Georgia Institute of Technology College of Sciences

Many problems in number theory boil down to bounding the size of a set contained in a certain set of residue classes mod $p$ for various sets of primes $p$; and then sieve methods are the primary tools for doing so.

Chambert-Loir and Ducros have introduced a theory of real-valued smooth differential forms on Berkovich spaces that play the role of smooth forms on complex varieties.  We compute the associated Dolbeault cohomology groups of curves by reducing to the case of metric graphs.  I'll introduce smooth forms on graphs, and explain how the theory in CLD has to be modified in order to get finite-dimensional cohomology groups.

The Chebotarev density theorem is a powerful tool in number theory, in part because it guarantees the existence of primes whose Frobenius lies in a given conjugacy class in a fixed Galois extension of number fields.  However, for some applications, it is necessary to know not just that such primes exist, but to additionally know something about their size, say in terms of the degree and discriminant of the extension.  In this talk, I'll discuss recent work with Peter Cho and Asif Zaman on a closely related problem, namely determining the least prime with a given cy

Let $f$ be an endomorphism of projective space defined over a number field.  When counting rational points ordered by a certain "canonical" height function attached to $f$, we encounter a mysterious asymptotic constant in the main term.  This constant is a product of local factors over the primes of bad reduction of $f$; and these local factors (which take the form of $v$-adic integrals) are rather difficult to calculate explicitly.  In this talk I will present my partial progress towards evaluating these integrals.  No knowledge of arithmetic dynamics will be assu

I will present some recent work with Debmalya Basak and Alexandru Zaharescu on potential improvements to the Siegel—Walfisz upper bound on the greatest real zero of a Dirichlet $L$-function.