# Undergraduate Research

The Mathematics Department is one of the top math departments in the nation for undergraduate mentored research. CURM, a national organization dedicated to assisting university math departments in undergraduate research, was founded here (see CURM home page

Undergraduate students can pursue research in various exciting topics. Many of these undergraduate research projects have led to publication and an opportunity to travel and present at various conferences. A list of a few recent publications by BYU undergraduates can be found here.

Why Undergraduate Research?

- Provide out-of-classroom learning experiences and apply class room knowledge to solve new problems.
- Develop and foster an analytical approach to doing research
- Gain motivation and create new knowledge
- Excellent experience and preparation for graduate school
- Develop oral and written communication skills
- Promote interactions with faculty and graduate students
- Make better informed decisions about your future career

Funding for Undergraduate Research

Pay: $14/hr — Students can work up to 20 hrs/week. To apply, please talk to a professor you are interested in working with. See below for a summary of projects professors are working on with students. Contact info for professors can be found here

#### Finding a Research Mentor

**Mark Allen**

My research involves modeling spread of disease or wildfires with differential equations. I have several possible undergraduate research projects that involve fractional derivatives. In many applications, modeling with a fractional derivative (like a 0.5 or 1.5) derivative is more accurate than modeling with integer order derivatives (like the first and second derivative). In order to start these research projects, a student should have already taken Ordinary Differential Equations (Math 334).

**Nickolas Andersen**

I am primarily interested in analytic number theory, especially the theory of modular forms and L-functions. Student projects combine computational and theoretical methods to prove new results in number theory. I am also interested in formalizing proofs using the Lean proof assistant. Students must have completed Math 290 (with Math 371 and/or Math 352 recommended) and be interested in coding in Mathematica and/or Sage (no prior experience with those languages is necessary).

**Lennard Bakker**

We study problems in an area of dynamical systems known as Celestial Mechanics. This includes analysis of a binary asteroid model, restricted N+k problems, and the general N-body problem. Initial training includes fundamentals such as the circular restricted three-body problem (that NASA uses to design space missions) and the theory of Hamiltonian systems. After initial training undergraduate students are given a problem in which numerical investigations are combined with analytic theory to understand the nature of solutions. Recent problems have involved various aspects of motion of binary asteroids. Students must have successfully taken Math 213, Math 215 (or have some coding skills), Math 314, and Math 334.

**Blake Barker**

Math FIRE lab (mathematical fire and industry research experience lab): We use machine learning, scientific computing, and modeling to advance knowledge about wildfires in ways that can aid wildfire managers. We are interested in problems like wildfire risk analysis, perimeter prediction, and ecological effects of burn severity. Before working with the group, students need to take a course on linear algebra, ODEs, and have some experience programming with Python.

**Zach Boyd**

I work in applied math/data science/math modeling, especially with the tools of network science. I have possible undergraduate projects across a broad range of application areas, such as global supply chains, genealogy, social drinking, brain networks, and network structure detection, to name a few. In terms of “mathematical purity,” I touch on some very pure topics, such as graph theory or functional analysis, but spend lots of my time close to the data doing modeling, algorithm design, data exploration, and so forth. There are no strict prerequisites to work with me, although the more you know in advance the more agile you will be. Particularly good preparatory topics include linear algebra, computer programming (e.g. Python), and network science. Data science/machine learning, dynamical systems, statistics, and real analysis can also open more topics to work on with me. If you already have a particular project you want to work on, I am open to talking about it, or I can provide topic ideas.

**Jennifer Brooks**

My group conducts research in complex analysis. Specifically, we study zeros of complex harmonic polynomials. It is helpful if students who join the group have taken Math 341 and Math 352, but I encourage any interested student to come talk to me.

**David Cardon**

My research is currently in the area of complex analysis. I study operators on entire functions with only real zeros that preserve reality of the zeros. This topic is motivated by the Riemann hypothesis. Students must complete of all of Math 341 (real analysis), Math 352 (complex analysis), Math 371 (abstract algebra) and begin working with me at least a year before graduation

**Gregory Conner**

Over the last few years I’ve had several undergraduate students work with me on research projects in low-dimensional wild homotopy groups. Topics range from geometric — understanding how “fractal-like” objects in the plane can be deformed in to others, to algebraic — understanding infinitely stranded braid groups, to analytic — understanding how to prove very delicate continuity arguments on wild subsets of our universe. These undergraduate research projects have all turned into masters’ theses at BYU and have lead each of the students into a high-quality mathematics Ph.D. program such as Vanderbilt, Tennessee and BYU.

**John Dallon**

I am doing research modeling properties of collagen lattices and cell motion. Students must have completed Math 334 and have some computational skills.

**Michael Dorff**

Minimal surfaces and complex-valued functions:

We investigate minimal surfaces in R^3. In some ways, minimal surfaces can be thought of as soap films that form when a wire-frame is dipped in soap solution–they tend to minimize the surface area for a given boundary condition. Images of minimal surfaces can easily be displayed by using computers, and this lends itself nicely to student explorations. We will use results about analytic functions from complex analysis (Math 332) to investigate minimal surfaces. To help introduce students to this topic and begin to do research, we have received a grant to write two chapters in a book on this topic along with exploratory problems using applets.

**Darrin Doud**

Undergraduate research with Dr. Doud can include topics such as modular forms with connections to Galois representations, diophantine equations, elliptic curves, and LLL-reduced lattices. A prerequisite for all of this research is Math 371, and several topics would require Math 372.

