**Upcoming talks**

Speaker: Hannah Larson (University of California, Berkeley)

Date: Wednesday, April 10, 2024

Time/location: GAB 105 @4pm

Title: Moduli spaces of curves

Abstract: How many circles are tangent to 3 fixed circles? I'll explain how moduli spaces, a central topic in algebraic geometry, help shed light on the answer to this question and others like it. We'll start by discussing the moduli space of circles. This is an example of a moduli space of "embedded curves." Next, I'll explain the distinction between an "embedded curve" and its associated "abstract curve" (also called a "Riemann surface"). I'll finish by sharing some recent results about moduli spaces of abstract curves, which are joint work with Samir Canning.

Speaker: Dinakar Ramakrishnan (California Institute of Technology)

Date: Monday, April 15, 2024

Time/location: GAB 461 @5pm

Title: An Invitation to Diophantine Equations, Congruent Numbers and Beyond

Abstract: The basic Equations of Number theory are the Diophantine Equations, which are polynomial equations with integer coefficients. One looks for integral, or rational, solutions, and makes use of the structure of real or complex solutions. Simplest non-trivial examples are the Pythagorean and Pell's Equations. Congruent numbers, which are positive integers which arise as areas of rational right triangles, have been understood quite well due to some striking recent results (due to others). At the end of the lecture, we will discuss a generalization to the setting of right rational tetrahedra, which leads to some intriguing geometry.

**Past talks**

Speaker: Sohail Farhangi (University of Adam Mickiewicz)

Date: Monday, March 18, 2024

Time/location: GAB 461 @ 4pm

Title: Uniform pointwise ergodic theorems via ultraproducts

Abstract: We take a new point of view on weighted pointwise ergodic theorems by viewing the sequence (f(T^{n}x))n >= 1 as a vector in an ultraproduct space. This point of view allows us to directly transfer mixing properties of the underlying transformation to mixing properties of a shift operator on the ultraproduct space. In this new context, weighted pointwise ergodic theorems correspond to identifying functionals in the dual space that annihilate the starting vector. This allows us to not only recover the uniform Wiener-Wintner theorem of Lesigne, but to also prove a uniform pointwise ergodic theorem for ergodic, mild mixing, and strong mixing measure preserving transformations. Furthermore, our methods are general enough to apply to any L^{1}-L^{\infty} contraction on a Bochner space L^{p}(X,\mu;E) constructed from a reflexive Banach space .

Speaker: Xinfeng Liu (University of South Carolina)

Date: Monday, March 4, 2024

Time/location: GAB 461 @ 4pm

Title: Data-driven mathematical modeling, computation and experimental investigation of dynamical heterogeneity in breast cancer

Abstract:

Solid tumors are heterogeneous in composition. Cancer stem cells (CSCs) are a highly tumorigenic cell type found in developmentally diverse tumors that are believed to be resistant to standard chemotherapeutic drugs and responsible for tumor recurrence. Thus understanding the tumor growth kinetics is critical for development of novel strategies for cancer treatment. For this talk, I shall introduce mathematical modeling with computational investigation to study Her2 signaling for the dynamical interaction between cancer stem cells (CSCs) and non-stem cancer cells, and our findings reveal that two negative feedback loops are critical in controlling the balance between the population of CSCs and that of non-stem cancer cells. Furthermore, the model with negative feedback suggests that over-expression of the oncogene HER2 leads to an increase of CSCs by regulating the division mode or proliferation rate of CSCs, which has profound implications for drug development to efficiently treat cancer tumors.

Short Bio:

Dr. Xinfeng Liu currently is a Professor and serves as the Director for Undergraduate Studies at the Department of Mathematics of University of South Carolina. He obtained his Ph.D. from Stony Brook University in 2006. Prior to joining the University of South Carolina, Dr. Liu was a visiting assistant professor from 2006-2009 at University of California at Irvine. Dr. Liu's research is mainly focused on mathematical modeling and computational methods for various applications, including fluid dynamics, computational systems biology, cancer tumor growth, etc. His research has been supported through multiple grants from the National Science Foundation (NSF).

