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Geometry Overview (start here)



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. My favorite examples 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). 

When I discuss my interests with other geometric analysts, I say that I am interested in the classification and rigidity of self-shrinking and self-translating solitons of mean curvature flow - and more broadly, I am interested in minimal and constant mean curvature submanifolds as solutions to the Isoperimetric Problem in Euclidean and Gaussian spaces.