Research

Hello!, welcome to my research page, and to the area I am mostly interested in, which is called discrete geometry. Wait, what is it?

   Well, discrete, spelt E-T-E, is the opposite of smooth: it means, something that perhaps is not so round and silky, because it has distinguished corners, ridges, facets... If you are thinking of common objects like a cube, or a surface sewn out of triangles, or a pixelated image, you are totally right. Some of these objects even exist in nature!, like this gold crystal found in Venezuela or this domesticated tiger seen in Arizona.

   Objects of this type in mathematics are called polytopal complexes, and provide a general approach to the study of differential geometry and algebraic topology.

   Since the advent of computers, discrete geometry has grown well beyond pure mathematics. Simple questions on the structure of polytopes, such as the Hirsch conjecture, have foundational importance in convex optimization. Moreover, an advantage of discrete geometry over continuous geometry is the possibility of leveraging computational and inductive tools. For example, Discrete Morse Theory allows to faithfully simplify given high-dimensional shapes, while maintaning their homotopy and homology. (If you wish to skip the preamble, the definition is at minute 24 of the video.)

The study of polytopal complexes of dimension one is extremely useful and fascinating, to the point that it forms a subject by itself, called graph theory. I am currently interested in using graphs to capture intersection patterns of curves and algebraic varieties.

My research is currently supported by NSF. If you want to know more on this stuff, come to our U Miami Combinatorics seminar! It usually takes place on Mondays.

(See the publications for more, or return to main.)