Application Information: Interested applicants should submit their application materials, including:

Cover letter with contact information (email, phone, mailing address) and topic preference (top 1 or 2)

Expected date of graduation

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. The deadline for submitting all application materials is March 27, 11:59 p.m. Applicants must be US citizens or permanent residents, and plan to be enrolled in an undergraduate program in the Fall 2022 semester. 

Program is pending support from the National Security Agency.

Topics to Be Explored 

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) 

Using comorbidities, demographic, and socioeconomic data to predict onset of rheumatoid arthritis (Sumona Mondal, Clarkson University):   Rheumatoid arthritis (RA) is an autoimmune inflammatory joint disease with a complex pathophysiological basis. The chronic and debilitating nature of the disease requires diagnosis and management under close rheumatologist supervision, however, a severe shortage of rheumatologists in the rural area creates barriers to proper care. Smart devices provide the opportunity to monitor individuals for risk of RA. To make optimal use of the data gathered by modern smart devices in RA risk assessment, it is necessary to mine predictable factors that have high associations with RA. Preliminary statistical studies conducted by Prof. Sumona Mondal (Mathematics, Clarkson University) and his collaborator on this project Prof. Shantanu Sur (Biology, Clarkson University) have indicated that factors related to human lifestyles such as high body mass index and depression, and demographic factors such as gender and ethnicity show correlation with RA, providing the potential to use these factors in smart RA risk prediction. As part of this research, the REU student will investigate the association of socioeconomic factors with RA and will develop learning-based algorithms to improve rural RA care by identifying critical factors associated with the disease and building predictive models. A course in basic "probability and statistics" is required. Some preliminary knowledge of regressions and familiarity with the R programming language will be helpful. However, students with a willingness to learn R or Python are highly encouraged to apply. 

Hybrid inpainting method (Prashant Athavale, Clarkson University):   An image can be viewed as a function. The image data can be damaged and part of the image is destroyed. Inpainting is a problem of filling in missing part of an image. There exist various ways to fill in missing information in the image processing literature. These methods can be categorized into variational based and exemplar-based inpainting methods. In variational methods we solve a minimization problem to flow the information from the boundary into the missing region. The variational methods are successful when the missing regions are composed of large number of small disconnected regions. In exemplar based methods parts of the missing region are systematically replaced by a similar patch from the known part of the data. The exemplar-based methods are often employed when the missing regions are composed of small number of large regions. The order of filling the data is crucial in such methods. In this project, we would like to explore whether these two approaches could be combined to produce better inpainting results. We intend to use the variational method to decide the order of inpainting in the exemplar method. The students should have taken a course in partial differential equations, multivariate statistics, or equivalent courses. Computer skills needed are Python and Matlab.Some background in machine learning is a plus. 

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-splittable 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. 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).

 (*reference available on request)