Works in probability and statistics.
Grab two ends of a rope, tie any knot you'd like, and fuse the two ends together. This is a mathematical knot studied in Topology. If we did this a million times, what behavior would we expect from these knots? If we had a rigid way to assign evaluations to these knots (like numbers, polynomials, or groups), how would this randomness manifest? Students may wish to help me understand my current model for random knots or explore their own model.
How many "different" ways can n lines intersect in the plane? You can try this for small n=3,4,5, but it gets pretty complicated pretty quickly. These arrangements of lines can be studied in Algebra, Algebraic Geometry, Geometry, Combinatorics, Graph Theory, Matroid Theory, and Topology. What if I gave you 11 lines but gave you some rules about what the intersections looked like? Students may wish to help me complete a census or to help me examine the interesting cases that have arisen already from work by previous students. More information is available on the homepage of my website.
Students with a background in proof-writing and an interest in combinatorics, graph theory, algebra, geometry, or topology are welcome to set up an appointment with me via email.
Works in applied mathematics.
My first specialty was Universal Algebra, which is related to Group Theory and Logic. My research there requires a lot of background knowledge, and is not very accessible to students. Areas where I’d be comfortable mentoring a research project include Algebra, Logic, Number Theory, Computability, Geometry, and Combinatorics.
I’ve also been doing some work with partial differential operators, which is easier to describe. (For example, “take the input function, differentiate it once with respect to x and twice with respect to y, and multiple the result by z” is a partial differential operator.) The best thing to do would probably be to get together and talk about our interests, until we converged on a topic. And it does not have to be my topic—one research project grew out of a student’s observation of patterns in the digits of successive squares.
I am working on algebraic geometry (study of solutions of polynomial equations) and combinatorics (counting some interesting stuff in a more interesting way). Areas that I am comfortable mentoring a research project in are (1) Algebra, (2) Combinatorics, (3) Geometry, (4) Number theory, (5) Mixture of (1)-(4).
I plan to meet once or twice a week with students, and there are some concrete questions (in combinatorics) that I have in mind; in case that students are interested in these questions, we can play with these questions. If students are interested in a bit more advanced questions (or other topics), we can talk about those until we find a proper question.
My research area is probability theory and stochastic processes. Specifically, I am interested in random movements which move by jumps. Jump processes are more desirable than their continuous counter part, Brownian motions, since a lot of real world data exhibits heavy tail behavior and many system do not evolve continuously.
I can work with students with interests in probability theory and stochastic processes. Examples of possible topics include discrete time Markov chains, continuous time Markov chains, Brownian motions, and large deviation principles. More advanced students with adequate backgrounds can work on L'evy processes and stochastic differential equations. Students having flavors in applied fields may choose to study topics from statistics, actuarial science, and data science.
Works in dynamical systems and mathematical modeling.