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.
Jenia Tevelev: Algebraic Geometry, details can be found at https://websites.umass.edu/tevelev/mentoring/
Leili Shahriyari: Mathematical modeling of bone
Owen Gwilliam, Abstract algebra to tackle problems like matrix models or quantum error correcting codes
Matthew Dobson: Speeding up Markov Chain Convergence - Markov chains are widely used in modeling randomness and as a tool for sampling large data sets. There are many techniques for speeding up the sampling process, and this project will be looking at enhancing the sampling of finite state-space Markov chains using both numerical and analytical approaches.
Weimin 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.
Andreas Buttenschoen: Computational biology
Troy Wixson, 1) Neural Bayes Estimators for time series models that are developed to capture the dependence in the tail of the distribution. These models are used, for example, to assess the increased risk of wildfires. 2) Create a map of a county (or state or country) which demonstrates where wildfire risk has increased and by how much. 3) Predict which plant species will be most suitable in different regions across the US conditional on climate change and the presence of invasive species.
Kien Nguyen, The REU project investigates chaotic dynamics and ergodic behavior in planar billiard systems, where a particle moves freely and reflects elastically off curved boundaries. Students will study how geometric features influence hyperbolicity, Lyapunov exponents, and long-term statistical behavior. The project combines analysis of the billiard map with computational experiments to measure finite-time dynamical quantities. Students may also explore parameter dependence and intermittency phenomena in classical chaotic models..
Paul Hacking, Consider the complex projective plane P2 and algebraic curves C inside it, defined by an equation F = 0 where F is a homogeneous polynomial in 3 variables of some degree d. If C is smooth, then C is a Riemann surface of genus g = (d − 1)(d − 2)=2, that is, C is topolgically equivalent to a sphere with g handles attached. We consider the other extreme, however, where C is highly singular so that, when we resolve the singularities of C, we obtain the Riemann sphere (g = 0). There is a sequence of such curves discovered by algebraic geometers and studied recently by J´anos Koll´ar (in algebraic geometry) and Dusa McDuff (in symplectic geometry), among others. These singular curves have the further property that there is a unique singular point p on C that can be locally defined by an equation yn = xm in analytic coordinates x; y at p for some positive integers m and n. Motivated by this, we consider a so called weighted blowup of the projective plane P2 at the point p with respect to a choice x; y of analytic coordinates at p and weights (m; n). This is a surgery (or birational modification) of the projective plane which removes a neighborhood of the point p and replaces it by a neighborhood of a Riemann sphere E in a singular surface with two cyclic quotient singularities on E. This yields a projective singular surface X with a birational morphism to the projective plane P2, which contracts E to the point p and is otherwise an isomorphism. If C ⊂ P2 is a curve with local equation yn = xm at p as above, then the strict transform of C on X (the closure of the inverse image of C n fpg) is the resolution of C. The project will study the geometry of the surface X, and in particular the following question: which homology classes of (real) dimension 2 on X are realized by algebraic curves? The results mentioned above give a complete answer for 1 ≤ n=m < τ4 where τ = (1+p5)=2 is the golden ratio, and there are infinitely many cases as n=m approaches τ4 from below, but in general the answer is not known. This problem is closely related to the famous Nagata conjecture in algebraic geometry, which studies the geometry of the ordinary blowup of P2 at n general points, and the answer to the analogous question in that setting is unknown for n > 9 and n not a perfect square.
Chris Cox, No-slip Billiards: Billiard dynamical systems, in which particles move freely until colliding with a boundary, have been studied for over a century. The alternative model of no-slip billiards, incorporating angular momentum, has recently been a topic of interest, but many fundamental questions remain open. We will investigate these both analytically and numerically.
Qian Zhao (ds4cg): In this project, you will study how transportation access and local affordability influence food insecurity, nutritional insecurity and health of the population. You will analyze both community level data from County Health Ranking and individual survey data from National Health and Nutrition Examination Survey (NHANES) and Household Pulse survey.
Yao Li: Diffusion model is a popular class of probabilistic generative AI models. This project is a pilot study of applying Malliavin calculus into score estimation in diffusion models. Students will use Monte Carlo simulation to estimate derivatives of the Fokker-Planck equation and the Feynman-Kac equation. Working knowledge of C++ is required.
Weiqi Chu, This project studies operator splitting methods for solving the Liouville–von Neumann equation, with an emphasis on nonlinear and time-dependent quantum dynamics. Direct simulation of these models is often computationally challenging, but operator splitting provides a simple and efficient way to decompose the evolution into tractable steps while retaining key dynamical features. The project requires knowledge of numerical ODEs and programming. No prior knowledge of quantum physics is required.