For independent study courses, the best advice is to ask anybody who teaches upper division courses. If they're not comfortable doing the course, they can at least tell you who to ask about it.
For research courses, please look through the following statements to get a sense of what faculty are doing.
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.
My professional research sits in the intersection of differential geometry, algebraic topology, and category theory. When working with students, I tend to try and find discrete/digital settings where the geometric and algebraic methods which I use in my research can be applied to more down-to-earth applications. These applications usually take some form of “studying the cocycle condition” in various applications (i.e., persistent homology in data analysis or sheaf theoretic network structures).
Of course, I’m also happy to advise undergraduate research in fields outside of my expertise if we find something we are both excited about. A current project I’m working with students on combines with from linear algebra and category theory to mathematically model both the grammar and the meaning of sentences.
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.
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.
Teaching, Learning, and Teacher Education for K-12 classrooms. Recent topics: Mathematical Habits of Mind, Classroom Authority, Public Record, and Perseverance.
I am working on probability theory, especially jump-type Lévy processes. Recently, I have been working on studying the geometry of domains in the Euclidean space using spectral objects such as trace, heat content, or torsional rigidity with respect to jump processes. I am also interested in studying fractal structures related to Lévy processes. If there are students who are interested in doing research questions on probability and stochastic processes, please contact me and we can discuss further details.
I am interested in developing and applying mathematical methods to models arising in natural and applied sciences. In particular, my research involves differential equations, nonlinear dynamics, and numerical simulation. Recent research topics include the mathematical applications in image processing and biology, but I am open to discussing a wide range of topics for independent study.
Works in dynamical systems and mathematical modeling.
Please see https://www2.newpaltz.edu/~radulesa/mentoring.html for details.
I am a noncommutative algebraist by training, which means that I study sets endowed with operations where the order matters. If you like fancy words, I study finitary proto-exact functor categories and their associated Hall algebras. But most people won't find this description very helpful, and in some sense, it obscures the fact that much of what I do is graph theory and linear algebra!
My work lies at the intersection of representation theory and mathematics over the "field with one element." Representation theory is a branch of mathematics which studies objects by converting them into linear transformations acting on vector spaces. Practically, it is interested in sets of matrices closed with respect to certain operations. It is ubiquitous throughout pure mathematics and has important applications within chemistry and physics. The "field with one element" can refer to several different directions in contemporary mathematical research: for our purposes, it is an effort to use the tools of algebraic geometry to study well-behaved combinatorial objects. So, you can think of my work as an effort to describe finite structures which mimic the behavior of nice collections of matrices.
Tools I use regularly in my work include graph theory, linear algebra, abstract algebra, and category theory. I have projects which would be suitable for undergraduates with a background in mathematical proofs and an interest in linear algebra, combinatorics, or abstract algebra. Of course, I am also happy to explore mathematics with students beyond my particular expertise. If you are interested in learning more, schedule an appointment with me through email.
