Bruhat-Tits theory in the representations of p-adic groups, and post-quantum cryptography

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My research program centers on an overarching vision on how to apply new ideas in algebra to solve problems in both pure and applied mathematics. Primary theme: representation theory of p-adic groups. This area of mathematics has become a focal point of research largely due to the celebrated Langlands program (1970s-). This is a vast series of conjectures that bring the power of representation theory to bear on some of the most fundamental questions about the nature of numbers. Here, the p-adic numbers, for any prime p, enter the picture as the myriad of different lenses that bring this theory into focus. My work centers on branching rules: breaking up complex representations into an infinite collection of smaller pieces, and then identifying those that retain answers to the original number-theoretic questions. What makes my research exciting is my discovery and development of new tools, using a combinatorial object called the Bruhat-Tits building of G, to give a geometric structure to this infinite collection and thus potentially, to the number-theoretic questions themselves. This is a field rich in fascinating open questions that can be made accessible at multiple levels, and is ideal for training my students in abstraction, critical thinking, and conducting research. Secondary theme: post-quantum mathematical cryptography. Our internet economy relies utterly on public-key infrastructure (PKI), which is the mathematical cryptography that allows us to initiate secure communications over an insecure channel. As quantum computers become a reality, the vast majority of existing PKI will be rendered obsolete because they are based on problems vulnerable to attack by quantum algorithms. Post-quantum cryptography (PQC) refers to a class of cryptosystems intended to be secure against all known attacks, using either classical and quantum computers. The development and thorough scrutiny of these mathematically sophisticated algorithms is arguably the most urgent and compelling mathematical challenge of the coming decade. My work is focussed on meeting this challenge. In the past, I have trained students in the use of algebraic methods to analyse, evaluate or attack new PQC. Going forward, I will additionally embark of the development of quantum algorithms, accelerated by an interdisciplinary project through New Frontiers in quantum information and quantum computing. One particular new direction of my research relates to the so-called Hidden Subgroup Problem, which is strongly related to my first theme (representation theory) and is, excitingly, a source for both quantum attacks and new PQC. Although high-risk to its core, research in PQC serves students extremely well, both as an entry to more theoretical mathematical research, and as an excellent opportunity to develop highly marketable skills in coding and cybersecurity.