One-Level Density Of Zeros Of Cubic And Quartic Dirichlet L-Functions Over Function Fields 

Hua Lin

September  23rd, 2022  -  11:00am - 12:00pm - 440R

Abstract

We compute the one-level density of zeros of cubic and quartic Dirichlet L-functions over function fields \mathbb{F}_q[t] in the Kummer setting ($q\equiv1\Mod{\ell}$) and in the non-Kummer setting ($q\not\equiv1\Mod{\ell}$). In each case, we obtain a main term predicted by Random Matrix Theory (RMT) and a lower order term not predicted by RMT. We also confirm the symmetry type of the family is unitary, supporting the Katz and Sarnak philosophy. 

About the Speaker

Hua is a 6th year PhD student. 

About this talk

This talk is on analytic number theory, more specifically about computing the one-level density, a statistic of zeros of Dirichlet L-functions. Over function fields, the one-level density has been computed for quadratic L-functions. What happens for higher order L-functions? Hua will show some results for cubic and quartic L-functions in this talk. 

   Grade EXTpectations: 

What should a G-graded Calabi-Yau algebra be? 

Yasmeen Sophia Baki

June 9th, 2022  -  1:00 - 1:50pm - 340N

Abstract

Much work on graded Calabi-Yau algebras has focused specifically on the N-graded case. In this talk we propose a definition for a group-graded Calabi-Yau algebra and, building on work of Reyes and Rogalski, prove that our G-graded Calabi-Yau condition is equivalent to an algebra being G-graded and Calabi-Yau. We further discuss a number of constraints we should place on the grading when searching for interesting examples of such algebras, and provide a number of open questions we can ask which make use of our proposed definition. 

About the Speaker

Yasmeen is a 3rd year PhD student. 

Advisor and Collaborators

Yasmeen's advisor is Manny Reyes.

Transformed Alternating Gradient Flow and Transformed Primal-Dual Iterations for Nonlinear Saddle Point Systems

Jingrong Wei

June 3rd, 2022  -  11:00 - 11:50am - 340N

Zoom: 959 3724 9751

Abstract

A transformed alternating gradient (TAG) flow is developed for a class of nonlinear smooth saddle point system. Exponential decay of the error is obtained by showing the strong Lyapunov property. Several transformed primal-dual (TPG) iterations are derived by implicit Euler, explicit Euler, and implicit-explicit Euler discretization of the TAG flow. For strongly convex and non-strongly concave problem, the state-of-the-art linear convergence rates can be proved, and for non-strongly convex and non-strongly concave case, $O(1/k)$ sub-linear rate for a rescaled TPG can be established. The algorithm and convergence analysis depends crucially on appropriate inner products of the spaces for the primal variable and dual variable. The augmented Lagrangian method is treat as a preconditioning for the Schur complement. As a byproduct, a clear convergence analysis with nonlinear inexact inner solvers is developed. This is work supervised by Long Chen.

About the Speaker

Jingrong is a 3rd year PhD student interested in numerical analysis, optimization and Hearthstone. 

Advisor and Collaborators

Jingrong's advisor is Long Chen.

Detecting Species Interactions in Ecological Systems

Thomas Beardsley

May 31st, 2022  -  1:00 - 1:50pm - RH 340N

Abstract

In this talk, we investigate methods for detecting interactions between members of an ecological community. Often, qualitative descriptions of interactions are used to infer benefits or consequences for a member's fitness. We develop a purely quantitative model of interactions in an evolving ecological community, and use analytical and numerical tools to determine the existence of interactions among subpopulations.

About the Speaker

Thomas is a 4th year PhD student working on ecological dynamics. Outside of math, he mostly enjoys Pho and coffee, but not at the same time.

Advisor and Collaborators

Thomas's advisor is Natalia Komarova.

The Spread of Mutant Strains of Infection over a Network 

Samuel Lopez 

May 19th, 2022  -  11:00 - 11:50am - RH 340N

Zoom: 913 2738 8410

Abstract

We consider stochastic and deterministic models of infection spread over a network, by using the susceptible-infected-removed (SIR) framework.  We explore a set of the so called small world networks, which contain a tunable parameter that regulates the proportion of  local and global edges, with the two extremes being a regular ring lattice and a random graph. We observe that these structural changes in the network  change the spread of infection. In particular, we focus on the dynamics of an advantageous mutant, and show that there is an optical balance of global and local connections that optimize its spread.

About the Speaker

Samuel is a 4th year PhD student in the math department working on infection dynamics on a network. 

Advisor and Collaborators

Samuel's advisor is Natalia Komarova.

Mass and Geometry 

Tin Yau Tsang 

April 28th, 2022  -  2:00pm - 3:00pm - RH 340N

Abstract

In general relativity, it is not easy to define mass for local objects. And when one is defined, whether it is positive or not can be a difficult question. In this talk, we are going to see the progress on the search for a good notion of mass and its interaction with geometry on compact manifolds. 

About the Speaker

Tin Yau is a 4th year graduate student studying differential geometry.

Cohomology and Morse Theory on Symplectic Manifolds

David Clausen

April 4th, 2022  -  5:30pm - 6:30pm - RH 306

Abstract

On symplectic manifolds, there are intrinsincally symplectic cohomologies of differential forms that are analogous to the Dolbeault cohomology on complex manifolds.  These cohomologies are isomorphic to the de Rham cohomologies on odd-dimensional sphere bundles over the symplectic manifold.  In this talk, I will describe how we can use this sphere bundle perspective to define a novel Morse-type theory on symplectic manifolds associated with the symplectic cohomologies.  

About the Speaker

David is a 5th year graduate student.  This is joint work with Li-Sheng Tseng and Xiang Tang.