2022 (Pending) Research Experience for
Undergraduate Students in Mathematics (REU in Math Summer 2022)
SUNY
REU Web Apply by March 27, 2022 on
Mathprograms.org
Location and Time:
Clarkson
University and SUNY Potsdam, May 23, 2022--July 15, 2022
Program Objectives:
1.
Engage a total of 12_15
students annually including traditionally underrepresented groups or colleges
and universities with limited research opportunities and immerse students in
ongoing research projects in graph theory, probability and statistics methods
for image processing and biomedical or biological systems, numerical stochastic
differential equations related pure and applied mathematics fields;
2.
Cultivate talented students to effectively plan,
conduct, and communicate scientific research through meaningful and engaging
research projects, close and effective mentoring, weekly group meetings, mentor
training, and public presentations; and
3. Improve educational pathways to advanced
pure and applied mathematics related careers through student involvement in field
trips, expert speaker series, and additional professional development activities.
Possible
Research Topics:
Topics |
Advisor |
Links in embedded graphs |
Joel Foisy, SUNY
Potsdam |
Numerical solutions to
high-dimensional stochastic differential equations |
Guangming Yao, Clarkson University |
Using comorbidities, demographic,
and socioeconomic data to predict onset of rheumatoid arthritis |
Sumona Mondal, Clarkson University |
Hybrid inpainting method |
Prashant Athavale,
Clarkson University |
Application
Materials Required:
Submit the following
items online at mathprograms.org to complete your application:
·
Unofficial Transcript
·
Statement of Interest
·
Cover letter with Contact Info & Topic Pref.
·
Two reference letters
Participant
Activities:
Accommodations
& Travel:
·
Stipend: Participants will receive a stipend of $4,500.
·
Housing: Participants will receive free housing in an on-campus
apartment with access to cooking facilities
·
Travel & Field Trip: Up to $800 to support travel expenses to/from Potsdam/attending
conferences.
Contact:
Dr. Joel Foisy, SUNY Potsdam, foisyjs@potsdam.edu
Dr. Guangming Yao, Clarkson University,
gyao@clarkson.edu
Project
Description:
· 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)