Publications

5. (with Ricardo Diaz and Sinai Robins) Fourier transforms of polytopes, solid angle sums, and discrete volume, Accepted to Math. Annalen pending minor revisions (arXiv:1602.08593 [math.CO])

Given a real closed polytope P, we first describe the Fourier transform of its indicator function by using iterations of Stokes' theorem. We then use the ensuing Fourier transform formulations, together with the Poisson summation formula, to give a new algorithm to count fractionally-weighted lattice points inside the one-parameter family of all real dilates of P. The combinatorics of the face poset of P plays a central role in the description of the Fourier transform of P.We also obtain a closed form for the codimension-1 coefficient that appears in an expansion of this sum in powers of the real dilation parameter t. This closed form generalizes some known results about the Macdonald solid-angle polynomial, which is the analogous expression traditionally obtained by requiring that t assumes only integer values. Although most of the present methodology applies to all real polytopes, a particularly nice application is to the study of all real dilates of integer (and rational) polytopes.

4. (with Fedor Duzhin) The group of sphere-trivial braids, Journal of Knot Theory and Its Ramification, 2016

We give a presentation for the group of Artin braids that become trivial when considered over the 2-sphere and explore the relation to Brunnian braids and homotopy groups of spheres

3. A projective analogue of Napoleon's and Varignon's theorems, Accepted to Mathematical Intelligencer pending minor revisions (arXiv:1609.04513 [math.MG])

What do Napoleon's and Varignon's theorems have in common? We claim that they both are examples of immediately regularizing natural polygon iterations in different planar geometries and present a new and analogous result in projective geometry.

2. (with Sinai Robins) Macdonald's solid-angle sum for real dilations of rational polygons, Submitted (arXiv:1602.02681 [math.CO])

The solid-angle sum A_P(t) of a rational polytope P ⊂ R^d, with t ∈ Z was first investigated by I.G. Macdonald. Using the Fourier-analytic method, we are able to establish an explicit formula for A_P(t), for any real dilation t and any rational polygon P ⊂ R^2. Our formulation sheds additional light on previous results, for lattice-point enumerating functions of triangles, which are usually confined to the case of integer dilations. Our approach differs from that of Hardy and Littlewood in 1992, but offers an alternate point of view for enumerating weighted lattice points in real dilations of real triangles.

1. A family of projectively natural polygon iterations Preprint (arXiv:1602.02699 [math.DS])

The pentagram map was invented by R. Schwartz in his search for a projective-geometric analogue of the midpoint map. It turns out that the dynamical behavior of the pentagram map is totally different from that of the midpoint map. Recently, Schwartz has constructed a related map, the projective midpoint map, which empirically exhibits similar dynamics as the midpoint map. In this paper, I demonstrate that there is a one-parameter family of maps which behaves a lot like Schwartz’s projective midpoint map.