Department of Mathematics and Statistical Science Algebra Seminar Series
About the Seminar
The Department of Mathematics and Statistical Science Algebra Seminar is a graduate student led seminar series that provides graduate students and advanced undergraduate students in mathematics the opportunity to give professional mathematical talks. These talks are open to any who would like to attend, and their contents reflect a variety of mathematical disciplines.
Times and Locations
For Fall 2021, the Algebra Seminar meets on alternating Tuesdays, from 11:00am - 12:00pm, on Zoom (occasionally it will also be offered in-person in TLC 247; the in-person offerings will be noted in the schedule below).
- We will use the same Zoom Link each week.
- Check the schedule below for more information.
Speaker / Title / Abstract
August 24, 2021 at 10:00 am
Virtual meeting via Zoom
|September 7, 2021||
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.)
|September 14, 2021||
Dr. Jennifer Johnson-Leung
The shape of social contact networks and topological data analysis, Part Two
|September 21, 2021||
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.
|October 5, 2021||Cancelled|
|October 12, 2021||
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.
|October 19, 2021||
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.
|October 26, 2021||Dr. Alex Woo
Springer fibers, their cohomology rings, and symmetric group representations
The Springer correspondence states that the top degree piece of the cohomology ring of certain algebraic varieties known as Springer fibers give all the irreducible representations of the symmetric group. I will explain all the words in the previous sentence and also say something about the rest of the cohomology ring. This was originally work of Springer in the 1970s, though the explanations I give are simplifications by other mathematicians from the 1980s and 1990s.
|November 2, 2021||
Dr. Alex Woo
We introduce a family of varieties generalizing Springer fibers whose cohomology rings have a symmetric group action generalizing the Springer action. The top dimensional cohomology is an induced Specht module. In one case, the cohomology ring is the Haglund-Rhodes-Shimozono ring whose Frobenius characteristic is essentially the expression in the Delta conjecture at t=0.
This is joint work with Sean Griffin (UC-Davis) and Jake Levinson (Simon Fraser).
(This is practice for a talk at the Special Session on Combinatorial and Geometric Representation Theory at the virtual AMS Sectional on November 20. The above title and abstract were the ones submitted to the AMS.)
November 16, 2021
This talk is available in person and via Zoom. Attendees are encouraged to come in person to TLC 247 as there will be props involved in the presentation.
Viruses and Their Symmetries in Three and Six Dimensions
Spherical viruses use icosahedral symmetry to package their genetic material in a way that optimizes their stability and genetic economy. The important structures of these viruses can be mapped out in nested point sets displaying icosahedral symmetry at each individual radial level. These point sets can be generated in at least two ways. The first method is to use affine extensions of 3-dimensional representations of the icosahedral group, whereas the second uses projections of point sets derived from the symmetry group of the 6-dimensional hyperoctahedron. The latter is of particular interest as the hyperoctahedron contains other large intermediate subgroups bearing icosahedral symmetry, with the implication of this being that viruses actually contain symmetries beyond their 3D rotational symmetries, which is expressed by their tightly ordered radial mass distributions in both the protein capsid and their genetic cargo. Using the representation theory of symmetry groups and drawing inspiration from the study of quasicrystals, we examine the connection between shapes in higher dimensions and nested structures in 3-dimensions and illustrate how well these point sets correspond to important features of viral architecture. We further explore the relationship between these two schemes of constructing point sets and show surprising areas in which they differ.
(This work was created completed as part of an undergraduate Senior Individualized Project at Kalamazoo College under the advisory of Dr. David Wilson. This project was funded through a grant provided by the Sherman Fairchild Foundation).
|November 30, 2021||
Dr. Hirotachi Abo
Singular value decompositions: from an algebraic geometry perspective
The purpose of this talk is three-fold; the first is to review the concept of left and right singular vectors of rectangular matrices, the second is to discuss an algebro-geometric interpretation of left and right singular vectors and the third is to show how to extend this concept to tensors.