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

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Ultrafilters formalize a generalized notion of convergence based on a prescribed idea of "largeness" for subsets of the natural numbers, and underlie constructions like ultraproducts. In the study of moduli spaces, they provide a clean way to encode degenerations and to establish uniformity results that are difficult to obtain using ordinary limits. This talk will discuss applications of ultrafilters to uniformity theorems in dynamics and arithmetic geometry.

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.

In this talk, we consider numbers with multiple close factorizations like $99990000 = 9999 \cdot 10000 = 9090 \cdot 11000$ and $3950100 = 1881 \cdot 2100 = 1890 \cdot 2090 = 1900 \cdot 2079$. We discuss optimal bounds on how close these factors can be relative to the size of the original numbers. It is related to the study of close lattice points on smooth curves.

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.