# Seminars

## Colloquia, Seminars, and Conferences

### Colloquium

Friday, October 23 at **2:00pm**, Virtual Room (Zoom Meeting ID: 754 716 1091)

**"Limit Theorems for Dependent Random Fields"**

Dr. Cristina Tone, *University of Louisville*

Abstract:

Starting from the classical central limit theorem, we will try to introduce analogous properties, assumptions in order to obtain central limit theorems for sequences of dependent random fields. We will present different ways of measuring dependence and the corresponding mixing conditions. Then some limit theorems for dependent random fields will be stated and some of the main proving techniques will be described.

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

Contact: Dr. Evan Milliken

**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, October 9 at 3:30pm, Virtual Room (Zoom Meeting ID: 754 716 1091)

**"Asymptotic analysis and numerical computations of some boundary value problems"**

Dr. Gung-Min Gie, *University of Louisville*

Abstract:

In the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. Boundary value problems arise in physics and other fields, e.g., Black--Scholes equation in financial math, Navier--Stokes equations in fluid mechanics, and Keller--Segel equations in math biology. In this talk, we consider a simple 1D (or 2D) boundary value problem, which consists of a convection--(reaction)--diffusion equation, supplemented with appropriate boundary conditions. Concerning this model problem, we analyze the asymptotic behavior of solutions at the vanishing diffusivity, and discuss the related numerical computations. Some recent progress in this research direction will be introduced as well.

Friday, September 25 at 3:30pm, Virtual Room (Zoom Meeting ID: 754 716 1091)

**"Can A Barrier Zone Stop Invasion of A Population?"**

Dr. Bingtuan Li, *University of Louisville*

Abstract:

A barrier zone can be viewed as an area at the front of a population invasion where eradication or suppression activity is performed. The barrier zone concept of slowing or stopping range expansion has been implemented against a variety of species. In this talk, we will discuss mathematical models in terms of reaction-diffusions and integro-difference equations and investigate the effect of a barrier zone on invasion of a population with a strong Allee effect. We provide a formula for the critical width L* of barrier zone. We show that when a barrier zone is set at the front of a population, if the width of barrier zone is bigger than L* then the barrier zone can stop the population invasion, and if the width of barrier zone is less than L* then the population crosses the barrier zone and eventually occupies the entire space.

Friday, September 11 at 3:00pm, Virtual Room (Zoom Meeting ID: 754 716 1091)

**"A Spatiotemporal Analysis of Progression of COVID‐19 From Urban to Rural Areas in the United States"**

Dr. Dan Han, *University of Louisville*

Abstract:

There are growing signs that the COVID‐19 virus has started to spread to rural areas and can impact the rural health care system that is already stretched and lacks resources. To aid in the legislative decision process and proper channelizing of resources, we estimated and compared the county‐level change in prevalence rates of COVID‐19 by rural‐urban status over 3 weeks. We used crowdsourced data on COVID‐19 and linked them to county‐level demographics, smoking rates, and chronic diseases. We fitted a Bayesian hierarchical spatiotemporal model using the Markov Chain Monte Carlo algorithm. We mapped the estimated prevalence rates using ArcGIS 10.8, and identified hotspots using local statistics. Additionally, we identified hotspots based on estimated prevalence rates.

Friday, August 28, 2020 at 3:00pm, Virtual Room (TBA)

**"All the Seasons of MinRank"**

Dr. Daniel Smith-Tone, *University of Louisville*

Abstract:

The MinRank problem is the very natural question of determining the matrix of least rank in the span of a collection of matrices. (Of course we mean nonzero linear combinations.) The decisional version of this problem is known to be NP-complete. Nevertheless, the computational version is often tractable for practical instances. We will talk about how this problem arises and show how different types of instances admit efficient resolution with different techniques. About 80% of the talk will require nothing more than a 3rd grade understanding of linear algebra.

Speaker : Dr. Blyman, United States Military Academy

Date: January 24, 2020 (Friday)

Time: 3:00 – 4:00 pm

Room: NS 212C

*Title:* **Achieving, Assessing, and Evaluating Higher-Order Learning Goals in the College Mathematics Classroom**

*Abstract:* Technology has changed the world in which we live. Our students carry phones in their pockets that give them access to answers to all kinds of questions. With this evolution, the requisite mathematical skills of a college graduate are evolving as well. We live in a world full of complex and ill-defined problems. As college educators, we must fill the vital role of preparing the next generation to solve unforeseeable future problems. While we cannot know what these problems will be, we can be almost certain that solving them will require creativity.

