Math

I am a graduate student in mathematics at MIT, so I often think about math. The kinds of math I like best are

• visual: with drawing, pictures, or things to play around with;
• surprising: where different areas of math collide; and
• accessible: I can explain it to most anyone.

The specific area I research is called combinatorial, discrete, and convex geometry, which is all about patterns and structure of geometric objects. It checks all the boxes: I can explain the problems to other people (even if they aren't a mathematician), but the solutions are surprising and satisfying, often relying on connections to other fields of math, such as graph theory, probability, topology, and linear algebra. As you can see in the picture, I am occasionally a serious mathematician. (But not too often.)

Current projects

• Clandestine families of balls

  Given a family of sets in Euclidean space, we call a specific pair of sets in the family secretive if their intersection contains a point that is not contained in any other set in the family. (This point is, of course, the secret they share.) We call the family clandestine if every pair of sets in it is secretive. What is the maximum size of a clandestine family of balls in \(\mathbb{R}^d\)?
  If \(d=2\), then any clandestine family of disks has at most 4 members. János Pach and I are investigating this and related questions for \(d≥3\).

• Helly-type theorems

  Helly’s theorem is one of the most fundamental results in combinatorial convex geometry. Jiří Matoušek observed that “Helly’s theorem [has] inspired a whole industry of Helly-type theorems”, and I am one of many cogs in that industry. Right now, I’m thinking about quantitative and integer versions of Helly’s theorem.

• Layer number of the integer grid

  Given a set of points in Euclidean space, imagine removing the vertices of its convex hull, looking at what remains, removing the vertices of its convex hull, and repeating. Each step peels away a layer of the set; thus, the number of layers a set has—its layer number—is a measure of the depth of the set’s innermost points. My current work, with Gergely Ambrus, is on improving the current best bounds on the layer number of the grid \(\{1,2,…,n\}^d\).