M@X -- Math At Xavier

Mathematics Department
Xavier University of Louisiana

Spring 2025 seminar schedule

Date	Place	Speaker	Title
Tue Mar 25, 12:15pm	NCF 574	Aikaterini Gkogkou Tulane University	From the first soliton until today.
Summary The study of solitons and the nonlinear partial differential equations that govern them is an exciting and extremely active field of research. In this talk, we will explore the unique features of solitons, tracing their journey from the first observed soliton to the development of the mathematical framework that describes them. Key to this exploration will be the concepts of integrability, inverse scattering theory, and Riemann-Hilbert analysis - core pillars of my research - that are essential for understanding and analyzing these intriguing waves. Through this discussion, we will uncover how the foundational tools that drive the study of nonlinear wave phenomena have evolved over time.
	I am Aikaterini Gkogkou, a postdoctoral fellow in the Department of Mathematics at Tulane University, working under the guidance of Professor Kenneth McLaughlin. Originally from Greece, I completed both my bachelor's and master's degrees in mathematics there. I then moved to Buffalo, NY, where I earned my Ph.D. in mathematics in May 2023, under the supervision of Professor Barbara Prinari. My research focuses on the mathematical modeling, analysis, and numerical simulation of nonlinear wave phenomena, with a particular emphasis on integrability and dispersive partial differential equations. Key areas of my work include the inverse scattering transform, Riemann-Hilbert analysis, numerical solutions to Riemann-Hilbert problems, soliton solutions, and soliton gases.
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Date	Place	Speaker	Title
Fri Mar 28, 11:00am	NCF 574	Milen Yakimov Northeastern University	Geometry of Cluster Algebras
Summary Cluster Algebras were invented by Fomin and Zelevinsky 25 years ago. For a short time, they turned into an indispensable tool for studies in Algebra, Geometry, Combinatorics and Mathematical Physics. The talk will be a gentle introduction to the area based on examples and without assuming any background except Linear Algebra. In the last bit of the talk we will discuss the underlying geometry of Cluster Algebra through the lenses of concrete examples.
	Professor Yakimov is a leading expert in many areas of mathematics with a wide range of research interests in algebra, representation theory, and mathematical physics. He obtained his Ph.D. at UC Berkeley and completed his postdoc at Cornell University. He was a tenured professor at UC Santa Barbara and Louisiana State University before arriving to Northeastern University. He is internationally recognized for his contributions in mathematics with many prestigious awards and titles, such as the Alfred P. Sloan Foundation Research Fellow and the Fellow of the American Mathematical Society. His research has been funded by the NSF since 2004.

Date	Place	Speaker	Title
Tue Apr 1, 12:15pm	NCF 574	Jiaxin Jin University of LA Lafayette	Infinitesimal Homeostasis in Mass-Action Systems
Summary Homeostasis occurs in a biological system when a chosen output variable remains approximately constant despite changes in an input variable. In this work we specifically focus on biological systems which may be represented as chemical reaction networks and consider their infinitesimal homeostasis, where the derivative of the input-output function is zero. The specific challenge of chemical reaction networks is that they often obey various conservation laws complicating the standard input-output analysis. We derive several results that allow to verify the existence of infinitesimal homeostasis points both in the absence of conservation and under conservation laws where conserved quantities serve as input parameters. In particular, we introduce the notion of infinitesimal concentration robustness, where the output variable remains nearly constant despite fluctuations in the conserved quantities. We provide several examples of chemical networks which illustrate our results both in deterministic and stochastic settings.
	I am an assistant professor in the Department of Mathematics at the University of Louisiana at Lafayette. I received my Ph.D. in math from the University of Wisconsin-Madison, under the supervision of Gheorghe Craciun and Chanwoo Kim. Before joining UL Lafayette, I was a Zassenhaus Assistant Professor in the Department of Mathematics at The Ohio State University under the mentorship of Marty Golubitsky. My research lies in Reaction Systems, Homeostasis in input-output networks, Kinetic theory, and Reaction-Diffusion Systems.

Date	Place	Speakers	Title
Thu Apr 24, 12:15pm	NCF 574	Broden Caton, Olivia Coward, Julian Davis, Austin Teter Xavier University of LA	Generalizing the Affirmative Action Problem: Mixing Numbers and Integrated Colorings of Graphs
Summary Consider a graph whose vertices are colored in one of two colors: black or white. A white node is called integrated if it has at least as many black as white neighbors, and similarly for a black node. A coloring of a graph is said to be integrated if each of the nodes is integrated. A classic exercise in graph theory, known as the Affirmative Action Problem, is to prove that every simple graph admits an integrated coloring. The solution is very simple and can be neatly summarized with the one-liner: "maximize the number of balanced edges," that is the edges that connect neighbors of different colors. However, not all integrated colorings have the maximal number of balanced edges. In this talk, we will enumerate the possible integrated colorings over complete graphs, complete bipartite graphs, paths, and cycles. We will also provide the distribution of the mixing numbers (the number of balanced edges) across all integrated colorings over these classes of graphs. In particular, we will characterize the colorings for both the maximal and minimal mixing numbers.
Slides from this talk
From left: Broden Caton, Austin Teter, Olivia Coward, Julian Davis