I have a number of projects for students (both undergraduate and graduate) interesting in working with me. I list below some of these projects. I would also be happy to discuss any idea you might have that could lead to a project we can explore together.
Undergraduate students
There are two groups of projects for undergraduate students. The first group of projects is centered on the theory of frame theory in finite-dimensional Euclidean spaces. Frames are spanning sets for these spaces and have some very interesting properties and applications.
The second group of problems is related to developing a theory of calculus on fractals, such as the Sierpinski triangle. We are interested in defining functions such as polynomials, sine, and cosine, and to explore their properties.
Students with a class in linear algebra, multivariable calculus, and/or advanced calculus can work on some interesting problems in either of these two groups of projects. Some knowledge of Matlab or any other computing program will be useful but not necessary. If you are interested in either of these groups of problems, please reach out to me and we can discuss. Even if you do not have the required background, I would still love to talk with you and explore how we might design a project that would allow you to learn.
Graduate students
I have two main active research areas, namely time-frequency analysis, and analysis on graphs and fractals. With regards to time-frequency analysis, the projects I am currently working on are related to discrete and continuous energy minimization problems on spheres, Gabor systems, Wilson bases, the HRT conjecture, and the Zauner conjecture. In terms of analysis on graphs and fractals, my interests lie certain applications of the spectral analysis of the Laplacian operator in these settings. I am especially interested in the study of non-linear Schrodinger equations on the Sierpinski gasket.
In both cases, a background in real analysis, functional analysis, linear algebra would be required. A course in harmonic analysis would be useful, but not necessary, and could be learned through a reading course. Familiarity with some computing software (Matlab, Python) would also be useful.
If you are interested in any of these topics I will be pleased to talk with you and give you more details.