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 $d>3$.