Undergraduate Mathematics Projects

Note that the difficulty here is showing that edges must cross in any drawing of this graph. A remarkable fact is that these two graphs are essentially the only obstruction to planarity; any non-planar graph essentially contains one of these as a subgraph. A beautiful obstruction to graph planarity was defined by van Kampen in 1932. It involves studying the shape (i.e., topology) of a corresponding graphical configuration space, a two-dimensional complex built out of squares. This obstruction is better known to topologists than graph theorists, and this project will involve giving alternate proofs of various theorems in graph theory using this topological tool, and hopefully prove some new results!

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).

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.

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.

1. simulation of Markov processes arising in different contexts,
2. 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.

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.

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

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

"Shape-Up" Sima Sobhiyeh is leading six undergraduate students (plus other peripheral support from graduate and CS students) in using deep learning techniques to analyze 3D optical scanning data.

"Improving Reading Skills" Rick Mekdessie and Sudip Sinha are leading five undergraduate students in building a treatment program for young students with dyslexia or other reading disorders. They will be building an app with games to improve reading skills.

"Does Elvis (the dog) know calculus?" This project involves the calculation of optimal paths in media with different components, such as beach sand and water in the case of Elvis fetching a ball.

Students will learn some general background on periodic operators, get research experience, and take part in the writing of a journal paper.

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.

The code for this project is available.

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/

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.

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.

This project is devoted to the study of meromorphic connections on the projective line (known as opers) with regular and irregular singularities. The goal is to describe them in terms of Bethe-type equations and relate them to the deformation of the Gaudin model.