Georgia Institute of Technology College of Sciences

Given an action of a finite group $G$ on a complex vector space $V$, the Chevalley-Shephard-Todd Theorem gives a beautiful characterization for when the quotient variety $V/G$ is smooth. In his 1986 ICM address, Popov asked whether this criterion could be extended to the case of Lie groups. I will discuss my contributions to this problem and some intriguing questions in combinatorics that this raises. This is based on joint work with Dan Edidin.

I will introduce briefly the notion of Berkovich analytic spaces and certain metric graphs associated to them called the skeleton. Then we will describe divisors on metric graphs and a lifting theorem that allows us to find tropicalizations of curves in P^3. This is joint work with Philipp Jell.

Let K be a non-trivially valued non-Archimedean field, R its valuation subring. A formal Gubler model is a formal R-scheme that comes from a polyhedral decomposition of a tropical variety. In this talk, I will present joint work with Sam Payne in which we show that any formal model of any compact analytic domain V inside a (not necessarily projective) K-variety X can be dominated by a formal Gubler model that extends to a model of X. This result plays a central role in our work on "structure sheaves" on tropicalizations and our work on adic tropicalization.

The talk reports on joint work with Wayne Raskind and concerns the conjectural definition of a new type of regulator map into a quotient of an algebraic torus by a discrete subgroup, that should fit in "refined" Beilinson type conjectures, exteding special cases considered by Gross and Mazur-Tate.The construction applies to a smooth complete variety over a p-adic field K which has totally degenerate reduction, a technical term roughly saying that cycles acount for the entire etale cohomology of each component of the special fiber.

I will explain how to explicitly compute the syntomic regulator for varieties over $p$-adic fields, recently developed by Nekovar and Niziol, in terms of Vologodsky integration. The formulas are the same as in the good reduction case that I found almost 20 years ago. The two key ingrediants are the understanding of Vologodsky integration in terms of Coleman integration developed in my work with Zerbes and techniques for understanding the log-syntomic regulators for curves with semi-stable reduction in terms of the smooth locus.

We determine the average size of the $\phi$-Selmer group in any quadratic twist family of abelian varieties having an isogeny $\phi$ of degree 3 over any number field. This has several applications towards the rank statistics in such families of quadratic twists. For example, it yields the first known quadratic twist families of absolutely simple abelian varieties over $\mathbb{Q}$, of dimension greater than one, for which the average rank is bounded; in fact, we obtain such twist families in arbitrarily large dimension.

In this talk, I will summarize forthcoming work with Griffin, Ono, and Zagier. In 1927 Pólya proved that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for Riemann's Xi-function. This hyperbolicity has been proved for degrees $d\leq 3$. We obtain an arbitrary precision asymptotic formula for the derivatives $\Xi^{(2n)}(0)$, which allows us to prove thehyperbolicity of 100% of the Jensen polynomials of each degree. We obtain a general theorem which models such polynomials by Hermite polynomials.