# Seminars

Colloquia, Seminars, and Conferences

### Colloquium

Friday, November 15, 2019 at 3:30pm, NS 212E

**"Method of upper/lower solutions to reaction diffusion equations"**

Dr. Changbing Hu

Abstract:

In this talk we will introduce some fundamental upper and lower solution method in the study of partial differential equations (PDEs).

We will illustrate this method by using some classical PDEs, like the Laplace equation and heat equation. As for its applications, we will focus on the reaction diffusion equations arising (RDs) from mathematical ecology, chemical reaction and network. Topics include the existence of solutions with general initial data, or some traveling wave solutions, persistence and extinction for some species. Some recent results will be reported as well.

This talk will be presented at the level of Calculus.

Colloquium Series Information

The mathematics department holds regular department colloquia. For more information please contact:

**Some Past Mathematics Department Colloquia**

### Seminars

The following research seminars meet reguraly in the Natural Sciences Building. Everyone is welcome to attend. For more information, please contact corresponding coordinators.**Analysis Seminar**

Contact: Dr. Alica Miller

**Algebra & Combinatorics Seminar**

Contact: Dr. Jake Wildstrom

past Algebra and Combinatorics seminars

**Differential Equations and Applied Math Seminar **

Contact:** **Dr. Gungmin Gie

**Graduate Student Seminar**

Contact: Dr. Jake Wildstrom

**Mathematical Biology Seminar**

Contact: Dr. Bingtuan Li

**Probability & Statistics Seminar**

Contact: Dr. Ryan Gill

### Conferences

The mathematics department has hosted several mathematics conferences. Here are links to some of the recent past conferences:

24th Cumberland Conference on Combinatorics, Graph Theory, and Computing, May 12-14, 2011

Fall Southeastern Sectional Meeting of the American Mathematical Society, October 5-6, 2013

## Prior colloquia

Friday, November 1st, 2019 at 3:00pm, NS 212D

**" A phase transition for the contact process with avoidance on Z, Zn, and the star graph"**

Dr. Matthew Wascher

Abstract:

The (classical) contact process, or SIS epidemic, is a model for the spread of disease through a population. We model the population with a graph, where vertices represent individuals and directed edges represent potential pathways for infection to spread. At any given time, each vertex is either "infected" or "healthy," and given an infection rate parameter lambda, the process evolves according to the following dynamics. Each infected vertex infects each of its out-neighbors at rate lambda. Simultaneously, each infected vertex become healthy at rate 1. This model is known to exhibit a phase transition in lambda for many graphs, such as the nearest-neighbor lattice Z. This means that there exists a value lambda_c such that the long term survival behavior of the epidemic when lambda < lambda_c differs from this behavior when lambda > lambda_c.

We consider a modified contact process that we call the contact process with avoidance. The process retains the infection and recovery dynamics of the classical contact process, but in addition each healthy vertex can avoid each of its infected neighbors at rate alpha by turning off the directed edge from that infected neighbor to itself until the infected neighbor recovers. This model presents a challenge because, unlike the classical contact process (alpha = 0,) it has not been shown to be an attractive particle system. We study the survival dynamics of this model on the nearest-neighbor lattice Z, the cycle Z_n, and the star graph. On Z, we show there is a phase transition in lambda between almost sure extinction and positive probability of survival. On Z_n, we show that as the number of vertices n -> infinity, there is a phase transition between log and exponential survival time in the size of the graph. On the star graph, we show that as n -> infinity the survival time is polynomial in n for all values of lambda and alpha. This contrasts with the classical contact process where the survival time on the star graph is exponential in n for all values of lambda.

Friday, October 25th, 2019 at 3:30pm, NS 212D

**"The Minrank Problem in Post-quantum Cryptography"**

Dr. Javier Verbel

*Universidad Nacional de Colombia, Sede Medellin*

Since 1995, thanks to Shor's algorithm, it was clear that current methods for secure communication would not be secure if a large quantum computer was built. Since then *post-quantum* cryptography has gained relevance, and many proposals claiming to provide security in a world with quantum computers started to be considered.

The search version of the *Minrank* problem is concerned in finding a linear combination of a given set of matrices with rank less than a certain positive integer. Besides being a natural question in linear algebra, this problem has strong implications in post-quantum cryptography nowadays. The security of several proposals relies on the complexity of solving the Minrank problem in the average.

In this talk we are going to see the connection between the Minrank problem and some of the most popular post-quantum schemes. Also we are going to see most known results about the complexity of Minrank problem.

Friday, October 11th, 2019 at 3pm, NS 212E

"Singular perturbations in fluid mechanics: Analysis and computations"

Dr. Gung-Min Gie

Department of Mathematics, University of Louisville

Abstract: Singular perturbations occur when a small coefficient affects the highest order derivatives in a system of partial differential equations. From the physical point of view, singular perturbations generate thin layers located near the boundary of a domain, called boundary layers, where many important physical phenomena occur. In this talk, we introduce a methodology, based on the utilization of correctors as proposed by J.-L. Lions and R. Temam, and use systematically this method of correctors to analyze the boundary layers of certain singularly perturbed fluid equations. We also discuss how to implement effective numerical schemes for slightly viscous fluid equations where the boundary layer correctors play essential roles.

*Key words: fluid mechanics, partial differential equations, singular perturbations, correctors.*

In this talk we will introduce some fundamental upper and lower solution method in the study of partial differential equations (PDEs).

We will illustrate this method by using some classical PDEs, like the Laplace equation and heat equation. As for its applications, we will focus on

the reaction diffusion equations arising (RDs) from mathematical ecology, chemical reaction and network. Topics include the existence of solutions

with general initial data, or some traveling wave solutions, persistence and extinction for some species. Some recent results will be reported as well.

This talk will be presented at the level of Calculus.