# Mathematics Colloquium

## Upcoming Colloquium

**Exploring Eigenvalue Perspectives: Linear Elliptic Problems with Nonlinear Boundary Condition**

Dr. Maya Chhetri, UNC Greensboro

May 17, 2024

12:00pm-1:00pm in FO3-200A and via Zoom

Join 5/17 Zoom

Meeting ID: 829 2168 0595

Passcode: 314159

### Abstract

In this presentation, we explore linear elliptic problems with nonlinear boundary condition. While problems involving linear diffusion and nonlinear reaction in the interior, satisfying either zero or linear boundary conditions, have garnered considerable attention for their practical applications, there's a natural curiosity surrounding nonlinear boundary conditions. Our focus will be on a class of boundary value problems distinguished by linear diffusion term in the interior coupled with a nonlinear boundary conditions, where the bifurcation parameter is on the boundary. This allows us to look at this problem through the lens of an eigenvalue problem.

We will explore how the shape and the asymptotic behavior of the reaction term impact the number of solutions as the bifurcation parameter varies.

The Mathematics Colloquium is a unique opportunity for students to learn about new developments in mathematics and what mathematics and statisticians do after they graduate. Hosted by the Department of Mathematics and Statistics at California State University, Long Beach, the weekly meetings invite guests from universities, research laboratories, and industry to present and discuss current topics in mathematics. All students are encouraged to attend.

## Schedule

The following is the schedule for Spring 2024. Additional colloquia may be added as the semester progresses.

Dr. Maya Chhetri, UNC Greensboro

### Abstract

In this presentation, we explore linear elliptic problems with nonlinear boundary condition. While problems involving linear diffusion and nonlinear reaction in the interior, satisfying either zero or linear boundary conditions, have garnered considerable attention for their practical applications, there's a natural curiosity surrounding nonlinear boundary conditions. Our focus will be on a class of boundary value problems distinguished by linear diffusion term in the interior coupled with a nonlinear boundary conditions, where the bifurcation parameter is on the boundary. This allows us to look at this problem through the lens of an eigenvalue problem.

We will explore how the shape and the asymptotic behavior of the reaction term impact the number of solutions as the bifurcation parameter varies.

Dr. Stephan Ramon Garcia, Pomona College

### Abstract

A simple question about chicken nuggets connects everything from analysis and combinatorics to probability theory and computer-aided design. Linear algebra is a recurring theme: piecewise-polynomial interpolation, "random norms," trace polynomials, and operator algebras all make an appearance. Students are invited to attend!

### Biosketch

Stephan Ramon Garcia is W.M. Keck Distinguished Service Professor and chair of the Department of Mathematics and Statistics at Pomona College. He is the author of six books and over 100 research articles in operator theory, complex analysis, matrix analysis, number theory, discrete geometry, combinatorics, and other fields. This includes dozens of papers published with students, many of whom have gone on to top graduate programs. He has served on the editorial boards of the Proceedings of the AMS, Notices of the AMS, Involve, and The American Mathematical Monthly. He has received five NSF research grants as principal investigator and six teaching awards from three different institutions. He is a Fellow of the AMS and the inaugural winner of the AMS Dolciani Prize for Excellence in Research.

Dr. Isaac Harris, Purdue University

### Abstract

In this talk, we will investigate the acoustic transmission eigenvalue problem associated with an inhomogeneous media with a conductive boundary. These are a new class of eigenvalue problems that are not elliptic, not self-adjoint, and non-linear, which gives the possibility of complex eigenvalues. The talk will consider the case of an Isotropic and an Anisotropic scatterer. We will discuss the existence of the eigenvalues as well as their dependence on the material parameters. Because this is a non-standard eigenvalue problem, a discussion of the numerical calculations will also be highlighted. This is joint work with: O. Bondarenko, V. Hughes, A. Kleefeld, and J. Sun.

### Bio Sketch

Isaac Harris is an Associate Professor of Mathematics at Purdue University. He graduated from The University of Delaware with a Ph.D. in Applied Mathematics under the advisement of Fioralba Cakoni in 2015. In 2018, he joined the Department of Mathematics at Purdue University as an Assistant Professor. From 2015-2018, he was at Texas A&M University as a Visiting Assistant Professor continuing research in Inverse Problems. Isaac’s research interests include inverse problems and spectral methods for PDEs, especially those arising in acoustic and electromagnetic scattering. His research employs techniques from a multitude of areas in mathematical analysis, such as Functional analysis, Analysis of PDEs, and Asymptotic methods.

