unQVNTS (Vermont)

In 2015, Adiprasito, Huh, and Katz settled the famous Heron-Rota-Welsh conjecture that the absolute value of the coefficients of the characteristic polynomial of a matroid are log-concave. The approach of AHK was to show that these coefficients can be interpreted as intersection numbers in the Chow ring of a matroid previously introduced by Feichtner and Yuzvinsky. They then establish a Kahler package for the Chow ring of a matroid: Poincare duality, the Hard Lefschetz theorem, and the Hodge-Riemann relations, and show that the desired log-concavity follows from the degree 1 part of the Hodge-Riemann relations (the Hodge index theorem). The AHK proof of the Kahler package for matroids is inspired by earlier work of McMullen on simple polytopes and thus utilizes a notion of "flipping" which provides a fine interpolation between matroids and projective space. For achieving their goal, AHK prove that the Kahler package respects flipping. This impressive program comes at a cost of working with more general objects than matroids which can obscure some combinatorial and geometric information. I will describe joint work with Chris Eur and Connor Simpson where we introduce a new presentation for the Chow ring of a matroid and apply this presentation to obtain a new proof of the Hodge index theorem for matroids which eschews the use of flipping and thus does not leave the realm of matroids.

A large enough quantum computer will break most of the currently deployed public-key cryptography, like RSA or elliptic curve cryptography. To prepare for this, NIST is running a public process to evaluate and standardize one or more of several proposed 'post-quantum' cryptosystems. These cryptosystems are thought to be secure even against an adversary with a quantum computer. One proposal, SIKE, bases its security on the hardness of computing isogenies between supersingular elliptic curves. In this talk, I will give a brief introduction to isogeny-based cryptography, its connection to the problem of computing endomorphism rings of supersingular elliptic curves, and sketch a new algorithm for computing these endomorphism rings. This is joint work with Eisentraeger, Hallgren, Leonardi, and Park.

All Boolean algebras satisfy the finite distributive laws, as part of their definition. However each atomless (complete) Boolean algebra will fail to satisfy some infinite distributive law. I will describe the relationship between some of these laws, in particular how they relate to the existence of a model of ZFC with a measurable cardinal. Then I will present a result of mine about how one of these distributive laws implies another.

This talk will explore some of the recent advances related to Goldfeld's Conjecture due to Daniel Kriz and Chao Li. Notably, these results draw upon Iwasawa theory in a rather remarkable way. We'll discuss the approach of Kriz-Li, putting it in the context of more classical results in Iwasawa theory, and we'll discuss some future lines of inquiry suggested by their work.

Recently, there has been an effort to study the torsion subgroups of elliptic curves defined over Q, base-extended to infinite extensions of Q. In this talk, we will start with a survey of the necessary background material and introduce a class of infinite extensions where this question is well-defined. Afterwards, we will give a survey of what is known and the techniques used to prove these results. If time permits, we will discuss an ongoing project.

To an algebraic curve C over the complex numbers one can associate a non-negative integer g, the genus, as a measure of its complexity. One can also associate to C, via complex analysis, a g by g symmetric matrix Omega called a period matrix. Because of the natural relation between C and Omega, one can obtain information of one by studying the other. Therefore, it makes sense to consider the inverse problem: Given a period matrix Omega, can we compute a model for the associated curve C?
In this talk, we will give a method that deals with this problem in the case of some superelliptic curves, in particular the genus-3 family of Picard curves y^3 = a_4 x^4 + ... + a_1 x + a_0, and the genus-6 cyclic plane quintic curves y^5 = a_5 x^5 + ... + a_1 x + a_0.