The following math faculty mentor MA 491 independent study and research projects and some have outlined some possible projects for honors students. Since MA 491 is one of the courses that can be used to satisfy the writing requirement for NCSU’s General Education Program, a student taking MA 491 is expected to write a paper summarizing the results of his/her work. Students will present their work in the semesterly Honors presentations may also wish to present their results at the Undergraduate Research Symposium sponsored by Sigma Xi and held in April.

This list is prepared as an aid for students considering possible MA 491 projects but students should not consider themselves limited to working with these faculty nor on these projects. In particular, projects (with the approval of the student’s advisor and the Math Honors Director) may be done with faculty in other departments.

E. Bates
I work in probability theory, with the goal of understanding large complex systems governed by random inputs.  My research often intersects with analysis, combinatorics, and/or mathematical physics.  Student projects can be either computational or theoretical.  For computational projects, programming experience is necessary.  For theoretical projects, a course in probability is usually necessary, e.g. 421 or (even better) 546.  In any project, students should bring their curiosity and attention to detail. 

M. Gilman
Satellite-based radar imaging is the only remote sensing modality that can provide 24/7 all-weather monitoring of the Earth surface. For a long time, the scarcity and high cost of radar images hindered development in this area. However, because of the recent advances in electronics and space launch technology, we are now living through a period of explosive growth of the available imagery, and new possibilities for the analysis are opening up.

There is a range of projects: some are completely specified while others may need formulating the details of varying importance, up to and including the research question. Some projects deal with synthetic data, whereas others deal with real imagery. There are projects that don’t require any prerequisites besides the very basic statistics, programming skills, creativity, and common sense. A familiarity with machine learning technologies will be an advantage with other projects, while having completed Calculus 1 and the familiarity with complex numbers and elementary geometry will be beneficial in many situations.

M. Farazmand

M. Kang
With numerous applications to a wide range of other disciplines, probability theory has emerged as one of the major branches of mathematical research. There are also some interesting connections between probability theory and other areas of mathematics. I would like to suggest a few probability theory-based projects for our undergraduate students who want to work on problems involving some degree of randomness. These projects naturally connect to other fields, such as statistical physics, financial mathematics, mathematical biology, combinatorics, and quantum computing. The project will be chosen based on the student’s interests, mathematical abilities, ambition, and the time constraints. The necessary mathematical background for the project may include basic analysis courses (rigorous), basic probability courses, and some linear/abstract algebra.

S. Paul
llustration plays many roles in mathematics—in research, education, artistry, and outreach. Projects in mathematical illustration will include both learning about a mathematical concept and the design and creation of the illustration, possibly using digital design software. All such projects require at least Calc 3. Other background and interests that may be helpful include experience with analysis, differential geometry, programming, art and design, and education.

A. Saibaba
I work in the intersection of scientific computing and data science. A big focus of my work has been on developing algorithms for imaging devices (e.g. x-rays, seismic) and quantifying the uncertainty associated with the image reconstructions. I am happy to tailor a project based on the student’s interest in theory, algorithmic development, or applications. Students may work with me individually or in a group. A background in linear algebra is necessary, some background in (a subset of) probability, numerical analysis, and scientific computing is preferable. 

Y. Shin
Research Focus: My research sits at the intersection of applied mathematics, numerical analysis, and machine learning — a field often called scientific machine learning. I study how neural networks can be used to solve and learn the differential equations that model physical systems (fluid flow, plasmas, heat transfer), and how to build fast, reduced “surrogate” models that capture the essential behavior of expensive simulations. A central question is how to design learning methods that respect known physics and come with mathematical guarantees on their accuracy, rather than treating the network as a black box.

Potential Project Types:
– Implementing and testing physics-informed neural networks (PINNs) to solve a PDE, and investigating where they succeed or fail and how to make it robust.
– Agentic SciML: using LLM-based agents to help automate parts of the scientific ML pipeline — proposing model structures, generating and debugging code, or steering numerical experiments — and studying how to keep such agents reliable and grounded.
– Digital twins via data assimilation and data-driven dynamical systems: learning a dynamical model from data and continually correcting it against incoming measurements using filtering methods (e.g., ensemble Kalman or score-based filters), building a “living” model that tracks a physical system in real time.

Prerequisites & Background:
Linear algebra, multivariable calculus, and differential equations. A course in numerical analysis or scientific computing is strongly preferred. Real analysis and PDEs are helpful for the theory-leaning projects. On the computational side, programming experience in Python is important; familiarity with a scientific/array library (NumPy, and ideally JAX or PyTorch) and LaTeX is a plus.

Mentorship Expectations: I typically meet with student researchers weekly to set goals, troubleshoot code, and discuss progress, with the aim of producing a concrete outcome such as a working implementation, a poster, or a written report by the end of the project. I look for students who are curious, persistent with debugging, and comfortable learning independently between meetings.

R. Smith