Representation and structure theory of p-adic groups via Bruhat-Tits buildings, and applications to cryptography

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Algebra is a fundamental domain of the mathematical sciences. Its richness manifests itself in its diversity of applications, from representation theory to communications theory. The diversity of the projects in my research program is a reflection of the universality of the language and tools of algebra. A first, major part of my research program concerns the representation theory of p-adic groups. Representation theory seeks to understand groups by linearizing them, that is, characterizing their actions on linear vectors spaces over the (well-understood) complex numbers. Here, the groups are matrix groups over the p-adic numbers, depending on a prime p.  Representations of p-adic groups are fundamental to some of the biggest open questions in number theory today, including the so-called Langlands program. My research seeks to deepen our understanding by determining how these representations decompose when they are restricted to a smaller subgroup. For example, if the subgroup is a maximal compact open subgroup, then there are infinitely many finite-dimensional components, which are unknown except in very few cases. One of the exciting open problems in this area is to discover when two apparently unrelated representations of the group share common constituents upon restriction, suggesting a deeper connection between them. A second aspect of my research relates to the action of a p-adic group on its Lie algebra, which is a vector space over the p-adic numbers. This action decomposes the Lie algebra into orbits, of which the finitely-many nilpotent orbits are the most interesting. DeBacker has recently proven a theoretical classification of these orbits in terms of a combinatorial and geometric object closely related to the p-adic group, called the Bruhat-Tits building. The important open problem which I am working on is to realize this classification concretely, by producing representatives of each orbit (for a fixed group), and deriving the dimension and attributes of the various nilpotent orbits, including their proximity to each other, from the DeBacker parameters. This has myriad applications; for example, the Harish-Chandra—Howe character formula parametrizes (characters of) representations by what are essentially weighted sums over a set of nilpotent orbits. Finally, a third aspect of my research, arising in part from my expertise with groups and their associated lattices, is mathematical cryptography. Cryptography is the art and science of obfuscating messages.  Our research in this area centers on analyzing and extending cryptographic protocols based on algebraic systems---such as NTRU, elliptic curve cryptography and homomorphic encryption systems. The goal is to gain a deeper understanding of a cryptographic algorithm, and thus of its potential unintended loopholes, by learning how it changes as a function of the algebraic object on which it is based.