**Scott Glasgow**

Undergrads in this program either work in Mathematical Finance, including Extremal Events in Insurance and Finance, or in certain components of mathematical physics—symmetries, conservation laws, integrability. These topics require interest in probability theory, differential equations, and/or complex variables, and students will have had success in courses 334, 343, and/or 332.

**Chris Grant**

**Denise Halverson**

**Mark Hughes**

My research is in low-dimensional topology, where I study things like knots, surfaces, and 4-dimensional spaces called manifolds. Recently I have been working with undergraduates on a particular representation of knots called petal diagrams, which provides a connection between knot theory and the algebra of the symmetric group. Familiarity with some abstract algebra is helpful with this research. I’m also interested in studying knots and topological objects using machine learning. I’ve been working with students to apply deep learning models (including generative deep learning and deep reinforcement learning) to answering difficult questions in knot theory. This research requires calculus, linear algebra, and some experience with programming (preferably in Python).

**Stephen Humphries**

I have mentored many students in various Abstract Algebra subjects including: Group theory, Difference sets, Representation Theory, Combinatorics, semi-simple rings.

I am happy to consider doing mentored research with anyone who has obtained a good grade in 371.

**Tyler Jarvis**

Project 1: This project is focused on numerical algebraic geometry and multivariable root-finding. Students must have completed Math 341 and CS 235 and preferably have completed CS 240, Math 320-321

Project 2 :This project is concerned with physiological signals. It includes analyzing spectrometer and bioimpedance signals to identify blood analytes noninvasively. Student must have completed Math 320-321 and preferably have completed CS 235 and Math 322-323

**Paul Jenkins**

We study problems in number theory related to modular forms and their coefficients. Students who have successfully mastered the concepts in Math 371 and 352 will be better prepared to do research in these areas. Problems in computational elementary number theory are also available. More information on papers written by students in this group is available here

**Mark Kempton**

I work in the area of spectral graph theory, which examines how matrices and their eigenvalues can help us understand graphs and networks. Specific projects include: understanding the mixing rate of non-backtracking random walks on graphs; studying quantum state transfer phenomena, especially using isospectral reductions; studying Kemeny’s constant and effective resistance in graphs; finding bounds for eigenvalues of the Laplacian and normalized Laplacian. I am always willing to talk to students interested in getting involved in research. Requirements to work in this area are Math 213, with Math 290 strongly recommended. Extra experience with linear algebra is also nice.

**Xian-Jin Li**

1. Research on spectral theory of automorphic forms:

In 1956, A. Selberg introduced trace formulas into the classical theory of automorphic forms, a theory whose origins lie in the work of Riemann, Klein, and Poincar\’e. The theory of automorphic forms is intimately connected with questions from the theory of numbers, and is one of the most powerful tools in number theory. The discrete spectrum of the non-Euclidean Laplacian for congruence subgroups is one of the fundamental objects in number theory. My research interests are Selberg’s trace formula, Selberg’s eigenvalue conjecture, and the multiplicity of the discrete eigenvalues.

2. Research on Beurling-Selberg’s extremal functions:

In 1974, A. Selberg used the Beurling-Selberg extremal function to give a simple proof of a sharp form of the large sieve. By using the large sieve, E. Bombieri proved in 1965 a remarkable theorem on the distribution of primes in arithmetical progressions that may sometimes serve as a substitute for the assumption of the generalized Riemann hypothesis. The large sieve is closely related to Hilbert’s inequality. An open problem is to prove a weighted version of H. L. Montgomery and R. C. Vaughan’s generalized Hilbert inequality. A weighted large sieve can be derived from the weighted Hilbert inequality, and is fundamentally more delicate than the large sieve. It has important arithmetic applications. My research interest is to attack the open problem on the weighted Hilbert inequality. “

**Pace Nielsen**

My current projects are focused on abstract algebra. To work with me, I usually require Math 371.

**Nathan Priddis**

My research is inspired the the physics of string theory. Mostly I study a phenomenon called Mirror Symmetry, which basically is that in string theory, there is a choice along the way, that shouldn’t make any difference. But it does, and so you get two different kinds of mathematical objects, that should be the same. Mirror symmetry is a way to see how these objects are the same. My research requires a solid understanding of abstract algebra, and I will ask you to learn some things about algebraic geometry.

**Jared Whitehead**

1. We use Bayesian statistics to determine the location and magnitude of historical (prior to 1950) earthquakes in Indonesia using historical records of the resultant shaking and tsunami that resulted. This is a very interdisciplinary project that has students participating from 3-4 departments on campus at any time. Primary pre-requisites are a basic knowledge of Python programming, and some basic understanding of probability and linear algebra.

2. Another project is determining parameters of high dimensional dynamical systems from sparse observations of the system. This project is more theoretical currently, but has application to experimental fluid dynamics (with colleagues from Mechanical Engineering), weather prediction, and climate modeling. Pre-requisites are familiarity with Python programming, and at least one course (preferably more) in differential equations.

**Vianey Villamizar**

Project 1. This project is concerned with the development of 3-D grid generators with nearly uniform cell volume and surface spacing, respectively. The proposed algorithm will be based on recently developed 2-D quasi-linear elliptic grid generators with similar features. It requires knowledge of boundary value problems of partial differential equations (Math 347), numerical iterative methods for linear and non-linear systems, interpolation techniques (Math 311), and good programming skills.

Project 2. We propose to obtain numerical solution for the Helmholtz equation in locally perturbed half-plane with Robin-type boundary conditions. This problem is motivated by a system sea-coast where each media is represented by a half-plane. Knowledge about partial differential equations (Math 347), numerical solution of partial differential equation (Math 511), and numerical methods in general is desirable.