Number Theory

Consider a collection of forms of odd degree with rational coefficients. Birch proved in 1957 that if the number of variables is sufficiently large, then the forms must have a nontrivial rational zero. The bounds resulting from Birch's proof, however, are so large that he has described…

I will explain a construction of p-adic variations of twistor structure and how it relates to other recent developments in relative p-adic Hodge theory including: geometric Sen theory, the p-adic Simpson correspondence, and analytic prismatization. After introducing the basic theory and…

The "fundamental curve" of arithmetic over nonarchimedean local fields is the Fargues–Fontaine curve. You'd think the "fundamental curve" of arithmetic over global function fields is... just the associated curve over F_q. In this talk, I will instead advocate for a different "curve", in the…

Despite a lot of progress, the question of whether all the finite groups PSL(n,q) of all n by n matrices with coefficients in F_q and of determinant 1, modulo center, occur as Galois groups of a Galois extension of Q remains open. We study the cohomology of curves with an automorphism with F_p-…

Abstract:   In this talk we review results on several types of harmonic weak Maass forms that are related to newforms of even integer weight.  

Starting with an integral weight newform, we briefly present different constructions of integral weight harmonic weak Maass forms via (…

The sum-product phenomenon concerns an incongruence between additive and multiplicative structure over finite sets of integers. Results of this type often involve showing that a finite set of integers either gives rise to many pairwise sums or many pairwise products. Such results have been…

For smooth manifolds, the Gysin map of a closed immersion is defined as the cohomology applied on the Pontryagin–Thom collapse map, which collapses the ambient manifold to the one-point compactification of the tubular neighborhood of the closed submanifold. In this talk, I will present a version…

In the 1980s Beilinson conjectured the existence of motivic cohomology, together with certain precise expected relations to algebraic K-theory, étale cohomology, algebraic cycles, and the category of mixed motives. Apart from the still conjectural category of mixed motives (and its consequences…

In 1972, Borel proved that every holomorphic map from a product of punctured unit disks to a complex Shimura variety extends to a map from a product of disks to its Baily--Borel compactification.  Recently, Oswal--Shankar--Zhu and Patel proved the corresponding p-adic statement over…

Are rational points distributed randomly near curved manifolds? After introducing a simple random model, we focus on this question in hyperbolic regions, which arise naturally in multiplicative Diophantine approximation. We then sketch how tools from homogeneous dynamics and harmonic…