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.

Kiran Kedlaya of the University of California, San Diego

Title: Frobenius structures on hypergeometric equations

Abstract: Hypergeometric equations are a class of ordinary differential equations with strong ties to geometry and arithmetic. In particular, each hypergeometric equation parametrizes a family of motives with associated L-functions; the minimal example of this is the Gaussian hypergeometric equation corresponding to the Legendre family of elliptic curves. We sketch an algorithm, based on work of Dwork, to compute these L-functions using the existence of p-adic analytic “Frobenius structures” on the equation. This is expected to be useful for building tables of hypergeometric L functions for the LMFDB.

For a positive integer $n$, we compute the shape of a totally real multiquadratic extension of degree $2^n$ in which the prime $2$ does not ramify. From this calculation, we see that the shape of such a number field is parametrized by the generators of its $2^n-1$ quadratic subfields. Restricting to the case $n=3$, we use this parametrization to count the number of triquadratic extensions of bounded discriminant and bounded shape parameters. We then show that, as the discriminant goes to infinity, these shapes become equidistributed in a regularized sense in the subset of the space of shapes of rank $7$ lattices that contains them.

Taylor Markham of the University of Calgary

Title: Integer Factorization

Abstract: The security of many modern day cryptosystems are impacted by the fact that it is computationally difficult to factor large numbers. This talk will give an introduction to the general number field sieve, which is currently the most efficient algorithm for factoring large integers.

Speaker: Stuart White (U. of Oxford)

Title: Amenable Operator Algebras

Abstract: Operator algebras arise as suitably closed subalgebras of the bounded operators on a Hilbert space. They come in two distinct types: von Neumann algebras which have the flavour of measure theory, and C*-algebras which have the flavour of topology. In the 1970’s Alain Connes obtained a deep structural theorem for amenable von Neumann algebras, leading to a complete classification of these objects. For the last 25 years the Elliott classification programme has been seeking a corresponding result for simple amenable C*-algebras, and now, though the efforts of numerous researchers worldwide, we have a definitive classification theorem. In this talk, I’ll explain what this theorem says, and the analogies it makes to Connes work, using examples from groups and dynamics as motivation. I won’t assume any prior exposure to operator algebras or functional analysis.

Speaker: Asaf Hadari (UHM)

Title: In search of a representation theory of mapping class groups.

Abstract:

Mapping class groups are nearly ubiquitous in low dimensional topology. They’ve been studied for over a century. Various results discovered during the past few decades it has become quite clear that there is much to gain by studying them via their linear representations.

Somewhat surprisingly, many such representations are known. Unfortunately, until recently there was almost no representation theory, that is – no underlying structure that allows you to say anything about the class of representations as a whole. It is precisely such an understanding that is necessary for studying mapping class groups.

In this talk I’ll talk about the major source of representations of mapping class groups, and talk about new results in their emerging representation theory.

Title: Analysis of partisan gerrymandering tools in advance of the US 2020 census

Abstract: Over the last decade, mapmakers have precisely gerrymandered political districts for the benefit of their party. In response, political scientists and mathematicians have more extensively investigated tools to quantify and understand the mathematical structure of redistricting problems. Two primary tools for determining whether a particular redistricting plan is fair are partisan-gerrymandering metrics and stochastic sampling algorithms. In this work we explore the Declination, a new metric intended to detect partisan gerrymandering. Within out analyses, we show that Declination cannot detect all forms of packing and cracking, and we compare the Declination to the Efficiency Gap. We show that these two metrics can behave quite differently, and give explicit examples where that occurs.