Combinatorics and Number Theory Seminar

Fall 2019: Fariborz Maseeh Hall (FMH) 462, every other Monday or so at 3:15pm–4:15pm and one special Wednesday at 3:30pm–4:30pm (all quarters)

October 7, 2019

Daniel Taylor-Rodriguez, Portland State University
The matryoshka doll prior for Bayesian regression

We present a (fairly) general construction of model space priors with a focus on regression problems. The graph of the model space can be used to build a distribution that views each model as a “local” null hypothesis whose alternatives are the set of models that nest it. This construction provides a natural isomorphism of model spaces induced by conditioning on a particular model and leads to the Poisson distribution as the unique limiting prior over model dimension.

(Joint work with Andrew Womack and Claudio Fuentes.)

October 14, 2019

Sean Griffin, University of Washington
Labeled binary trees, subarrangements of the Catalan arrangements, and Schur positivity

In 1995, Gessel introduced a multivariate formal power series G tracking the distribution of ascents and descents in labeled binary trees. In addition to showing that G is a symmetric function, he conjectured that G is Schur positive. In this talk, we'll see how to expand G positively in terms of ribbon Schur functions. Moreover, we'll see how certain specializations of G relate to actions on hyperplane arrangements. As an application of our work, we get a proof of gamma-positivity of the distribution of right edges over the set of local binary search trees.

October 28, 2019

Elisa Bellah, University of Oregon
Norm form equations and linear divisibility sequences

Finding integer solutions to norm form equations is a classic Diophantine problem. Using the units of the associated coefficient ring, we can produce sequences of solutions to these equations. It turns out that such a sequence can be written as a tuple of integer linear recurrence sequences, each with characteristic polynomial equal to the minimal polynomial of our unit. After an expository introduction to norm form equations and linear recurrence sequences, I will discuss how one might use this observation to investigate when these sequences satisfy a certain divisibility property.

November 13, 2019 (3:30pm–4:30pm)

Andrei Jorza, University of Notre Dame
Computing with modular forms

In this talk geared towards non-experts I will describe about various computational aspects of modular forms and elliptic curves. Questions such as “How does one enumerate elliptic curves and modular forms?” and “What are the arithmetic properties of the Fourier coefficients of modular forms?” remarkably rely on a collection of commuting linear maps called Hecke operators, whose eigenvectors are intimately connected to algebraic geometry and arithmetic. Towards the end of the talk I will explain some recent results with Chiriac on the Newton polygon of Hecke operators. I will focus on examples and I won't assume prior knowledge of modular forms.

December 2, 2019

Liubomir Chiriac, Portland State University
Summing Fourier coefficients over polynomials values

Functions of number-theoretic interest are often studied on average. This talk is concerned with mean values of Fourier coefficients of modular forms over polynomials. While much progress has been done in this direction for polynomials of degree at most two, rather little is known beyond that. Here we will present an approach to obtain upper bounds for sums involving polynomials of arbitrary degree. We will also discuss specific examples to illustrate our results.