## Overview

- My principal area of interest is the representation theory of algebraic groups. One of my long-standing goals is to apply the method of coadjoint orbits (a philosophy for classifying unitary representations of real and complex Lie groups) to p-adic algebraic groups. See below for a "non-technical" description of this area.
- Over the past several years, I have been exploring branching rules, specifically, decompositions of representations of reductive p-adic groups upon restriction to a maximal compact subgroup. The goal is to gain insight into the structure of these representations, and their types. We have recently made major headway on the classification of types for tame supercuspidal representations, using novel methods based in Bruhat-Tits theory.
- My secondary interests are in applications of representation theory and algebra to coding theory and cryptology. Most recently my focus is on post-quantum cryptography, which concerns classical cryptographic schemes that are secure against both classical and quantum adversaries. In this work I am part of the group QUASAR at the University of Ottawa (Quantum Security via methods from Algebra and Representation theory), that was initially created through a 2019 grant from the Tri-Council New Frontiers in Research Fund.

## A non-technical description of my research area in pure mathematics

(Summer 2018) Logan Toll, an aspiring mathematician, is doing a series of podcast interviews with research mathematicians under the broad title of "Group Theory"; our conversation about Number Theory and Cryptography is on his YouTube Channel.

My field of study is in an area of Pure Mathematics called the Representation Theory of Lie Groups.

A Lie group is an abstract mathematical object which embodies the great many symmetries of such objects as circles or spheres. It turns out, that what we want to study is not so much the groups themselves, as the way they act on linear spaces -- their representations. Physicists, for example, are interested in certain representations of Lie groups to explain electron spin in an atom. One can picture representations as rotations of the plane, for example, though in general the spaces are infinite dimensional, and the actions are much more complicated.

We don't yet know how many of these representations there are, or what they all look like -- and that is the big open problem in representation theory today: the classification and construction of representations.

One promising approach is a geometric one, called the Orbit Method. It has been studied extensively, and is giving an answer to this classification problem for many Lie groups over the real and complex numbers (the numbers used most commonly by scientists today).

I'm interested in applying this Orbit Method to the infinitely more fascinating case of p-adic numbers.

The p-adic numbers have a lot of the same arithmetic properties as the real numbers, but look much different (like a Cantor set on the real line). Mathematically, they provide a complement to the real numbers, and together with them provide a more complete picture of the whole. Moreover, some physicists now believe that p-adic numbers can describe what's going on at the tiniest level of the atom, where notions of real distance no longer make sense.

So I hope to understand the representation theory of p-adic Lie groups, both to complement the corresponding problem over the real numbers, and to provide some tools for physicists to use eventually, in their ongoing serach for the deepest secrets of the atom.

## Potential research projects

I am interested in applications of representation theory and p-adic groups to a variety of areas, and am looking for interested students to pursue these topics. I am also willing to supervise undergraduate students through the NSERC USRA or the Work-Study programs. The following are some sample problems you might work on in graduate studies (but note that I do not update this page often.)

#### Characterizing representations via nilpotent coadjoint orbits

Background to acquire: algebraic groups, reflection groups, harmonic analysis, p-adic numbers.

Project: Recent work of DeBacker gave a classification of nilpotent orbits of an algebraic group via objects in the associated Bruhat-Tits building. This in turn is crucially related to our understanding the local character expansion of a representation in terms of nilpotent orbits, in a neighbourhood of the identity. Some recent master's projects have related to describing this classification concretely. Another master's project, which is more combinatorial in nature, is to give an enumeration of the different associativity classes of r-facets (where associativity is defined via essentially linear algebra). A doctoral project would be to characterize special, admissible or other classes of nilpotent orbits within the parameters of DeBacker's classification, and to elaborate on the construction of representations of compact open subgroups of p-adic groups built from nilpotent orbits, using the tools from the Bruhat-Tits building.

#### Branching Rules for Representations of p-adic groups

Background to acquire: algebraic groups, representation theory of finite groups, p-adic numbers.

Overview: determining how an irreducible admissible representation of a p-adic group decomposes upon restriction to a maximal compact subgroup. Doctoral project: study the equivalence relation on irreducibles of the maximal compact generated by: two reps are equivalent if they occur in the restriction of the same irreducible of the group, in general or for low-rank examples. More generally there is lots one could ask about any particular group, where things can be worked out very explicitly.

#### Algebra in Cryptography

Background to acquire: Algebra and Algebraic Geometry or Number theory, Discrete Mathematics.

Undergraduate or Master's Project: Cryptography is an exciting and dynamic area of research, which draws on many different areas in mathematics and engineering. Its importance to the successful functioning of our modern electronic economy cannot be overstated. Contributions to this field include anything from developing new ideas for cryptographic schemes or attacks, to testing and improving those ideas which appear in the current literature. Some recent students have worked on variants of NTRU, group-based cryptography, and code-based cryptosystems.