# Undergraduate Student Seminar Archives

**Spring 2018**

Undergraduate Seminar/Pi Mu Epsilon talk

Speaker: Stan Mintchev

Date and time: Friday, April 6, 4-6 p.m.

Place: Monroe Hall, #B32

Title: A friendly introduction to slow-fast systems and their importance in mathematical neuroscience.

Abstract: A variety of differential equations models for physical systems exhibit the so-called separation of time scales, where phase space trajectories are not traversed uniformly, but rather at (at least two) vastly different speeds. Such models are prevalent in mathematical neuroscience where the dynamics of neural depolarization-polarization cycles suggest a rapid discharge followed by a gradual recovery period. Since the late 1960’s, a growing body of work in dynamical systems known as geometric singular perturbation theory has yielded a framework for understanding systems featuring time scale separation, the simplest variety of which is colloquially referred to as slow-fast systems.

This talk will feature an illustration of some basic phenomenology of slow-fast systems through two simple examples — the classical van der Pol model and the FitzHugh-Nagumo model for neural dynamics; of note, we will restrict attention to ODEs in both instances. We will note first the similarities between these systems, which place them in a context where the results of the theory are applicable. We will then discuss a major difference between the two, namely, the equilibrium featured in the latter model and its importance for capturing neural excitability. The talk will finish with a brief mention of our current research project on signal propagation in networks of FitzHugh-Nagumo neurons.

Bio: Dr. Mintchev is Associate Professor of Mathematics at The Cooper Union in New York. He received his B.S. in Mathematics and Physics from GWU in 2002, and subsequently his Ph.D. from the Courant Institute at NYU in 2008. His research interests lie in applied dynamical systems and relations to mathematical physics, biology, and neuroscience.

**Fall 2017**

The GW Chapter of the Pi Mu Epsilon Mathematics Honor Society Lecture Series

Speaker: Professor Elizabeth Drellich, Swathmore College

Date and time: Friday, November 3, 4-5 p.m.

Place: Duques Hall, Room 151

Title: Cars, Cartoons, and Cohomology

Abstract:

How do you program a 3D printer to get the sleek curves on your model of your favorite car? How do animators make sure that complicated surfaces appear smooth on screen? And what do these questions have to do with algebra?

The surface of your model car or your favorite Pixar character's face is made up of a collection of polynomials that fit together smoothly called a spline. These splines are models, but they are also algebraic objects that can be added and multiplied. Some splines contain deep information about the structure of groups and algebraic varieties.

This talk will introduce you to splines and show you some of their many wonderful uses.

Professor Drellich earned her Bachelor of Science in mathematics from GW in 2009 and her Ph.D. from the University of Massachusetts at Amherst in 2015.

**Spring 2017**

__Honors Thesis Defense__

**Title**: Thesis Defense/presentation session

S**peaker:** Shigeng Sun, GWU**Date and Time:** Monday, May 8, 11:10–12:10pm**Place:** Rome 771

**Shigeng Sun Senior Honor Thesis Project Abstracts:**

**Part I. **MODELING THE INTERNATIONAL LINKS BETWEEN INTERBANK OFFERED RATES AMONG DIFFERENT MARKETS THROUGH A WAVELET ANALYSIS APPROACH

SHIGENG SUN SVETLANA ROUDENKO

Abstract. This project investigates the links and interactions of the interbank offered lending rates among different types markets, through the implementation of a wavelet multi-scale approach. The data used includes USD-Libor rate, which is long been well established, CHYShibor rate and RUB-Mosprime rate. The latter two are relatively new, but already have close interactions with the USD counterpart because of the globalization of the money and financial markets. In our studies, we employ the wavelet multi-scale approach, which has various advantages over a direct application of the traditional econometrics methods. The wavelet approach allows us to decompose the data into multiple (specific) time scales instead of being limited to only the short-run and the long-run scales. The wavelet method provides the unique versatile ability to separate the local dynamics from the global one. We fully exploit in this study the fact that different participants of the market react to changes diversely, however, persistently (within themselves) in terms of time. We perform the wavelet transform on the data using various Daubechies bases that have different lengths of oscillations. As a result, we decompose the data into different time scales and then perform the Granger Causality Test. This provides the evidence of causality across different time scales. The wavelet variances are computed in turn to show that the short-term rates are more volatile than the longer-term rates. We also discuss causality between the offered rates from different types of markets, how it varies across different time scales, among other results.

