LSU
Mathematics

Here are descriptions of currently active projects in the LSU Department of Mathematics, in which undergraduate students are involved. Some of them may be seeking undergraduate participants.

## Modeling, analysis, and simulation of liquid crystals with applications in material science and biology

Description:

Background: Liquid crystals (LCs) combine the fluidity of liquids with long-range order usually seen in solids. LCs exhibit orientational order which gives rise to their anisotropic material aspects, and leads to their interesting properties (manipulation of light, easy actuation by physical forces such as electric and magnetic fields). Applications of LCs range from displays (LCD) to self-assembled materials and novel types of mechanical actuation (e.g. shape changing materials). This research project will involve three main sub-areas: modeling, analysis, and simulation of LC phenomena for various applications. Depending on the student’s background, the project can be tailored to focus on one or more of these sub-areas. Some example applications are better numerical methods for the Landau-deGennes Q-tensor model, optimal material design, and simulation of active matter as a model of bacterial swarms.

Student expectations: Students will learn some general background material, explore the mathematical model through simulation and/or mathematical analysis, do scientific visualization (in 3-D), and take part in the writing of a journal paper.

Skills required: Students should at least know multi-variable calculus, some ordinary differential equations, and some basic (sophomore level) physics. They should also have some experience in programming, e.g. Matlab, or Python, or C, etc. Knowledge of mathematical analysis and/or advanced numerical methods is a plus, but not required.

Prior mentoring experience: I have mentored a total of 8 undergraduates (5 have been REU students). The most recent two students, Angelique Morvant and Ethan Seal worked on a LC project that resulted in a published journal paper.

References:

1. Mottram, N. J. & Newton, C. J. P., “Introduction to Q-tensor theory,” ArXiv e-prints, 2014.
2. Virga, E. G., “Variational Theories for Liquid Crystals,” Chapman and Hall, London, 1994, 8, 376.
3. Ball, J. M., “Mathematics and liquid crystals,” Molecular Crystals and Liquid Crystals, Taylor & Francis, 2017, 647, 1-27.
4. de Gennes, P. G. & Prost, J., “The Physics of Liquid Crystals,” Oxford Science Publication, 1995, 83.
5. Lagerwall, J. P. & Scalia, G., “A new era for liquid crystal research: Applications of liquid crystals in soft matter nano-, bio- and microtechnology,” Current Applied Physics, 2012, 12, 1387-1412.

## A variation of the stable marriage problem

Current students: Tobias Bork

Description:

Problem description. In the stable marriage problem there are two disjoint sets M and W (men and women), where each member of M has a preference (linear order) of members of W and, similarly, each member of W has a preference of members of M. The problem is to find a matching such that the following unstable scenario does not happen: some m and w prefer each other than their matched partner.

We consider a variation, which allocates residents to hospitals. Again, assume that each hospital and each resident has a preference. In addition, assume that some residents are married to each other and the couple has a joined preference. The problem is to find a stable assignment.

It is known that the problem is NP-hard if each couple is allowed to have two or more choices. So we only consider the case when each couple has only one choice. Under this assumption, the problem was solved when each hospital and each resident has at most two choices. Tobias is trying to generalize this result, by considering the case when each resident has many choices. It seems he has obtained an algorithm for this case. It it works out, he should have a nice short paper.

## Quantitative Nevanlinna-Pick interpolation

Recruiting: Interested students may contact Prof. Zimmer.

Description:

The Nevanlinna-Pick interpolation theorem is a vast generalization of the Schwarz lemma in complex analysis. In particular, it gives necessary and sufficient conditions for the existence of a holomorphic map of the disk to have prescribed values at certain points. This theorem was developed in the early 20th century and has a number of important applications in pure and applied mathematics. The purpose of this project is to develop new quantitative variants of this classical result. Analytic techniques and non-Euclidean geometry (e.g. real hyperbolic geometry) will play an important role in this investigation.

Background Required: MATH 4036 Complex Variables and MATH 4031 Advanced Calculus I.

