Explanations of my research interests for different audiences appear below. Alternatively, you can browse my publications.

For the public

I am an algebraic geometer. Abstract algebra studies the manipulation of symbols according to well-defined rules going far beyond the solution of equations from grade school. Modern geometry appeals to one’s physical intuition, even though most of the spaces are too exotic to be drawn or sculpted. There is a deep connection between these two areas that, over many centuries, has developed into a versatile bridge where problems from one domain are solved with insights from the other. My research ranges across this bridge but tends to linger on objects exhibiting unusual symmetry.

For undergraduate students

At its core, algebraic geometry studies varieties, which are the solution sets to systems of polynomial equations. For example, the solutions to \[ \begin{array}{rl} x^2+yz-3z\!\!\!\!&=5\\ 3y^3z+4x^2\!\!\!\!&=4 \end{array} \] form a variety. I am especially interested in their groups of symmetries. Some of the relevant concepts you might know are finite groups, ideals, fields and Galois theory. At UofSC, the abstract algebra sequence (Math 546-547) and computational algebraic geometry course (Math 548) cover some of these ideas.

For graduate students

I do classically-flavored algebraic geometry over arbitrary base fields. Group actions tend to show up a lot (finite, algebraic, or Galois). Adjacent areas include algebraic groups, algebraic number theory, commutative algebra, Galois theory, and representation theory.

For mathematicians

I am an algebraic geometer who uses geometric techniques to study problems in number theory and algebra. I have interests in birational geometry, derived categories, algebraic groups, and Galois cohomology. I am actively thinking about Cremona groups, arithmetic toric varieties, del Pezzo surfaces, and essential dimension.