In this paper we prove the equidistribution of horospherical flows on infinite volume hyperbolic 3-manifolds (see movie above and stills below):
The manifold in the pictures above is a Schottky domain, which is the region lying above the ground and outside of the fixed geodesic hemispheres. We show that as the plane (horosphere) falls to the ground and gets reflected into the fundamental domain, it spreads out in all directions. Moreover, the amount of time its image spends in any given region is proportional to the "size" of the region. (Since the domain has infinite volume, the meaning of size must change for the previous statement to make sense -- one must measure size relative to the Patterson-Sullivan base eigenfunction in the spectral resolution of the corresponding Laplace-Beltrami operator.)
As an application, we can now give an asymptotic formula for the number of circles in an Apollonian packing whose curvatures (the numbers appearing in the pictures below) are bounded by some parameter.
We also consider questions of how many primes curvatures appear in the pictures above, as well as pairs of tangent circles whose curvatures are prime -- the circles marked "11" and "23" in the middle picture above are twin prime circles (so are "11" and "47").
Almost Prime Pythagorean Tripes in Thin Orbits, with Hee Oh, in preparation.
In this paper we study Diophantine properties of thin orbits of Pythagorean triples, shown below on the bottom half of the cone x^2 + y^2 - z^2 = 0:
In the pictures above, the dots represent Pythagorean triples which have been generated via a group action by an infinite co-area Fuchsian subgroup of SL(2,Z). The red dots in the first picture highlight those triples whose hypotenuse is prime. The black dots in the second picture denote triples whose area has at most 4 prime factors. We prove various theorems about sets of triples having few prime factors. We prove here the 2-dimensional version of the 3-dimensional equidistribution theorem described above, that is, the long horocycle (dark blue line below) spreads out uniformly as it flows down and gets reflected into the light blue region in the hyperbolic plane, see the movie above and pictures below:
My thesis (see "The Hyperbolic Lattice..." below) solves this problem in the case when there is a parabolic stabilizer.
The First Non-Vanishing Quadratic Twist of an Automorphic
L-series, with Jeff Hoffstein, submitted.
On Integral Transforms for Higher Rank Groups, with Dorian Goldfeld, in preparation.
First and Second Moments for Quadratic Twists of Half-Integral Weight Forms, with Gautam Chinta and Jeff Hoffstein, in preparation.
This paper is my PhD thesis at Columbia, supervised by Dorian Goldfeld and Peter Sarnak. I develop novel techniques for spectral methods on infinite volume hyperbolic surfaces; the layman's content of this is described above in the paper "Almost Prime Pythagorean..."
Though I did not know it at the time, this work also proves the equidistribution of certain horocycle flows for an infinite-area hyperbolic surface which has a cusp. In the movie above and stills below, the cusp at infinity has width 4, so the fundamental domain lies in the vertical strip -2 < Re[z] < 2.
Other methods were developed in "Almost Prime Pythagorean..." to handle the case when the horocycle has infinite length.
The Dirichlet Series for the Exterior Square L-function on GL(n), submitted.
This project was my junior paper at Princeton, which stemmed from a summer spent at the Weizmann Institute. In it we relate the combinatorial game of Nim to a generalization of the Sierpinski triangle and to Pascal's triangle. It is quite an elementary read as is, so hope you enjoy!
Structure Theorem for (d,g,h)-Maps, with Yakov G. Sinai, Bulletin of
the Brazilian Mathematical Society, New Series 33(2), 2002, pp. 213-224.
This was my senior thesis at Princeton, supervised by Prof. Ya. G. Sinai. In it we show that, in a certain sense, the ensemble of paths of the (3x+1)-map are a geometric Brownian motion with drift. One of the pictures below is of random drunken walk of +1s and -1s.
For the other picture, take the (3x+1)-path of length 100 starting with the number 1,234,567,891,011; take logs and normalize by the drift log(3/4)=-0.287682... Can you tell which picture is which? The theorem says "No, you can't." :) We generalize this phenomenon to a class of Collatz-like maps, which we call (d,g,h)-Maps.