Dr. Gunhee Cho, UC Santa Barbara

### Abstract

The family of 1-dimensional normal distributions reveals a distinct differential geometric structure known as the Poincaré Upper Half Plane, arising from the Fisher-Information metric. The widely applied Cramér-Rao lower bound can be understood as a reflection of the Fisher-Information metric's decreasing nature. Furthermore, this metric's decreasing property resonates with the differential geometric interpretation of the Schwarz-Pick lemma within the Poincaré Disk of complex analysis. In this presentation, our goal is to extend the insights from the Poincaré Disk to encompass bounded domains in C^n as statistical models, leveraging observations from normal distributions and the Schwarz-Pick lemma. By treating bounded domains as statistical models, we establish a coherence between the Bergman metric and the Fisher-Information metric, introducing a natural decreasing property. Additionally, the establishment of an equality condition of two Bergman metrics corresponds to the identification of sufficient statistics. This recent work is a collaborative effort with J. Yum.

### Bio Sketch

Dr. Gunhee Cho received his Ph.D. degree in 2021 from the Department of Mathematics at the University of Connecticut, and now he is a visiting assistant professor in the Department of Mathematics at the University of California, Santa Barbara. His research areas are Complex Geometry, sub-Riemannian Geometry, Stochastic Differential Geometry, and Information Geometry.

Dr. Danny Luecke, North Dakota State University

### Abstract

Every Indigenous nation, and specifically Standing Rock Nation, has rich ways of knowing passed down for generations. Specifically, mathematical ways of knowing are embedded within the language and culture. However, calls for Indigenous language and culture to be more integrated into the math classrooms at Sitting Bull College (along with other TCUs) have been met with epistemological challenges as well as a dearth of math and local culture resources. The Dakota/Lakota Math Connections research project addresses both of these challenges through applying an Indigenous research paradigm. From the outset, this study chose to center Indigenous values and ways of knowing. This presentation will story my understanding, application, and results of an Indigenous research paradigm to math education at Sitting Bull College.

### Bio Sketch

Danny Luecke is the developer/instructor for the bachelor’s degree in Secondary Math Education at Turtle Mountain Community College. He earned his PhD in math and math education at North Dakota State University. His research focuses on Indigenous Math and specifically the Dakota/Lakota Math Connections project at Sitting Bull College. He was born and raised in Fargo, North Dakota. He is also a citizen of the Choctaw Nation of Oklahoma and has ancestry from multiple European nations. He believes he is honoring his ancestors and Creator through his life and work with the Standing Rock and Turtle Mountain communities.

Dr. Sandra Laursen, University of Colorado Boulder

### Abstract

From education research, we know much about teaching approaches that offer undergraduates engaging and inclusive learning experiences and can strengthen their success and persistence toward their goals, and that support all students to achieve these good outcomes. More difficult to come by is high-quality evidence about whether and how we support teachers to take up these teaching approaches. Well-designed professional development, offered in the discipline or locally at institutions, is widely thought to be one powerful lever for change in promoting evidence-based teaching, yet it has been challenging to determine whether such programs make a positive difference in enough classrooms to benefit substantial numbers of students. I will outline why these studies are important and difficult. Then I will discuss recent work in our research group to develop and test measures of teaching that seek to capture instructors’ movement toward evidence-based teaching, using examples from recent studies of national-scale professional development programs. Finally, I will offer some comments about what we do know about what works in professional development.

### Bio Sketch

Sandra Laursen maintains interests in both research and practice in science education. As senior research associate and director of Ethnography & Evaluation Research (E&ER), she leads research and evaluation studies focusing on education and career paths in science, technology, engineering, and mathematics (STEM) fields. Particular research interests include the visibility of women, people of color, disabled people and queer people in the sciences; professional socialization and career development of scientists; teacher professional development; and organizational change in higher education. She is also interested in inquiry-based teaching and learning, and the opportunities and challenges for strengthening STEM education in the classroom, in the lab and the field, and across organizations.

Dr. Gunther Uhlmann, Southern Methodist University

### Abstract

Inverse problems arise in all fields of science and technology where causes for a desired or observed effect are to be determined. By solving an inverse problem is in fact how we obtain a large part of our information about the world. An example is human vision: from the measurements of scattered light that reaches our retinas, our brains construct a detailed three-dimensional map of the world around us. In the first part of the talk we will describe several inverse problems arising in different contexts.

