I consider myself a geometer. I try to keep my interest broad and read about all aspects of geometry, be it algebraic geometry, geometric group theory, discrete geometry or geometric topology. My current focus is dynamical systems in geometry and latticepoint enumerating functions. I love to write Java programs and use CAS software to (literally) look at geometric 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: Iterations on polygonsWe can consider the iteration of a map on polygons as a discretetime dynamical system and study its longterm 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 1parameter 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 1dimensional complex dynamics. It is also possible to enlarge this 1parameter family to a 2parameter one with more possibilities for the dynamics.
Project 2: Latticepoint 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 ongoing collaborations with Ricardo Diaz, James Pommersheim, Sinai Robins and Nick Salter, we focus on Macdonald's solidangle sums from the Fourieranalytic perspective.
Project 3: Polytopeexchange transformations on zonotopesPolytopeexchange transformations (PETs) are piecewise translations and higher dimensional generalizations of interval exchange transformations, which have been studied extensively for the past 40 years or so.Recently, many systems of PETs have been discovered to be renormalizable. For instance, Richard E. Schwartz investigated a family of PETs on parallelograms, known as octagonal PETs, which possesses a renormalization scheme in which the (2, 4, ¥)triangle groups acts on the parameter space as a renormalization symmetry group. In this project, we hope to search for renormalizable PETs that arises from a pair of cell decompositions of a zonotope, which are is a lowerdimensional projection of a higherdimensional cube.
