For $\ell$ an odd prime number and $d$ a squarefree integer, a notable problem in arithmetic statistics is to give pointwise bounds for the size of the $\ell$-torsion of the class group of $\mathbb{Q}(\sqrt{d})$. This is in general a difficult problem, and unconditional pointwise bounds are only available for $\ell = 3$ due to work of Pierce, Helfgott—Venkatesh and Ellenberg—Venkatesh. The current record due to Ellenberg—Venkatesh is $h_3(d) \ll_\epsilon d^{1/3 + \epsilon}$. We will discuss how to improve this to $h_3(d) \ll d^{0.32}$. This is joint work with Peter Koymans.

Let G be a connected reductive group defined over a non-Archimedean local field k. Endow G with a k-involution and take H to be the fixed-point subgroup in G of that involution. In this talk I will report on some of my recent results regarding Chabauty limits of H(k) inside G(k). Although the results are similar to the real and complex cases, the techniques are totally different and with a strong geometric flavor. Some of the main actors are the Bruhat—Tits building associated with G(k) and basic methods from CAT(0) geometry.

Let g be the Lie algebra of a simple algebraic group over an algebraically closed field of characteristic p. When p=0 the celebrated Jacobson-Morozov Theorem promises that every non-zero nilpotent element of g is contained in a simple 3-dimensional subalgebra of g (an sl2). This has been extended to odd primes but what about p=2? There is still a unique 3-dimensional simple Lie algebra, known colloquially as fake sl2, but there are other very sensible candidates like sl2 and pgl2. In this talk, Adam Thomas from the University of Warwick will discuss recent joint work with David Stewart (Manchester) determining which nilpotent elements of g live in subalgebras isomorphic to one of these three Lie algebras. There will be an abundance of concrete examples, calculations with small matrices and even some combinatorics.