Learning Objectives: learn about topics in analysis and non-Euclidean geometry, learn about doing original research in pure mathematics

## Predicting outcomes in college football using machine learning

Recruiting: Interested students may contact Prof. Zimmer.

Description:

Recently sports news organizations like ESPN and 538, have developed mathematical models to predict the outcomes of sports games. The goal is this project is develop our own models that can compete with the professionals.'' The types of models will depend on the participant's background, but can range from adaptations of the ELO ranking in chess to sophisticated neural networks.

Background required: MATH 2085 Linear Algebra, MATH 2057 Multidimensional Calculus, and a basic knowledge of Python would be ideal, but MATH 1550 Calculus I, a basic knowledge of any programming language, plus a willingness to learn a lot is sufficient.

Learning Objectives: develop skills in mathematical modeling and computer programming, learn about methods in machine learning and how to use the standard machine learning libraries in Python

## Flow Semigroups: Global Linearization of Nonlinear Problems

Current students: Amy Adair, Arun Banjara, Rohin Gilman, Logan Hart

Recruiting: Interested students may inquire about becoming involved in this project.

Description:

In this project, we investigate how Koopman’s global linearization method can be used to (a) extend exponential operator splitting methods like the Lie-Trotter or Strang Product Formulas from linear semigroup theory to approximate solutions of non-linear integral, delay, or differential equations such as x′(t) = F(t, x(t)), x(s) = x, t ≥ s; (b) study qualitative properties of the solutions of the underlying nonlinear problem (flows) in terms of spectral properties of the generator of the associated linear flow semigroup (Lie generator).

This project builds on previous work by late LSU Professor J.R. Dorroh (1937 - 2015) and his colleague J.W. Neuberger from the University of North Texas (see also Chapter 19 in J.W. Neuberger, A Sequence of Problems on Semigroups, Springer 2011).

## Nodal sets of eigenfunctions in balls

Current students: Zilin Li

Recruiting: Interested students may contact Prof. Zhu.

Description:

Eigenfunctions of a ball refer to the vibrational modes, or harmonics, of a ball, that vibrate at a certain frequency. The nodal set of an eigenfunction refers to where the material (maybe air) is stationary during the vibration.

Students are working on determining the size (technically the "Hausdorff measure") of nodal sets of eigenfunctions in the ball in 2 dimensions. A famous conjecture of Yau states that the Hausdorff measure of nodal sets of eigenfunctions on a compact manifold is bounded below and above by square root of eigenvalues. In the projects, students are verifying this conjecture explicitly in balls. This involves solving eigenvalue problems with different boundary conditions on the surface of the ball. Based on the solutions, one can characterize the graph and shape of nodal sets in the balls and then explicitly compute the measure of the nodal sets.

Students will broaden their knowledge on partial differential equations, use mathematical software such as MATLAB, LATEX, and familiarize themselves with scientific writing.

## Bethe equations for the Gaudin model and Wronskian relations

Recruiting: Interested students are encouraged to contact Prof. Zeitlin.

Description:

Integrable systems form an important part of modern mathematics; they are systems with enough conserved quantities that allow one to determine the state of the system by knowing these quantities along with other associated data. Apart from being interesting on their own, they appear to be crucial in the study of many other areas of mathematics, geometry in particular. While their applications can be quite complicated, such as those that appear in modern enumerative geometry, the underlying structures within integrable systems are quite simple. This project is devoted to a typical example of this kind, called the "Gaudin model." We are interested in the relation between two mathematical entities that arise in studying this model--certain equations called the "Bethe" equations and "Wronskian" matrices for polynomials. While the solution of this problem addresses a cutting-edge topic of modern mathematics, known as the geometric Langlands correspondence, the tools needed for its solution are rather basic--calculus, linear algebra and a bit of complex analysis, and the necessary knowledge from the latter can be easily attained on the way.

## Informetric indicators for citation networks

Faculty advisors: Profs. Larry Smolinsky and Dan Sage and LSU Librarian Aaron Lercher

Current students: Aaron Cao

Description:

Informetric indicators, particularly based on the network of citations, have become important in scientometrics, which is an interdisciplinary area in information science. There is a substantial literature grappling with the meaning of citations and comparing them to other metrics. This project is examining citations in light of a decade of the American Mathematical Society's identification of featured reviews.

