Georgia Institute of Technology College of Sciences

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 cycle type.  We develop a new, comparatively elementary approach for thinking about this problem that nevertheless frequently yields the strongest known results.  We obtain particularly strong results in the case that the Galois group is the symmetric group $S_n$ for some $n$, where determining the cycle type of a prime is equivalent to Chebotarev.

We explore the modular properties of generating functions for Hurwitz class numbers endowed with level structure. Our work is based on an inspection of the weight $\frac{1}{2}$ Maass--Eisenstein series of level $4N$ at its spectral point $s=\frac{3}{4}$, extending the work of Duke, Imamo\={g}lu and T\'{o}th in the level $4$ setting. We construct a higher level analogue of Zagier's Eisenstein series and a preimage under the $\xi_{\frac{1}{2}}$-operator.  We deduce a linear relation between the mock modular generating functions of the level $1$ and level $N$ Hurwitz class numbers, giving rise to a holomorphic modular form of weight $\frac{3}{2}$ and level $4N$ for $N > 1$ odd and square-free. Furthermore, we connect the aforementioned results to a regularized Siegel theta lift as well as a regularized Kudla--Millson theta lift for odd prime levels, which builds on earlier work by Bruinier, Funke and Imamo\={g}lu. I wil lbe discussing joint work with Andreas Mono and Ngoc Trinh Le.

Assuming the Riemann Hypothesis, Montgomery established results concerning the pair correlation of zeros of the Riemann zeta function. Rudnick and Sarnak extended these results for all level correlations of automorphic $L$-functions. We discover surfaces associated with the zeros of automorphic $L$-functions. In the case of pair correlation, the surface displays Gaussian behavior. For triple correlation, these structures exhibit characteristics of the Laplace and Chi-squared distributions, revealing an unexpected phase transition. This is joint work with Debmalya Basakand Alexandru Zaharescu.