In the second part of the lecture we will discuss invisibility. Can we make objects invisible? This has been a subject of human fascination for millennia in Greek mythology, movies, science fiction, etc including the legend of Perseus versus Medusa and the more recent Star Trek and Harry Potter. In the last 20 years or so there have been several scientific proposals to achieve invisibility. We will describe in a non-technical fashion a simple and powerful proposal, the so-called transformation optics, and some of the progress that has been made in achieving invisibility.

### Bio Sketch

Gunther Uhlmann is the Robert R. and Elaine F. Phelps Professor of Mathematics at the University of Washington. His research focuses on inverse problems and imaging, microlocal analysis, partial differential equations and invisibility.

Professor Uhlmann was an Invited Speaker at The International Congress of Mathematicians in Berlin in 1998 and a Plenary Speaker at the International Congress on Industrial and Applied Mathematics in Zurich in 2007. He was elected to the American Academy of Arts and Sciences in 2009 and a SIAM Fellow in 2010. In 2011 he was awarded the American Mathematical Society Bôcher Memorial Prize and the Society for Industrial and Applied Mathematics Kleinman Prize. Professor Uhlmann delivered the American Mathematical Society Einstein Lecture in 2012. In 2013, he was elected Foreign Member of the Finnish Academy of Science and Letters. He gave a Plenary Lecture at the International Congress on Mathematical Physics in 2015. In 2017 he was awarded the Solomon Lefschetz Medal by the Mathematical Council of the Americas. In 2021 he was awarded by AMS and SIAM the George David Birkhoff Prize. In 2023 he was elected as a member of the National Academy of Sciences.

Dr. Thomas Hagstrom, Southern Methodist University

### Abstract

Although many mathematical models in wave theory lead to hyperbolic initial-boundary value problems which are inherently local due to the finite domain-of-dependence of the solution at any point on past values, there are also examples where nonlocal operators, in particular space-time integral operators, arise. A primary example is the radiation boundary condition needed to truncate an unbounded domain to a finite one to enable numerical solutions, as well as closely-related operators for unidirectional propagation. In this work we show how to leverage results from rational function approximation theory to construct spectrally-convergent local algorithms for evaluating such operators. For an important class of problems - systems equivalent to the scalar wave equation, such as acoustics or Maxwell’s equations in homogeneous, isotropic media, we will explain the construction, analysis, and implementation of our complete radiation boundary conditions, which are in a certain sense optimal. We will also discuss other applications of these approximation methods, as well as the fundamental barriers to extending the successful methods to other systems.

Daniel Lévesque, Polytechnique Montreal

*Details being finalized.*

Dr. Heyrim Cho, Arizona State University

### Abstract

Recent advances in biotechnology and genome sequencing, resulting in a surge of data, are bringing in new opportunities in mathematical modeling of biological systems. However, the amount of data that can be practically collected in everyday patients in the clinic is limited due to various reasons including the cost and the patient’s burden. Especially the amount of data that can be collected in the time domain is limited. This motivates us to transfer the mathematical and computational models to meet the challenges in clinical setting, to guide patient therapy via prediction. In this talk, I will discuss modeling approaches on the two ends of the spectrum of data. In the first part, I will discuss a Bayesian information-theoretic approach to determine effective scanning protocols of cancer patients. We propose a modified mutual information function with a temporal penalty term to account for the loss of temporal data. The effectiveness of our framework is demonstrated in determining patient scanning scheduling for prostate cancer patients. In the second part, I will discuss modeling work using single-cell gene sequencing data. Due to the high cost of obtaining gene sequencing data, temporal data also lacks. We show that our cell state dynamics model can be used to incorporate genetic alteration with low cost, where we show an example of modeling hematopoiesis system and simulating abnormal differentiation that corresponds to acute myeloid leukemia.

### Biosketch

Heyrim Cho received her Ph.D. degree in 2015 from the Applied Math department at Brown University, then was a Brin-Postdoc fellow in the Math department at University of Maryland, College Park, and an assistant professor in the Math department at University of California Riverside. She recently moved to Arizona State University as an assistant professor in the School of Mathematical and Statistical Sciences. Her research areas are: Mathematical and computational biology especially focusing on oncology and medical applications, Stochastic modeling and simulation, Uncertainty quantification, and Scientific Computing.

## Previous Colloquia

The Mathematics Colloquium Archive has the Colloquia from previous semesters.

## Colloquium Committee

For Fall 2023:

- Dr. Brian P Katz (BK)
- Dr. Yann-Meing Law
- Dr. Seungjoon Lee
- Dr. Rolando de Santiago