A well-known open problem for representations of symmetric groups is the Saxl conjecture. In this talk, we put Saxl's conjecture into a Lie-theoretical framework and present a natural generalisation to Weyl groups. After giving necessary preliminaries on spin representations and the Springer correspondence, we present our progress on the generalised conjecture. Next, we reveal connections to tensor product decomposition problems in symmetric groups and provide an alternative description of Lusztig’s cuspidal families. Finally, we propose a further generalisation to all finite Coxeter groups.

June Huh proved in 2012 that the Betti numbers of the complement of a complex hyperplane arrangement form a log concave sequence.  But what if the arrangement has symmetries, and we regard the cohomology as a representation of the symmetry group?  The motivating example is the braid arrangement, where the complement is the configuration space of n points in the plane, and the symmetric group acts by permuting the points.  I will present an equivariant log concavity conjecture, and show that one can use representation stability to prove infinitely many cases of this conjecture for configuration spaces.
 

Let $K$ be an unramified extension of $\mathbb{Q}_p$ for a prime $p > 3$. The reduced part of the Emerton-Gee stack for $\mathrm{GL}_{2}$ can be viewed as parameterizing two-dimensional mod $p$ Galois representations of the absolute Galois group of $K$. In this talk, we will consider the extremely non-generic irreducible components of this reduced part and see precisely which ones are smooth or normal, and which have Gorenstein normalizations. We will see that the normalizations of the irreducible components admit smooth-local covers by resolution-rational schemes. We will also determine the singular loci on the components, and use these results to update expectations about the conjectural categorical $p$-adic Langlands correspondence. This is based on recent joint work with Ben Savoie.

Smooth generic representations of $GL_n$ over a $p$-adic field $F$, i.e. representations admitting a nondegenerate Whittaker model, are an important class of representations, for example in the setting of Rankin-Selberg integrals. However, in recent years there has been an increased interest in non-generic representations and their degenerate Whittaker models. By the theory of Bernstein-Zelevinsky derivatives we can associate to each smooth irreducible representation of $GL_n(F)$ an integer partition of $n$, which encodes the "degeneracy" of the representation. By using these "highest derivative partitions" we can define a stratification of the category of smooth complex representations and prove the surprising fact that all of the strata categories are equivalent to module categories over commutative rings. This is joint work with David Helm.

In his final paper in 1954, Alan Turing wrote `No systematic method is yet known by which one can tell whether two knots are the same.' Within the next 20 years, Wolfgang Haken and Geoffrey Hemion had discovered such a method. However, the computational complexity of this problem remains unknown. In my talk, I will give a survey on this area, that draws on the work of many low-dimensional topologists and geometers. Unfortunately, the current upper bounds on the computational complexity of the knot equivalence problem remain quite poor. However, there are some recent results indicating that, perhaps, knots are more tractable than they first seem. Specifically, I will explain a theorem that provides, for each knot type K, a polynomial p_K with the property that any two diagrams of K with n_1 and n_2 crossings differ by at most p_K(n_1) + p_K(n_2) Reidemeister moves.