Speaker: Rohit Nagpal (U. Wisconsin)

Title: S_{infty}-equivariant modules over polynomial rings in infinitely many variables

Abstract: Let R be the polynomial ring k[x_1, x_2, ldots] in countably many variables. Cohen proved that S_{infty}-stable ideals in R satisfy the ascending chain condition. This makes the category of smooth equivariant R-modules a noetherian category. Smooth modules over R appear naturally in algebraic statistics and chemistry, and so it makes sense to study this category in detail. The first step in this direction is to understand its spectrum. In this talk, we describe this spectrum. In particular, we show that the S_{infty}-stable ideal I_n generated by n-variable discriminants is in this spectrum. We show that every nonzero S_{infty}-stable ideal must contain I_n for some large n, and so these ideals are of primary interest. We also mention some new algebraic properties of discriminants. This is a part of an ongoing project with Andrew Snowden.

Speaker: Yash Lodha (Ecole Polytechnique Federale de Lausanne)

Title: Group actions on 1-manifolds: A plethora of treasures.

Abstract: The study of group actions on 1-manifolds has seen some striking developments in recent years. In this talk I will describe some concrete examples that illustrate the richness of the theory. These groups are relatives of the famous groups F and T discovered by Richard Thompson in the 1970s.

Speaker: Yago Antolin (U. Autonoma de Madrid)

Title: Growth in graphs with symmetry

Abstract: In the early 1980′s Jim Cannon showed that the Cayley graph of a

group acting properly co-compactly and by isometries on a hyperbolic

space had a growth function (i.e. counting how many vertices at distance

n from a base point) satisfied a linear recursion. This property is now

known as rational growth of the graph. Cannon’s ideas were fundamental

for the development of the theory of automatic groups. In this talk I

will review Cannon’s ideas and I will explain how they can be used to

show the rationality of other growth functions. Part of the talk will

be based on joint works with L. Ciobanu.

Formalization of a Deontic Logic Theorem in the Isabelle Proof Assistant

<a href=”https://math.hawaii.edu/home/depart/theses/MA_2019_Fennick.pdf”>Draft project report</a>

Title: Single- and Multivariable (φ, Γ)-Modules and Galois Representations

Abstract: I will introduce the notion of a (single-variable) (φ, Γ)-module and explain the relationship of (φ, Γ)-modules to representations of Gal(Q̅_p | Q_p), the absolute Galois group of the p-adic numbers. I will then describe joint work with Kiran Kedlaya and Gergely Zábrádi which extends this relationship to multivariate (φ, Γ)-modules.

Let $K$ be a number field. We will show that any bicritical polynomial $f(z) in K[z]$ is conjugate to a polynomial of the form $amathcal{B}_{d,k}(z) +c in K[z]$ where $mathcal{B}_{d,k}(z)$ is a normalized single-cycle Belyi map with combinatorial type $(d; d-k, k+1, d)$. We use results of Ingram to determine height bounds on pairs $(a,c)$ such that $amathcal{B}_{d,k}(z) +c$ is post-critically finite. Using these height bounds, we completely describe the set of post-critically finite cubic polynomials over $Q$, up to conjugacy over $Q$. We give partial results for post-critically finite polynomials over $Q$ of arbitrary degree $d>3$.

Alina Bucur of the University of California, San Diego

Title: Effective Sato-Tate under GRH

Abstract: Based on the Lagarias-Odlyzko effectivization of the Chebotarev density theorem, Kumar Murty gave an effective version of the Sato-Tate conjecture for an elliptic curve conditional on the analytic continuation and the Riemann hypothesis for all the symmetric power L-functions. Using similar techniques, Kedlaya and I obtained a similar conditional effectivization of the generalized Sato-Tate conjecture for an arbitrary motive. As an application, we obtained a conditional upper bound of the form O((logN)^2(loglogN)^2) for the smallest prime at which two given rational elliptic curves with conductor at most N have Frobenius traces of opposite sign. In this talk, I will discuss how to improve this bound to the best possible in terms of N and under slightly weaker assumptions. Our new approach extends to abelian varieties. This is joint work with Kiran Kedlaya and Francesc Fite.