University of Oxford

Waldspurger proved maximality and minimality results for certain generalised Springer representations of $\text{Sp}(2n,\mathbb{C})$. We will discuss analogous results for $G = \text{SO}(N,\mathbb{C})$ and sketch their proofs.

Let $C$ be a unipotent class of $G$ and $E$ an irreducible $G$-equivariant local system on $C$. Let $\rho$ be the generalised Springer representation corresponding to $(C,E)$. We call $C$ the support of $\rho$. It is well-known that $\rho$ appears in the top cohomology of a certain variety. Let $\bar\rho$ be the representation obtained by summing the cohomology groups of this variety.

We show that if $C$ is parametrised by an orthogonal partition consisting of only odd parts, then $\bar\rho$ has a unique irreducible subrepresentation $\rho^{\text{max}}$ whose support is maximal among the supports of the irreducible subrepresentations of $\rho^{\text{max}}$. We also show that $\text{sgn}\otimes\rho^{\text{max}}$ is the unique subrepresentation of $\text{sgn}\otimes\bar\rho$ with minimal support. We will also present an algorithm to compute $\rho^{\text{max}}$.

In this talk we will present some work in progress generalising Lubin-Tate formal group laws to higher dimensions. There have been some other generalisations, but ours is different in that the ring over which the formal group law is defined changes as the dimension increases. We will state some conjectures about these formal group laws, including their relationship to the Drinfeld tower over the p-adic upper half plane, and provide supporting evidence for these conjectures.

An occupational hazard of mathematicians is the investigation of objects that are "optimal" in a mathematically precise sense, yet may be far from optimal in practice. This talk will discuss an extreme example of this effect: Gauss-Hermite quadrature on the real line. For large numbers of quadrature points, Gauss-Hermite quadrature is a very poor method of integration, much less efficient than simply truncating the interval and applying Gauss-Legendre quadrature or the periodic trapezoidal rule. We will present a theorem quantifying this difference and explain where the standard notion of optimality has failed.

For the first time, a method has become available for fast computation of near-best rational approximations on arbitrary sets in the real line or complex plane: the AAA algorithm (Nakatsukasa-Sète-T. 2018).  After a brief presentation of the algorithm this talk will focus on twenty demonstrations of the kinds of things we can do, all across applied mathematics, with a black-box rational approximation tool.
 

I will outline a new algorithm for unknot recognition that runs in quasi-polynomial time. The input is a diagram of a knot with n crossings, and the running time is n^{O(log n)}. The algorithm uses hierarchies, normal surfaces and Heegaard splittings.

Studying quasi-isometries between groups is a major theme in geometric group theory. Of particular interest are the situations where the existence of a quasi-isometry between two groups implies that the groups are equivalent in a stronger algebraic sense, such as being commensurable. I will survey some results of this type, and then talk about recent work with Daniel Woodhouse where we prove quasi-isometric rigidity for certain graphs of virtually free groups, which include "generic" cyclic HNN extensions of free groups.

Hierarchically Hyperbolic Groups (HHGs) were introduced by Behrstock—Hagen—Sisto to provide a common framework to study several groups of interest in geometric group theory, and have been an object of great interest in the area ever since. The goal of the talk is to provide an introduction to the theory of HHGs and discuss the advantages of the unified approach that they provide. If time permits, we will conclude with applications to growth and asymptotic cones of groups.

I will introduce the Friedl-Tillmann polytope for one-relator groups, and then discuss how it can be generalised to the Friedl-Lück polytope, how it connects to the Thurston polytope, and how we can view it as a convenient source of intuition and ideas.

In joint work with Jacob Tsimerman we study when the primitive
of a given algebraic function can be constructed using primitives
from some given finite set of algebraic functions, their inverses,
algebraic functions, and composition. When the given finite set is just {1/x}
this is the classical problem of "elementary integrability".
We establish some results, including a decision procedure for this problem.