# Math Department Colloquium

The colloquium features recent research in the mathematical and statistical sciences. The colloquia are scheduled for Tuesdays from 4pm-5pm, unless noted otherwise. This semester all colloquia will be held virtually on Zoom.

If you wish to be added to the department colloquium mailing list, or if you wish to give a colloquium talk, please contact the organizer Grady Wright.

Archive of past math department colloquium abstracts

## Schedule for 2020–2021

### Spring 2021

#### March 30 – Asif Zaman, University of Toronto

Title: Small primes on average in the Chebotarev density theorem

Abstract: First established in 1926, the Chebotarev density theorem is a broad generalization of the prime number theorem describing the asymptotic distribution of prime splitting behavior in number fields. It has many applications to modular forms, elliptic curves, binary quadratic forms, and much more. The most interesting applications require small primes with a prescribed splitting behaviour. While existing unconditional results are useful in some contexts, they cannot usually produce small enough primes. By averaging over a family of number fields, you might hope to prove a stronger form of the Chebotarev density theorem “on average” but, until recently, there was little progress in this direction. I will introduce the theorem with some basic examples, describe a bit more of its history, and present recent results on averaging which unconditionally establish estimates commensurate with the Grand Riemann Hypothesis. This talk is based on joint work with Robert Lemke Oliver and Jesse Thorner.

### Fall 2020

#### September 8 – John Clemens, Boise State University

Title: Definable cardinalities and complexity of classification problems

Abstract: The cardinalities of infinite sets can be compared via injections: We say a set B is at least as large as a set A if there is an injection from A into B. In descriptive set theory, we can consider a more nuanced comparison, where we require the injection be suitably definable, e.g., not obtained using the Axiom of Choice. Typically this involves comparing quotient spaces of definable equivalence relation on Polish spaces using injections arising from Borel-measurable maps on the underlying spaces. This presents a rich structure of different definable cardinalities. One can study this structure for its own sake, as well as use it to gauge the complexity of natural classification problems from various areas of mathematics. I will survey some recent work showing the richness of this structure of definable cardinalities, as well as applications to classification problems in areas such as analysis and dynamical systems.

#### September 11 (Friday, 3:00-4:00pm) – Zach Teitler, Boise State University

Title: Recent advances in Waring rank and symmetric tensor complexity

Abstract: The Waring rank of a homogeneous form is the number of terms needed to write the form as a sum of powers of linear forms. I will give an introduction and overview of the subject,

centered on complexity of the determinant and permanent. I will describe progress in the last ten years on lower and upper bounds for Waring rank; on Strassen’s conjecture, which asserts that Waring rank is additive; and the new study of high-rank loci, initiated in the last 4 years.

#### September 15 – Jens Harlander, Boise State University

Title: 2-Complexes

Abstract: Cell complexes are topological spaces that are formed by gluing together Euclidean balls (cells). An n-complex is a cell complex that does contain cells of dimension n, but not higher dimensional ones. Graphs are 1-dimensional cell complexes, the vertices being the 0-cells and the edges being the 1-cells. My talk will be about 2-complexes. On one hand 2-complexes are visually approachable, being just one dimension away from graphs. On the other hand dimension 2 does not place restrictions on the fundamental group, an invariant of central importance in algebraic topology.

I will talk about the status of 2-dimensional conjectures concerning homotopy classification, cohomological dimension, and asphericity.

#### September 22 – Sasha Wang, Boise State University

Title: An ACE Model for Mathematics Instruction

Abstract: An ACE (Applying, Connecting, Experiencing) model for mathematics instruction is designed to create an active learning environment engaging prospective elementary teachers mathematics learning by (1) *Applying* learned knowledge and skills to the practice of teaching; (2) *Connecting* crosscutting concepts through STEM inquiries and practices; and (3) *Experiencing* community-based experiential learning to increase their attitudes and beliefs toward STEM education. The potential of the ACE model to leverage students learning experience, to build a knowledge base for identifying best practices, and to promote Science, Technology, Engineering, and Mathematics (STEM) learning in our community will be discussed.

#### October 13 – Bryan Quaife, Florida State University

Title: Transport in Viscous Eroded Media

Abstract: Flow in a fixed porous media is synonymous in many geophysical, medical, and industrial applications. However, when the geometry erodes, the porous media develops preferred flow directions and anisotropic permeability. By combining a high-order integral equation method to solve the fluid equations with tailored time stepping methods for the interface dynamics, I will demonstrate how erosion can be simulated. Then, I will use Barycentric quadrature methods to simulate transport through eroded geometries and examine how erosion affects transport properties. This is joint work with Nick Moore and Shang-Huan Chiu.

#### October 27 – Adrianna Gillman, University of Colorado, Boulder

Title: Fast direct solvers for boundary integral equations

Abstract: The numerical solution of linear boundary values problems play an important role in the modeling of physical phenomena. As practitioners continue to want to solve more complicated problems, it is important to develop robust and efficient numerical methods. For some linear boundary value problems, it is possible to recast the problem as an integral equation which sometimes leads to a reduction in dimensionality. The trade-off for the reduction in dimensionality is the need to solve a dense linear system. Inverting the dense N by N matrix via Gaussian elimination has computational cost of O(N^3). This talk presents solution techniques that exploit the physics in the boundary integral equation to invert the dense matrix for a cost that scales linearly with N with small constants. For example, on a laptop computer, a matrix with N=100,000 can be inverted in 90 seconds and applying the solver takes under a tenth of a second. The speed in which new boundary conditions can be processed makes these methods ideal applications involving many solves such as optimal design and inverse scattering. Extensions of the single body direct solver to select applications will also be presented.

#### November 3 – Go Vote!

#### November 10 – Chao Xu, University of Illinois at Urbana-Champaign

Title: The Shortest Kinship Description Problem

Abstract: Consider a person new to a language, who only learned some of the kinship terms (e.g. brother and daughter). Is it possible to describe a particular kin with a minimum number of those words? The problem is straightforward for American kinship system: other than ancestor and descendants, and a few well structured exceptions, it is always nth cousin mth removed for some positive integers n and m. We focus on a much more complicated kinship system, the Sudanese system. It turns out the problem is a special case of the submonoid membership optimization problem, where the monoid here is generated by taking products of the kin types. We show that for this special case, there is a polynomial time algorithm using tools from rewriting systems. We will also state an open problem on the length of the output.

This is joint work with Qian Zhang.