## Control of dynamical systems with engineering applications

Recruiting: Interested students may contact Prof. Malisoff. Description:

Background: Control systems arise in many important engineering applications, including bioreactors, human-computer interactions, marine robotics, motors, and the control of unmanned aerial vehicles. Control systems can be modeled mathematically as dynamical systems that can contain a state-dependent parameter called a control. Controls can represent forces that can be applied to the physical system being modeled, and then the feedback control problem entails finding formulas for the control such that all solutions of the dynamical systems satisfy certain prescribed behavioral goals. This project has been funded by research grants from the National Science Foundation Directorate for Engineering.

Student expectations: Students will learn basic concepts and results for control systems and apply feedback controls to mathematical models using software, in collaboration with engineering students or faculty.

Skills required: Students should first take sophomore level elementary differential equations and linear algebra and should have an interest (but not necessarily experience) in using MATLAB or Mathematica software.

Prior mentoring experience: I have co-led a marine robotics project that included undergraduates, and REU projects involving mechanical systems arising in the oil and gas industry and other contexts.

References:

1. Astrom, K., and R. Murray. Feedback Systems: An Introduction for Scientists and Engineers. Princeton University Press, Princeton, NJ, 2008.
2. Hespanha, Joao. Linear Systems Theory, Second Edition. Princeton University Press, Princeton, NJ, 2018.
3. Mukhopadhyay, S., C. Wang, M. Patterson, M. Malisoff, and F. Zhang, "Collaborative autonomous surveys in marine environments affected by oil spills," in Cooperative Robots and Sensor Networks 2014, A. Koubaa and A. Khelil, Eds., Studies in Computational Intelligence Series Vol. 554, Springer, New York, 2014, pp. 87-113.

## How to design an automobile insurance product for autonomous vehicles

Faculty advisors: Matthew Arnold, Blue Cross Blue Shield of Louisiana and Prof. Larry Smolinsky, LSU math

Current students: Mason Guillot, Mark Russo, Kaju Sarkar, Mason Guillot, Emily Turner Description:

It is anticipated that the current automobile insurance market may eventually decrease with future public adoption of autonomous vehicles. Such a shift from traditional automobile products to autonomous vehicle products is likely to usher in a new era of risk and liability considerations that will demand big data capabilities and new actuarial models. Some of the objectives for this project are to recommend a design for a new autonomous vehicle insurance policy, and develop loss cost estimates, also known as pure premiums, for the new policy. Recommend a launch date for the policy and provide a ten-year forecast of pure premium from the point of launch.

This project is part of the 2019 Student Research Case Study Challenge for the Society of Actuaries. https://www.soa.org/research/opportunities/2019-student-case-study/

## Stochastic algorithms

Current students: Osarumwense Adun and Soheil Motlagh

Recruiting: Interested students are welcome to contact Prof. Ganguly.

Description:

Stochastic Algorithms are indispensable tools in this modern age of fast computation. For example, algorithms like Markov Chain Monte-Carlo have revolutionized simulation of high-dimensional probability distributions which underlie many models in physics, biology, economics, and last but not the least data-science. In short, the project is about two things:

(a) simulation of Markov processes arising in different contexts,
(b) estimation of unknown parameters of Markov process from observed data.

Markov processes are some of the most popular and natural ways of modeling evolution of a variety of systems in time. For example, the movement of stock prices, dynamics of cellular reactions, motions of microscopic particles in force fields can be efficiently captured by suitable Markov processes. Typically, these processes are described as solutions of certain stochastic (random) differential equations. So it is important to devise numerical schemes which can simulate trajectories of these stochastic differential equations. Simulation can help us understand behavioral patterns of the systems these equations are modeling. For example, for cellular processes, we can use these types of simulation techniques to deduce distributions of certain protein molecules at equilibrium. Developing and implementing fast and efficient simulation algorithms for these types of equations is the objective of (a).

