The Specht modules are of fundamental importance to the representation theory of the symmetric group, and their 0th cohomology is understood through entirely combinatorial methods due to Gordon James. Over fields of odd characteristic, Hemmer proposed a similar combinatorial approach to calculating their 1st degree cohomology, or extensions by the trivial module. This combinatorial approach motivates the definition of universal $p$-ary designs, which we shall classify. We then explore the consequences of this classification to problem of determining extensions of Specht modules. In particular, we classify all extensions of Specht modules indexed by two-part partitions by the trivial module and shall see some far-reaching conditions on when the first cohomology of a Specht module is trivial.

A nice collection of problems in additive combinatorics focus on analysing solutions to additive equations over sequences that exhibit some flavour of convexity. This, for instance, includes genuine convex sequences as well as images of arbitrary sets under convex functions. In this talk, I will survey some of the literature surrounding these type of questions, along with some motivation from analytic number theory as well as the current best known results towards these problems.

 I will review some well-established relationships between four manifolds and vertex algebras that can be deduced from studying the M5-brane worldvolume theory, and outline some of the corresponding results in mathematics which have been understood so far. I will then describe a proposal of Gaiotto-Rapcak to generalize these ideas to the setting of multiple M5 branes wrapping divisors in toric Calabi-Yau threefolds, and explain work in progress on understanding the mathematical implications of this proposal as a complex network of relationships between the enumerative geometry of sheaves on threefolds and the representation theory of affine Lie algebras.

One of the consequences of Wiles' proof of Fermat's Last Theorem is that elliptic curves over rational numbers can be associated with modular forms, whose Fourier coefficients essentially count points on the curve. Generalisation of this modularity to higher dimensional varieties is a very interesting open question. In this talk I will give a physicist's view of Calabi-Yau modularity. Starting with a very simplified overview of some number theoretic background related to the Langlands program, I relate some of this theory to black holes in IIB/A string theories compactified on Calabi-Yau threefolds. It is possible to associate modular forms to certain such black holes. We can then ask whether these modular forms have a physical interpretation as, for example, counting black hole microstates. In an attempt to answer this question, we derive a formula for fully instanton-corrected black hole entropy, which gives an interesting hint of this counting. The talk is partially based on recent work arXiv:2104.02718 with P. Candelas and J. McGovern.

Following on from Christoph's talk last week, I will present a version of the supercooled Stefan problem with noise. I will start by discussing the physical intuition and then give a probabilistic representation of solutions. From there, I will identify a simple relationship between the initial heat profile and a single parameter for how the liquid solidifies, which, if violated, forces the temperature to develop a discontinuity in finite time with positive probability. On the other hand, when the relationship is satisfied, the temperature remains globally continuous with probability one. The work is part of a new preprint that should soon be available on arXiv.