We know that creative problem solving cannot be learned through a single experience, so we provide our students with a blend of experiences. Our Mathematical Modeling and Introduction to Calculus course structure at the United States Military Academy enables creative problem solving through class instruction, during class activities, during out-of-class assessments, and during in-class assessments. We have found that the combination of these elements within our course structure increases student’s creative problem-solving abilities. We believe that each of these components contributes to students’ gains in comfort with solving open-ended and ill-defined problems like those they will encounter in the real world.

Speaker : Dr. Amy Been Bennett, University of Arizona

Date: January 22, 2020 (Wednesday)

Time: 3:00 – 4:00 pm

Room: NS 212C

Title: **Active Learning across Grade Levels: From Mathematical Models to Collaborative College Classrooms **

Abstract: Teaching practices that emphasize student participation and collaboration during mathematics tasks are gaining attention at the university level. Mathematics education research shows that teaching via an active learning approach leads to greater student learning gains, higher engagement in mathematics during class, and increased persistence in STEM disciplines. In this talk, I will present research on active learning approaches in different settings across grade levels K-16. My early research in active learning emphasized the value and versatility of mathematical modeling tasks. My current study explores the norms of teaching introductory mathematics courses via active learning at the undergraduate level. This work highlights the factors that influence instruction, such as the physical learning space, and has implications for undergraduate mathematics instruction.

Speaker : Dr. Dalton D. Marsh, University of New Hampshire

Date: January 21, 2020 (Tuesday)

Time: 4:00 pm – 5:00 pm

Room: NS 212C

Title: **Why Do Students Choose Majors in STEM? **

Abstract:

The building of a more capable and diverse workforce in science, technology, engineering, and mathematics (STEM) is seen as a critical issue in the United States. But why do some students pursue these careers while others do not? In this talk, I will discuss how I am using statistics and nationally representative data to investigate this question. Specifically, I will show how I am applying structural equation modeling and data from the High School Longitudinal Study of 2009 to model the relationships between students’ educational experiences, attitudes towards mathematics, and college major choice. An overview of structural equation modeling will be included that highlights why this particular statistical technique is ideal for addressing the challenges associated with analyzing nationally representative survey data (e.g., categorical variables, multilevel data structure, missing data) and models with multiple populations and dependent variables. Implications for policy and practice will be discussed.

Speaker : Dr. Hyun-Jung Kim, Illinois Institute of Technology

Date: January 10, 2020 (Friday)

Time: 3:00 – 4:00 pm

Room: NS 212C

Title: Stochastic parabolic Anderson model: optimal regularity and parameter estimation.

Abstract: In this talk, we study the stochastic parabolic Anderson model driven by time-independent white noise with a drift parameter and a diffusion parameter. We first define a solution and establish its optimal regularity by two different approaches. We show that the obtained optimal regularity is in the line with standard parabolic PDE theory. Moreover, the optimal regularity suggests a way of estimating the parameters, assuming that one trajectory of the solution is observed discretely in time and space.

Speaker : Dr. Evan Milliken, Arizona State University

Date: January 8, 2020 (Wednesday)

Time: 3:00 – 4:00 pm

Room: NS 212C

Title: A new approximation technique for Markov chains with applications to epidemics and biodiversity

Abstract:

Continuous time Markov chains are frequently used to model biological systems in which interactions are driven by underlying randomness. Often, interesting and important questions can be investigated by formulating first passage problems with respect to such models. We may ask what is the likelihood that one event occurs before another, or how long, on average, does it take before the first event occurs. Unfortunately, analysis of these problems is impossible in all except the simplest cases. In this talk, presently available approximation techniques are revisited. A new technique called Local Approximation in Time and Space is presented along with applications to problems in epidemiology and ecology.

Speaker : Dr. Dang Nguyen, University of Alabama

Date: January 6, 2020 (Monday)

Time: 3:00 – 4:00 pm

Room: NS 212C

Title: Persistence and Extinction of Stochastic Kolmogorov Systems.

Abstract:

In recent years there has been a growing interest in the study of the dynamics of stochastic populations. A key question in population biology is to understand the conditions under which populations coexist or go extinct. Theoretical and empirical studies have shown that coexistence can be facilitated or negated by both biotic interactions and environmental fluctuations. We study the dynamics of n populations that live in a stochastic environment and which can interact nonlinearly (through competition for resources, predator-prey behavior, etc.). The environmental variation is modeled by white noise and/or color noise. We give sharp conditions under which the populations converge exponentially fast to their unique stationary distribution as well as conditions under which some populations go extinct exponentially fast. Examples are also given to illustrate how stochasticity can facilitate or inhibit persistence of populations.

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.

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.