In (a), it is tacitly assumed that the model parameters are known, otherwise simulation cannot be carried out. But in reality, not all of their values are known. It is then necessary to first estimate them from observed data points. For example, one might be interested in estimating the volatility of a stock from observation of its price over certain time period. Developing and implementing such inference schemes is the goal of (b). This can be thought of as an "inverse problem" of (a). Of particular interest are Bayesian algorithms, which naturally provide uncertainty quantification of the desired estimates.

Skills required: Students should have good knowledge of multivariable calculus, probability theory. Sone knowledge of stochastic processes, statistics and ordinary differential equations will also be helpful. They should also have experience in programming. Preferred programming language is Python or R, but this can be picked up easily if the student has knowledge of any other programming language (like Matlab).

## Local Langlands correspondence for SL2

Current students: Matthew Bertucci

## An online graphical user interface for electromagnetic waves in layered media

Current students: Noah Templet and Michael Sheppard

Recruiting: There is a need for a student to help write an exposition of the physics and mathematics behind the GUI--the student should be good at electromagnetics and explaining it in writing.

Description:

The development of this graphical user interface was supported by grant DMS-1411393 (2014-2018) from the National Science Foundation at Louisiana State University under the guidance of PI Stephen P. Shipman. As one of the projects covered by the grant, undergraduate students created the mathematical code and web interface to compute and simulate harmonic electromagnetic fields in layered media. The goal is to make available to the community a versatile online application for the computation of EM fields in media with any number of layers having arbitrary electric and magnetic tensors. The objectives are (1) to allow scientists to explore phenomena of scattering, guided modes, and resonance in the most general EM layered media and (2) to provide a pedagogical tool for students and professionals to learn EM in layered media.

The project is in its initial stages and is being further developed by undergraduate students through various funding sources, including President’s Aid from Louisiana State University. Further developments include a tutorial on the theory of electrodynamics in layered media, a user manual, computation of transmission and reflection coefficients as functions of frequency and angle of incidence, and computation of dispersion relations for pure and leaky guided modes.

## Schrödinger operators on graphs

Current students: Daniel Rockwell

Recruiting: Interested students may inquire about becoming involved in this project.

Description:

Schrödinger operators come from quantum mechanics and have constituted a large part of the mathematical study of spectral theory for almost a hundred years. There is differential equation involving the "Laplacian" and a "potential" (and maybe other things), and we are interested in its spectrum and eigenfunctions, as a vast generalization of eigenvalues and eigenvectors for matrices. This undergraduate project investigates some problems concerning Schrödinger operators on graphs (edges connecting vertices) that repeat periodically in space. The mathematics involves ordinary differential equations, linear algebra, polynomial algebra, and computational algebra.

This project is funded by the National Science Foundation under grant NSF DMS-1814902 entitled "Asymmetry, Embedded Eigenvalues, and Resonance for Differential Operators."

## Wave scattering by periodic structures: Surface modes and resonance

Recruiting: Interested students may inquire about becoming involved in this project.

Description:

When waves, such as electromagnetic or acoustic ones, interact with a periodically repeating structure (like a corrugation or perforated film), they get reflected in different directions and can produce waves that run along the surface of the structure. We have developed mathematical algorithms and numerical code for computing these waves. We are interested in using this code to explore surface-wave phenomena in a variety of structures that are of interest from the phenomenological and applied points of view.

This project is funded by the National Science Foundation under grant NSF DMS-1814902 entitled "Asymmetry, Embedded Eigenvalues, and Resonance for Differential Operators."

## Title: Visualizing Trends in Teacher Production Using Title II Data

Current students: Daniel J Brignac, Nicholas Angelo, Garrett A Harvey, Kirby J Moore

Recruiting:

Description: Detailed data on K-12 teacher production is collected by the U.S.Department of Education and published on title2.ed.gov. At present, data spanning years 2010--2015 is available. All teacher-eduation programs in the USA are listed, togther with their demographics, program features, and the numbers of teachers produced in all possible areas, subjects and majors. Data is stored in separate files by year. The web site does not provide any way of detecting or viewing trends. This project aims to remedy this by determining the most impirtant trends across the years represented and providing informative graphical summaries.