Research Program

Broadly, I am interested in number theory and its applications and connections to other area of mathematics and science. Number theory is the theoretical study of arithmetic, that is, addition and multiplication of integers, as well as higher arithmetic of other number rings. Research problems in number theory include, for example, characterizing integer solutions to algebraic equations and understanding the statistical distribution of prime numbers. Long treated as a “pure” field of mathematics, number theory has found many modern applications to computer science and physics, where important numbers are often “quantized,” that is, forced to take discrete or integer values.

More specifically, I am interested in analytic formulas for algebraic objects. Classic examples of such formulas include the Dirichlet class number formula and the formula for elliptic units in an imaginary quadratic field in terms of theta functions. Both of those (proven) examples occur special cases of the Stark conjectures, which connects the leading term of an Artin -function at to a generalized regulator. In some cases, the Stark conjectures can be used to numerically find “Stark units” that generate abelian extensions of number fields. The Stark conjectures thus provide a framework for trying to build an explicit class field theory (that is, solving Hilbert’s 12th Problem).

I am especially interested in unconventional approaches to the Stark conjectures and explicit class field theory. Conventional approaches aim to relate special -values to celebrated arithmetic-geometric objects like automorphic forms and Shimura varieties. Unconventional approaches aim to find algebraic structure via relationships to more exotic objects, such as quantum designs, conformal field theories, and noncommutative rings.