The main objects of study in this project are word maps of groups, that is, maps that are obtained by substitution into a fixed element in the free group on d generators by elements of a given group G. The first part of the project is a study of distribution of values of general word maps in groups. The broad goal of this part is to give quantitative results about how random these word maps can be. More specifically, the goal is to prove a bound on the Fourier coefficients of such distributions in finite and p-adic groups. These Fourier coefficients are non-commutative analogs of Gauss sums. The theme of the second part of this project is to study the distribution of values of certain words built from commutators and its connections to the moduli space of G-local systems on a curve, the representation theories of lattices in higher rank algebraic groups, and a new kind of Topological Field Theory.
Groups are mathematical objects measuring (and expressing) symmetry. They are fundamental objects of mathematics and are ubiquitous in many other sciences. This project has two parts, each dealing with equations in groups. The first part deals with general equations, but from a quantitative angle: instead of asking whether an equation has a solution, we ask how many solutions the equation has. The second part of the project will focus on a specific equation, for which the answer to the question of how many solutions there are (for different groups) is related to representation varieties, which are well-studied objects from mathematical physics and number theory. The goal of this part is to use this connection to answer questions about groups using representation varieties and vice versa.
