Research Experiences for Undergraduates (REU)
Hoewoon Kim: Navier-Stokes
Equations and Stability Problems in PDEs.
Dr. Kim is a researcher
in partial differential equations, mathematical physics, and probability.
Project 1: Navier-Stokes equations. This project involves mathematically and physically
understanding the motion of incompressible fluids (e.g. water) in certain domains, such as
a ball in three dimensional space, through basic differential equations such as diffusion
and Laplace equations. In this project we'll do this analytically and/or probabilistically.
Project 2: Stability Problems. This project deals with the following interesting question:
“Under what conditions must a mathematical object that satisfies a certain property
approximately, e.g. a differential equation, be close to satisfying the property exactly?"
Probability and Stochastic Processes.
Dr. Kovchegov is a researcher in mathematics of data science, mathematical models of
statistical mechanics, interacting particle systems, models of mathematical biology,
stochastic self-similarity, and quantum computation.
Project: Analyzing the generalized notions of skewness and standardized central moments.
We will use Frechet means and the stochastic dominance method in the analysis of
generalized skewness. Additionally, skweness, excess kurtosis, and other standardized
central moments will be considered using the Stein operator. Students are expected to be
familiar with advanced calculus, upper division probability and/or statistics, and know
either MATLAB or R.
Patrik Nabelek: Rational Functions
in Mathematical Physics.
Dr. Nabelek is a researcher in applied mathematics specializing in linear and nonlinear
waves, physics, geometry, differential equations, probability, and numerical analysis.
Project: Completely integrable systems. Completely integrable partial differential
equations appear in classical mechanics, quantum mechanics, fluid mechanics and oceanography.
The goal of this project is to do computational studies on simple models in soliton physics
and quantum mechanics using methods that require only a knowledge of rational functions.
Instead of being hard to implement, these methods are hard to derive, and the reason it
is possible to use them comes from the theory of completely integrable systems. The
mathematics and physics of completely integrable systems relate to some important recent
discoveries in combinatorics. Thus, this project would be beneficial to most students
hoping to pursue modern mathematics and physics research.
This project will require proficiency with rational functions, series, complex numbers
and matrix operations typically found in courses on calculus, series, differential
equations and linear algebra. A conceptual understanding of Fourier series as well as
familiarity with MATLAB or Mathematica and/or a course that involves coding in a scientific
context (such as numerical analysis) would also be helpful.
Mike Rosulek: Secure
Computation in Cryptography.
Dr. Rosulek is a researcher in cryptography, with specific interest in secure computation
protocols. A secure computation protocol allows n parties, each of which has a private
input, to learn some agreed-upon function of their inputs, while learning nothing else.
Project: Secure Computation. Suppose n users each hold private data, and they would like
to compute some agreed-upon function of that data. Secure multi-party computation (MPC)
is a cryptographic tool that allows them to do so, in a way that releases *only* the
output of the computation, and leaks nothing else about the private inputs. In this
project we will explore characterizations of which functions can be securely computed
in various settings (security definition, number of users, communication channels between
Samaneh Yourdkhani: Phylogenetics
in Mathematical Biology.
Dr. Yourdkhani is a researcher in mathematical biology with
specific interest in developing new mathematical methods of inferring evolutionary
relationships between species from their sequenced genomes. This involves mathematics,
statistics, computer science, and biology, and is fundamental for many other areas of
science and medicine.
Project: Inferring a metric species tree from topological gene trees. There are many ways
to infer a species tree relating populations from gene trees relating biological sequences.
But most of these methods do not give a metric species tree, meaning that we just get a
topology of a tree, not its branch lengths. Also, the conflict between gene trees and
species trees makes the inference problem challenging as standard phylogenetics methods
applied to different gene trees for the same collection of species often give different
trees. In this project, we will investigate the relationship between gene trees and species
trees, and develop a statistically consistent method to infer a metric species tree from
topological gene trees under a specific model. We will work both theoretically and
practically. Students with a background in either mathematics and/or computer science
are encouraged to apply.
In general, the proceedings
from the past few years will give an idea of the
variety and general levels of the projects.