Research Experience for Undergraduates (REU) in Mathematics with support from the National Security Agency
About the Program
After a hiatus, we plan to reboot the REU program for Summer 2021 (May 31 - July 23) either in-person (if deemed safe) or virtually. Students will likely work in groups of 2 (or more, pending local funding) with a faculty adviser from SUNY Potsdam or Clarkson University. Participants will receive a stipend of $4,400. If the program is in-person, participants will receive free housing in an on campus apartment or dorm room with access to cooking facilities, as well as $375 in funds to support travel expenses to/from Potsdam. We are seeking applicants, in particular from students from groups traditionally under-represented in mathematics.
Interested applicants should submit their application materials, including:
- Contact information (email, phone, mailing address)
- Expected date of graduation
- Preference of topic
- 300-500 word statement of interest
- 2 letters of recommendation (that address your interest and positive experiences in mathematics, work ethic, and ability to work in a group)
- Unofficial transcript
Please send application materials, including letters of reference, via Mathprograms.org. Applications already submitted via email do not need to be resubmitted.
The deadline for submitting all application materials is March 29, 11:59 p.m. Applicants must be US citizens or permanent residents, and plan to be enrolled in an undergraduate program in the Fall 2021 semester.
Topics to Be Explored
- Links in embedded graphs (Joel Foisy, SUNY Potsdam)
A spatial embedding of a graph is a way to place a graph in space, so that vertices are points and edges are arcs that meet only at vertices. Mathematicians have studied graphs that are intrinsically linked: that is, in every spatial embedding, there exists a pair of disjoint cycles that form a non-splitable link. Sachs and Conway and Gordon showed that the complete graph on 6 vertices is intrinsically linked. More recently, people have studied graphs that have non-split links with more than 2 components, as well as knotted cycles, in every spatial embedding. Building off a previous REU group’s work, we’ll first examine certain graphs that can be embedded in the plane that can help us better understand graphs that have a non-split 3 component link in every spatial embedding. We will use tools from graph and knot theory. Experience in these areas is not required. (minimum requirement: good experience in at least one proof intensive math class)
- Numerical solutions to high-dimensional stochastic differential equations (Guangming Yao, Clarkson University)
Mathematical models described by partial differential equations (PDEs) have been a necessary tool to model nearly all physical phenomena in science and engineering. Due to the growth of the complexity in emerging technologies, the increase in the complexity of the PDEs for realistic problems become inevitable. Some of the complexities are, for example, complicated domains, high-dimensional spatial domains, multiscale, large-scale problems, etc. This project will develop a new algorithm for solving partial differential equations (PDEs) in high dimensions by solving associated backward stochastic differential equations (BSDEs) using neural networks [*], as is done in deep machine learning. Another option is to employ radial basis functions [*] to reduce the dimensions in the numerical simulation. The project can be future enhanced by adding complicated computational domains, large scale problems with or without multiscale feature. If a particular student became interested in parallel computing, there could be a productive a collaboration between this REU site and the NSF REU Site: High Performance Computing with Engineering Applications, led by the Department of Electrical and Computer Engineering, Clarkson University, Potsdam, NY.
The process of dealing with realistic PDE models with various behaviors of the solutions will help students to understand the key concepts in computational science, including accuracy, efficiency, convergence and stability. Numerical simulation requires programming in MATLAB or Python to test efficiency and accuracy of the proposed algorithms by solving various applied problems such as the Allen-Cahn equation[*], and nonlinear pricing models for financial mathematics[*], the Black-Scholes equations [*], the Boltzmann transport equations [*] for modeling phonon distribution functions in high dimensional space (higher than 6 dimensions), and/or more advanced PDE models for COVID-19[*]. Fundamental concepts in computational mathematics and numerical analysis can be introduced at beginning, followed by particular focuses of students’ choices of PDE models.
(A course in differential equations required. Familiarity with MATLAB or similar software recommended, though students with a willingness to learn some coding are encouraged to apply)
(*reference available on request)
Dr. Joel Foisy, Department of Mathematics
44 Pierrepont Avenue
Potsdam, New York, U.S.A. 13676
Phone: (315) 267 - 2084