# Research

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.