Arithmetic geometry involves the interplay between algebra, number theory, and geometry to study spaces of solutions to systems of polynomial equations. Computers have an important role to play in exploring these spaces, whether it's numerically approximating solutions or counting solutions over finite fields. I have engaged in undergraduate research on topics such as: noncommutative algebras (especially Clifford algebras), error-correcting codes (especially those coming from algebra), enumerating orbit spaces of large group actions (especially those coming from polynomials), point counting and the Weil conjectures (for solving “finding hay in the haystack” problems), and cryptography. I am also interested in working on projects spanning the history of mathematics and in projects combining art and mathematics. Most projects would require some prerequisite background in algebra, such as Math 71. If algebra and number theory have piqued your interest and you are interested in learning more, come talk to me!

Professor Chernov advises students on various knot theory projects and the interactions of knot theory and general relativity. The knowledge of Math 54 Pointset Topology is required. The possible projects include Presidential Scholar research and Senior Theses.

I sometimes supervise undergraduate research projects in combinatorics, including senior theses, Presidential Scholars, WISP, and other independent projects. For most of the projects, having some background in combinatorics (such as math 28, 38 or 68) and algebra (24 and 31/71) is helpful, along with some programming skills. View some of the projects that I've been working on.

I am working on problems in an area of mathematics called homological mirror symmetry. The goal of this field is to understand a surprising relationship between two fields of mathematics that was first observed by string theorists. This duality predicts a correspondence between geometric shapes determined by polynomials (algebraic geometry) and the more flexible geometry of two-dimensional areas (symplectic geometry). In particular, mirror symmetry can point to new unexpected facts about combinatorial objects such as polytopes and hyperplane arrangements. I currently have a few problems in mind of this flavor.

To get started, students need at least to be comfortable with mathematical proofs and linear algebra, but additional background knowledge certainly would be useful. If you're interested but unsure if you're prepared, please reach out to discuss.

There are numerous opportunities to perform undergraduate research in my group on the mathematics of networks (how things are connected) and network representations of data. Previous students have developed new methods and written code for clustering networks into “communities” and other network measures, developed simulations for studying the spread of behaviors on networks, and studied network representations of a wide variety of application data sets from brain connectomes to political systems, and many other examples.

I have a number of projects accessible to undergraduate students in Combinatorics, Algebra and Graph Theory. These projects can lead to a senior thesis for honors or high honors. The ideal student should have taken Math 28 and Math 24 (or Math 22) if interested in a combinatorics project, Math 38 and Math 24 (or Math 22) if interested in a graph theory project, and Math 28 and Math 71 (or Math 31) if interested in an algebra project. Ideally the student should have some programming skills. For more details schedule an appointment. Here is a sample project with undergraduate student Geoffrey Scott.