Here is an evolving list of the projects proposed by mentors for this summer. You do not need to choose a project in your application. The committee will coordinate with mentors and match with students.
Exploring Neural ODEs in Quantum System Simulations. Neural ODEs combine machine learning with traditional ODE frameworks, offering a flexible approach to modeling complex systems. The goal is to use Neural ODEs to approximate and simulate quantum mechanical systems, such as the evolution of a particle in a potential well. Students will use Neural ODEs to model quantum and classical systems from data, without needing prior quantum mechanics knowledge. A solid foundation in Python is essential, as the project involves implementing and training models in Python. By Weiqi Chu
Use machine learning methods to detect credit card fraud and predict stock prices. ByJinguo Lian
Research projects on algebra, combinatorics. By Alexei Oblomkov
PINNs for solving differential equations. 1) For hyperbolic conservation laws, we'll focus on 1D inviscid Burgers equation. The goal is to compare the classic methods like TVD methods to PINNs with a focus on the accuracy and entropy condition. 2) For parabolic/elliptic equations, we'll solve the 2D diffusion equation with a very strong anisotropic diffusion tensor which doesn't align with the direction of the mesh. The goal is to study whether PINNs will behave just like other ad hoc methods, which may lead to maximum principle violating numerical solutions. By Qian-Yong Chen
The REU project is concerned with constraints on rational cuspidal curves in the complex projective plane, a classical topic in algebraic geometry. We aim to investigate an alternative approach to this classical problem, which is computer aided and based on tools from symplectic geometry and 4-manifold topology. By Weimin Chen
Data mapping for food insecurity analysis: In this project, you will explore datasets containing features related to food insecurity through aspects such as availability, access and affordability. Examples of datasets include National Health and Nutrition Examination Survey (NHANES), National Health Interview Survey (NHIS), American Community Survey (ACS) and NeilsenIQ scanner data. You will identify and create relevant features by geographic identifiers and explore methods to quantify causal contributions of each feature on food insecurity. By Qian Zhao, Chaitra Gopalappa, and Eleni Christofa.
Analytical and Numerical Simulation of the Nonlinear Schrödinger Equation: The nonlinear Schrödinger equation (NLS) is a fundamental equation in mathematical physics that describes wave propagation in various contexts, including optics, Bose-Einstein condensates, and water waves. Under appropriate physical conditions, the NLS equation admits special solutions, whose stability plays a crucial role in understanding the asymptotic dynamics of the wave. In this project, we will study the stability of these solutions in the exterior of an obstacle, combining theoretical analysis and computational simulations. By Oussama Landoulsi
Alternative Models of Mathematical Billiards: The basic collision model of "angle in equals angle out" has proved to be remarkably useful for a wide range of applications. Recently, mathematicians have considered new models of billiards, for example incorporating particle spin or defraction, opening up many new questions. We will conduct numerical simulations, a first step in understanding the behavior of the new models. By Chris Cox
Mathematics and Statistics Education Research, by Adena Calden