Research Overview

In my doctoral thesis I studied rings of differential operators (in characteristic p and on singular varieties). I continued to study rings of differential operators D(X) in cases where the variety X has extra stucture. For example, X might be a toric scheme or a geometric invariant theory quotient. I helped Karen Smith write a nice introduction to algebraic geometry, but I continue to feel that every day I learn more about the subject.

Often I'm learning with my students and colleagues at USNA. For instance, two students, Andy Bashelor and Tom Paul, wrote Trident projects (senior-year projects counting for 8 courses) on enumerative geometry. Andy's project (supervised with Amy Ksir) eventually appeared in the American Mathematical Monthly and won both the Ford and Hasse Prizes for exposition. Tom's project (supervised with Max Wakefield) dealt with the enumerative geometry of hyperplane arrangements. Max and I are currently extending Tom's results and writing them up for publication.

Another Trident scholar, David Zane (supervised with Chris Brown), studied optimization in the context of scheduling classes and final exams. Along these lines, I've recently written some notes on the network interdiction problem and I continue to be interested in operations research and optimization.

Other joint work with my colleagues at USNA involves coding theory (with Amy Ksir and Dave Joyner) and graph theory (with T.S. Michael). I'm always keen to discuss mathematics and I've been fortunate to collaborate with researchers all over the world, including Mutsumi Saito (Hokkaido, Japan), and Kia Dalili and Sara Faridi (Halifax, Canada). Recently I've become very interested in data science and machine learning. I've taught courses in both topics and I run an interdisciplinary seminar for faculty interested in these areas.