Overview

My mathematical interests lie primarily in the world of geometry.

Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Geometers take different approaches to study these objects -  including visual images, lists of properties, and descriptive equations.

For example, a middle school student might look at objects such as circles, squares, and triangles, examining properties of these objects (such as area and perimeter) and learning how to distinguish between similar objects (with results such as side-side-side determining congruent triangles). As we become more mathematically mature, we can examine more advanced classes of objects (such as curves, surfaces, or manifolds), and study interesting properties of these objects.

I am interested in understanding the shapes and properties of "nice" geometric objects. By "nice," I mean that the objects I am interested in optimize certain geometric quantities under various constraints. “Optimization” suggests a type of analysis that is calculus-based, and this is accurate: The field of mathematics that my research sits in is Differential Geometry, and smooth, calculus-based geometry is my favorite geometric “flavor.”

My favorite “nice'“ geometric objects are bubbles - geometric objects (surfaces) that optimize a given quantity (minimize surface area) under a certain constraint (a fixed boundary, or a fixed amount of enclosed volume). Much of my mathematical research can be described, loosely, as studying the geometry of bubbles and similar geometric objects.

When I discuss my interests with other geometric analysts, I say that I am interested in the classification and rigidity of solutions to the to the Isoperimetric Problem in Euclidean and Gaussian spaces, as well as spaces with prescribed density functions. I maintain an interest in Minimal Surfaces, Mean Curvature Flow, and related subfields of Geometric Analysis.