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). In recent years, hundreds of students have participated in undergraduate research mentoring.
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.
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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
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Funding for Undergraduate Research
Pay: $12/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.
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Finding a Research Mentor
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.
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Topics: Number theory, algebra, and logic.
Prerequisities: Usually Math 371
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.
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.
Undergraduates in this program work in dynamical systems, specifically in the areas of celestial mechanics, complex dynamics, or toral automorphisms. Students who have successfully mastered the concepts in Math 314, 332, 334, and 343 are sufficiently prepared to engage in research in these areas.
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. Interested students are invited to attend meetings of the Computational Number Theory research group at 10 AM on Thursdays during fall and winter semesters.
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.
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.
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.
Students would study point lattices — discrete, full-rank subgroups of R^n, as applied to physics and cryptography. Possible topics include lattice algorithms and periodic colorings of lattices. Some minimal knowledge of group theory may be helpful.
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. ”
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).
Pressure recovery problem: Numerical solution of non-Newtonian fluid flows is a theoretically and computationally challenging subject that also has much application in many practical areas. One approach to handle such problems is to employ divergence free finite elements to compute the velocity field. With the velocity is known, it is also of interest to find the pressure. How one recovers the pressure leads to several interesting problems that will be addressed in this senior undergraduate or graduate level research project.
GUI for sincpack: The sinc method is a theoretically advanced method for finding approximate solutions of differential equations. Dr Stenger from the University of Utah has recently released a set of matlab routines called sincpack. The project, which is accessible to any undergraduate with a background in programming and basic numerical methods, is to develop a GUI to aid novice users to solve classical problems without having to know the details of matlab or sincpack.
Image processing problems: Several interesting computational problems arise from image deburring and edge detection that are of interest in several application areas. One approach is to apply alternating direction methods to compute the solution of a nonlinear partial differential equation related to image processing. The research project is at senior undergraduate or graduate level.
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.
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.
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.
My research is in the area of spectral graph theory, which involves understanding graphs and networks by way of matrices that can be associated to a graph. In particular, eigenvalues of matrices reveal many interesting structural properties of a graph. Students working with me have investigated how spectral properties give information about various kinds of random walks on graphs, how to measure how well-connected a graph is, and other various structural properties. Research in this area requires a thorough mastery of Math 213 topics, and competence with writing mathematical proofs.
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