Undergraduate Research Experiences

Waves, trapped energy, and resonance in combinatorial and quantum graphs provide and excellent medium for research projects for undergraduate mathematics students at all levels. The subject touches many areas of mathematics, such as partial and ordinary differential equations, linear and nonlinear algebra, Fourier analysis, geometry, operators, and spectral theory, and benefits from a rich ideological symbiosis with phenomena of physics.

Over the years, I have worked with many undergraduate students in summer REUs, vertically integrated courses, and honor theses, on projects involving waves and resonance in graphs. Below are links to descriptions of some of these projects and posters created by students.

2022 Discover Day and Spring TX-LA Undergraduate Mathematics Conference

2022 Discover Day and TX-LA Undergraduate Mathematics Conference

This project was presented at LSU's 2022 Discover Day and the Third Annual meeting of the TX-LA UG Math Conference.

		Lillian is a mathematics and physics major at LSU. She investigated the free oscillatory frequencies of system of strings coupled together at their ends and produced animations of the oscillating modes. The computations involve finding eigenvalues and eigenfunctions of a Schrödinger operator on a metric graph.

2019 TX-LA Undergraduate Mathematics Conference

This project was presented at the Third Annual meeting of this conference: LINK.

Noah is a CS major and Michael is a math major at LSU.

2017 LSU Discover Day

In the spring of 2017, two students participated in Discover Day 2017 at LSU.

Stacey Wieseneck is a senior student of chemical engineering at LSU.

Calla Harper is a senior mathematics major who is in the Geaux Teach program at LSU.

2014 Summer Group on Quantum Graphs

In the summer of 2014, four undergraduate students participated in a research group on quantum graphs. They presented three posters at LSU's SURF, the Summer Undergraduate Research Forum.

Alexa Martin is a mathematics major at LSU and participates in LSU's Geaux Teach program for mathematics teachers. She investigated the vibrational frequencies of binary tree graphs. The spectrum of frequencies can be computed for a binary tree of any size n from the roots of a polynomial of degree n in the cosine of the frequency. These polynomials can be computed recursively in n as the determinant of an nxn matrix.

Michelle Laskowski is a mathematics major at LSU and participates in LSU's Geaux Teach program for mathematics teachers. She investigated the band-gap structure of the spectrum of 1D-periodic quantum graphs and how the bands depend on the complexity of the cell structure.

Melanie Dutel is a mathematics major at LSU; Morgan Landry is also a mathematics major and participates in LSU's Geaux Teach program for mathematics teachers. They investigated waves in a string with defects that are not regularly spaced. The problem is converted into a discrete dynamical system that exhibits initial exponential growth depending on the mass of the defects and fractal behavior.

2012 Summer Group on Resonance in Lattices

In the summer of 2012, four undergraduate students participated in a research group on lattice models of resonance phenomena in photonics. They presented four posters at LSU's SURF, the Summer Undergraduate Research Forum.

		Jeremy is a mathematics major at LSU. He investigated a one-dimensional model of a discrete bimodal transmission line. In certain frequencies, one mode is propagating and the other is evanescent. This is made possible by next-nearest neighbor interactions between sites on the discrete line. A carefully chosen defect in the line allows energy to be trapped in the evanescent modes. A perturbation of this system instigates a resonant interaction between the propagating mode and the trapped-energy state.

		Emelie is a mathematics major and participates in LSU's Geaux Teach program for math teachers. She investigated a lattice model for a phenomenon in optics known as "Fabry Perot" resonance, in which an optical mode of a photonic crystal cavity is strongly excited by radiation modes that weakly penetrates into the cavity.

		Grace is a mathematics major and participates in LSU's Geaux Teach program for math teachers. She investigated a lattice model for optical frequency selection, or filtering, by exploiting a resonant cavity. An unfiltered source field in an incoming transmission line strikes a resonant cavity, which captures energy at specifically tuned frequencies. An outgoing transmission line is excited a precisely these frequencies and therefore is able to extract, or filter, them for use in a frequency-specific device.

		Tyler is a mathematics major and participates in LSU's Geaux Teach program for math teachers. He investigated a lattice model for metamaterials. By tuning a periodically dispersed defect within a lattice host material, an isolated frequency band of propagation can be made to be arbitrarily narrow. This effectively creates a material that inhibits the propagation of waves at all frequencies of a source field and allow a single frequency to pass, or filter, through.

2009 Summer High School Math Circle

For the High School Math Circle in 2009, I presented a purely geometric way to construct the spectrum and trapped-energy modes for a simple one-dimensional lattice. Two students, Hira Khan and Joyce Ward, mentored by graduate student Rick Barnard, wrote an exposition of this problem and presented it in the last day of the Math Circle. The following fall semester, they travelled with a cohort from the LSU Mathematics Department to San Francisco to present their work in a Math Circle session of the joint meetings of the American Mathematical Society and the Mathematical Association of America.