**Part II.** NUMERICAL STUDY OF PERIODIC MIGRATION OF ONE DIMENSIONAL CELL

SHIGENG SUN YANXIANG ZHAO

Abstract. In this project, we study the one-dimensional cell migration on micro- patterned substrates. This work is an extension of the study proposed in a paper of Camley et.al, in which the proposed model (later referred as ‘the model’) couples cell morphology with the polarizations of actin and myosin molecules. In this project, we use the model to study the effects of system parameters on the cell migration periodicity. More specifically, we study the protrusion and contraction forces and their effects on the amplitude and frequency of the periodical migration behaviors. For the numerical simulations, we show that periodic motion emerges naturally from the coupling of cell polarization and cell shape by running simulations with different values of system parameters. Both parameters are determined by different sharp interface results. We also show via simulations that there are important bifurcation points resulted from difference in protrusion and contraction forces. Furthermore, we have discovered an emergent phenomenon from our simulation, which shown that the protrusion stress increase resulted in an abrupt change on the amplitude of the periodic migration.

The George Washington University Department of Mathematics

**Spring 2015**

Math Major Undergraduate Talk

Title: "Thresholds for Solutions Existence in the Focusing Nonlinear Schroedinger equation"

Speaker**:** Changkai Sun (our math major undergraduate)

Date and Time: Tuesday, April 28, 4-5pm

Place: Monroe 267

Abstract: We study the nonlinear Schroedinger equation with focusing nonlinearity in various space dimensions. We consider finite energy and finite variance initial data in the so called mass-supercritical regime. One of the goals in such studies is to understand whether solutions evolved by the nonlinear Schroedinger evolution exist globally in time, or could form a singularity, and thus, `break down' (in a certain sense) in finite time. There have been much research done in this direction recently, and there are various theoretical thresholds available now. However, all of them lack the full picture, i.e., there are theoretical gaps in classification of the above initial data. In this numerical work, we consider Gaussian and super Gaussian initial data, we let it evolve by the nonlinear Schroedinger flow with power nonlinearities p=3,5,7 and we are able to obtain numerical thresholds which identify the borderline between the globally existing solutions and solutions which blow up in finite time. We then compare our results with the known theoretical ones and show how the "gaps" in long-time existence of solutions should be addressed.

This research is a part of Changkai Sun's senior thesis.

**Fall 2014**

Senior Honors Project Defense

Speaker: Jiayuan Wang

Title: A computational method for solving exponential-polynomial Diophantine equations

Date and Time: Friday, December 19, 12:45-1:45

Place: Monroe 267

Abstract:

Combining number theory with computer programming, we developed a novel computational method for solving Diophantine equations of the form f(m) = k*Q^n with respect to integers m and n, where Q>0 and k>0 are fixed integers and f(m) is a second-degree polynomial. Our method involves solving generalized Pell equations and computing periodic zeros of the solution modulo some powers of Q and employs computer algebra system PARI/GP. We use our method for systematic study of such equations and present many numerical results. As an example, we prove that the only solutions to the equation 2m^2 + 1 = 3^n are (m,n) = (0,0), (+/- 1,1), (+/- 2,2), and (+/-11,5).

(Project advisor: Max Alekseyev)

**Undergraduate Talk**

**Speaker:**Prof. Lushnikov, Univ of New Mexico (Math Dept) and Landau Physics Institute

**Title:** Derivation of basic nonlinear wave equations

**Date and Time:**Friday, Nov 21, 3:45-5pm

**Place:**Government 102

**Abstract:**

We will study the derivation of basic nonlinear wave equations from the practical systems. Focus will be on derivation of Nonlinear Schrodinger equation (NLSE) from the propagation of intense laser beam in Kerr media. Typical example of such medium is usual window glass or fused silica used in optical fibers. NLSE will be obtained in dimensions one, two and three. If time permit we will also derive Korteweg–de Vries equation from the nonlinear dynamics of atoms connected by nonlinear springs (Fermi-Pasta-Ulam problem).