific instance). A number of applications in various areas follow from our main results including: Latin squares, high dimensional permutations, and rainbow matchings. This is joint work with Luke Postle.

A graph has a pair $(m,f)$ if it has an induced subgraph on $m$ vertices and $f$ edges. We write $(n,e)\rightarrow (m,f)$  if any graph on $n$ vertices and $e$ edges has a pair $(m,f)$.  Let  $$S(n,m,f)=\{e: ~(n,e)\rightarrow (m,f)\} ~{\rm and}$$     $$\sigma(m,f) =   \limsup_{n\rightarrow \infty}\frac{ |S(n,m,f)|}{\binom{n}{2}}.$$ These notions were first introduced and investigated by Erdős, Füredi, Rothschild, and Sós. They found five pairs $(m,f)$ with  $\sigma(m,f)=1$ and showed that for all other pairs $\sigma(m,f)\leq 2/3$.  We extend these results in two directions.

First, in a joint work with Weber, we show that not only $\sigma(m,f)$ can be zero, but also $S(n,m,f)$  could be empty for some pairs $(m,f)$ and any sufficiently large $n$. We call such pairs $(m,f)$ absolutely avoidable.

Second, we consider a natural analogue $\sigma_r(m,f)$ of $\sigma(m,f)$ in the setting of $r$-uniform hypergraphs.  Weber showed that for any $r\geq 3$ and  $m>r$,  $\sigma_r(m,f)=0$ for most values of $f$.  Surprisingly, it was not immediately clear whether there are nontrivial pairs $(m,f)$,  $(f\neq 0$, $f\neq \binom{m}{r}$,  $r\geq 3$),  for which $\sigma_r(m,f)>0$. In a joint work with Balogh, Clemen, and Weber we show that $\sigma_3(6,10)>0$ and conjecture that in the $3$-uniform case $(6,10)$ is the only such pair.