Research

I consider myself a geometer. I try to keep my interest broad and read about various aspects of geometry, be it algebraic geometry, geometric group theory, discrete geometry or geometric topology. My current focus is on applications of Fourier analysis in combinatorial geometry and on convexity in analysis and algebraic geometry. Another direction that I'm passionate about is the study of iterations on polygons. I love to write Java programs and use CAS softwares to (literally) look at problems, as well as, to assist in proving results. Here is my research statement and below you can find the summaries of my current research.

Project 1: Lattice-point enumerating functions of polytopes

This is a classical subject of discrete geometry that has found numerous connections to number theory, Diophantine geometry, toric varieties and even knot theory. Basically, it studies functions that count the number of lattice points inside a polytope, with or without weights.

Traditionally, the theory is limited to rational polytopes and their integer dilations. However, a more recent approach using Fourier analysis has allowed for an extension of the classical theory to real dilations of real polytopes. In on-going collaborations with Ricardo Diaz, James Pommersheim, Sinai Robins and Nick Salter, we focus on Macdonald's solid-angle sums from the Fourier-analytic perspective.

Project 2: Convexity in algebraic geometry

Project 3: Iterations on polygons

We can consider the iteration of a map on polygons as a discrete-time dynamical system and study its long-term behaviors. Although there is a large literature on such systems in affine geometry with very different dynamics, the study of analogous systems in projective geometry is, however, more limited due to the lack of linearity, and therefore, tons of tools from linear algebra.

To the midpoint map, whose dynamical behavior is related to the heat equation, there is a projective analogue introduced by Richard E. Schwartz. Its construction is related to the pentagram map, which is dynamically different and known to be a discrete integrable system.

In this project, I explore a 1-parameter family of projectively natural maps that interpolates the projective midpoint map and the pentagram map. Surprisingly, according to computer simulations, all maps in this family, except for the pentagram map and its inverse, behave dynamically more or less in the same way as the projective midpoint map. These systems also contain more dynamically intricate subsets, apparently of measure zero, which resemble the Julia sets that feature prominently in the theory of 1-dimensional complex dynamics. It is also possible to enlarge this 1-parameter family to a 2-parameter one with more possibilities for the dynamics.