Speaker: Shuang Liu, Department of Mathematics, UNT

Date: Monday, November 13, 2023

Time/location: GAB 461 @ 4pm

Title: Computational Moving Boundary Problems

Abstract:

Moving boundary (or often called "free boundary") problems are ubiquitous in nature and technology. A computational perspective of moving boundary problems can provide insight into the "invisible" properties of complex dynamics systems, advance the design of novel technologies, and improve the understanding of biological and chemical phenomena. However, challenges lie in the numerical study of moving boundary problems. Examples include difficulties in solving PDEs in irregular domains, handling moving boundaries efficiently and accurately, as well as computing efficiency difficulties. In this talk, I will discuss three specific topics of moving boundary problems, with applications to ecology (population dynamics), plasma physics (ITER tokamak machine design), and cell biology (cell movement). In addition, some techniques of scientific computing will be discussed.

Speaker: TBA

Date: Monday, November 27, 2023

Time/location: GAB 461 @ 4pm

Title: TBA

Abstract: TBA

Speaker: TBA

Date: Monday, December 4, 2023

Time/location: GAB 461 @ 4pm

Title: TBA

Abstract: TBA

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**Past talks**

Speaker: Paco Villarroya, Department of Applied Mathematics, Santa Clara University

Date: Wednesday, September 6, 2023

Time/location: GAB 461 @ 4pm

Title: A characterization of compactness for singular integral operators

Abstract: pdf

Speaker: Tamara Kucherenko, CUNY

Date: Monday, September 18, 2023

Time/location: GAB 461 @ 4pm

Title: Ergodic theory on coded shifts

Abstract: We discuss ergodic properties of coded shift spaces. A coded shift is defined as a closure of all bi-infinite concatenations of words from a fixed countable generating set. It turns out that many well-known classes of shifts are coded including transitive subshifts of finite type, S-gap shifts, generalized gap shifts, transitive Sofic shifts, Beta shifts, and many more. We derive sufficient conditions for the uniqueness of measures of maximal entropy based on the partition of the coded shift into its sequential set (sequences that are concatenations of generating words) and its residual set (sequences added under the closure). We will also outline some flexibility results for the entropy on the sequential and the residual set. (Joint work with M. Schmoll and C. Wolf)

Speaker: Natasha Sharma, Department of Mathematical Sciences, UT El Paso

Date: Monday, October 9, 2023

Time/location: GAB 461 @ 4pm

Title: Unconditional Energy Stability and Solvability for a C0 Interior Penalty Method for a Sixth-Order Cahn-Hilliard type equation modeling microemulsions

Abstract: Microemulsions are thermodynamically stable, transparent, isotropic single-phase mixtures of two immiscible liquids stabilized primarily by surfactants. Recently, microemulsion systems have emerged as an effective tool in capturing the static properties of ternary oil-water-surfactant systems with widespread applications such as enhanced oil recovery processes, the development of environmentally-friendly solvents, consumer and commercial cleaning product formulations, and drug delivery systems. Despite its applications, a major challenge impeding the use of these equations has been, and continues to be, a lack of understanding of these complex systems. Microemulsions can be modeled by means of an initial-boundary value problem for a sixth-order Cahn-Hilliard equation. In this talk, we present a numerical scheme for approximating the solutions to these sixth-order equations. To numerically approximate this sixth-order evolutionary equation, we introduce the chemical potential as a dual variable and consider a Ciarlet-Raviart-type mixed formulation consisting of a linear second-order parabolic equation and a nonlinear fourth-order elliptic equation. Here, the spatial discretization relies on a continuous interior penalty Galerkin finite element method while for the temporal discretization, we propose a first-order accurate time-stepping scheme. Theoretical properties of convergence, unique solvability, and unconditional stability of the proposed scheme will be discussed and through extensive numerical experiments, we will demonstrate the performance of the proposed scheme.

Speaker: Wesley Perkins, Lyon College

Date: Monday, October 23, 2023

Time/location: GAB 461 @ 2:30pm

Title: Modulational and Subharmonic Dynamics of Periodic Waves

Abstract: In this talk, we will explore some of the modern techniques used to study the stability and dynamics of periodic solutions to nonlinear partial differential equations (PDEs) arising from applications. We will develop some of the intuition for these techniques using ordinary differential equations (ODEs) and integral transforms. This intuition will then allow us to investigate the stability and dynamics of several PDEs when subjected to various types of perturbations.