Department of Mathematics and Statistical Science Algebra Seminar Series

 August 24, 2021 at 10:00 am

 Virtual meeting via Zoom

Dr. Jennifer Johnson-Leung

The shape of social contact networks and topological data analysis

My motivation for using topological data analysis has arisen in the context of understanding community-level susceptibility to infectious disease transmission via network and agent-based simulations. After a brief overview of this motivation, I will explain the construction of the Vietoris-Rips complex associated to a discrete metric space and the persistence diagrams that are constructed from it. We will then consider metrics on the space of persistence diagrams and look at a continuous embedding of this space, via tropical polynomials, into Euclidean space. I plan to conclude the talk with some speculation about separating diagrams under this embedding.

(This is ongoing work with Ben Ridenhour and Trevor Griffin. Erich Seamon and Tyler Meadows also contributed to the network modeling components. The work was funded through the National Institute Of General Medical Sciences of the National Institutes of Health under Award Number P20GM104420.)

Dr. Jennifer Johnson-Leung

The shape of social contact networks and topological data analysis, Part Two

Jordan Hardy

Explicit Definitions of Nonprincipal Polarizations on Abelian Surfaces with Complex Multiplication

The classical theory of complex multiplication is a beautiful, explicit theory by which the class field theory of a quadratic imaginary field can be developed explicitly. An elliptic curve is said to have complex multiplication if it has an endomorphism ring which is strictly bigger than the integers. In this case, it is isomorphic to an order of a quadratic imaginary number field. The class fields of these quadratic imaginary fields are generated by the coordinates of torsion points on the elliptic curves with complex multiplication. This theory elegantly describes all of the class field theory of these fields.

Elliptic curves are the dimension one case of the broader theory of abelian varieties, and these ideas can be generalized to higher dimensional abelian varieties, though with limitations. Abelian varieties A of dimension g have endomorphism rings which are Z-modules of rank at most 2g. In the case when the rank is 2g, the endomorphism ring is isomorphic to an order O of a totally imaginary number field K called a CM field. In this case, A is said to have complex multiplication by O.

An abelian variety of dimension greater than 1 comes with an extra structure called its polarization. In the case of abelian surfaces these can be classified by pairs of integers (m, n) with m|n. If the polarization is of type (1, 1), it has been well-studied, for instance by Shimura and Taniyama, when we can generate abelian surfaces with complex multiplication, and how to use these to generate class fields of CM fields. In this talk we will learn how to extend these ideas to abelian surfaces with non principal polarizations.

Josh Parker

On Left Coset Representatives for Paramodular Hecke Operators

In this talk I will give a brief overview of the work that I have done on left coset representatives for Hecke operators acting on Siegel paramodular forms. In particular I will present several results on the number of left coset representatives for two important Hecke operators that act on paramodular forms at primes that divide the level of the form exactly once. The focus of the talk will be on the use of lattices in counting the number of unique coset representatives for each Hecke operator.

Dr. Brooks Roberts

Calculating Hecke eigenvalues of paramodular newforms

In this talk we will describe some formulas relating the Hecke eigenvalues and Fourier coefficients of Siegel modular newforms defined with respect to the paramodular groups. These formulas apply to the case when the square of the prime divides the paramodular level. In this case the usual formulas for the Hecke operators are not of upper-block type and cannot be applied to the Fourier expansion. Using local theory, we developed a new approach to this problem. We will describe the resulting formulas and explain how we used a computer to verify that they indeed hold for known examples. This is joint work with Jennifer Johnson-Leung and Ralf Schmidt.