Analytic Number Theory; Harmonic Analysis; Automorphic Forms, Representations,
and L-functions; Hyperbolic Geometry and Spectral Theory; Group Actions and
Ergodic Theory; Sieve and Circle Methods
In this paper, we prove an almost local-global theorem for entries of thin matrix subgroups of SL(2,Z). The exact statement is hard to explain
quickly (read the introduction?) but here's a wordle that might help give the overall impression of what's involved:
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").
This paper is a generalization of my thesis (see
"The Hyperbolic Lattice..." below). In it we study Diophantine properties
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 solves this problem in the case when there is a parabolic stabilizer.
This paper is my PhD thesis. In it, I develop new techniques for spectral methods on infinite volume hyperbolic surfaces; the
layman's content of this is described above in the paper "Almost Prime Pythagorean..."
This also gives 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.
This was my senior thesis at Princeton, supervised by Prof. 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." :) (Actually, you could download the program below to find out.) We generalize this phenomenon to a class of Collatz-like maps, which we call (d,g,h)-Maps.
As part of our NSF FRG,
and I are organizing
an educational workshop at
Thin Groups and Super Approximation in Geometry and Arithmetic, March 28 - April 1st, 2016.
It is supported in part by the NSF, BSF, IAS, and Rutgers.