**Local Statistics of Orbits of Thin Groups¹:**

Consider the orientation preserving isometry group of hyperbolic half space, PSL(2,R). For a long time there has been much interest in the orbits of discrete subgroups when these subgroups are lattices. For instance the orbit of the point at infinity generates the rationals. Thus one can reformulate most problems in Diophantine Approximation as problems concerning the orbits of lattices.

Similarly, one can ask the same sort of questions when the discrete subgroups have infinite covolume (see the picture). This branch of my research is asking what can we say about the equidistribution and local statistics (e.g gap distribution) of such sequences. In this setting the orbits will resemble fractals and thus the local statistics cannot be studied using classical horospherical equidistribution techniques. This research links some problems in continued fraction theory, hyperbolic geometry, and classical geometry (e.g Ford circle packings).

[See for example this paper]

**The Random Lorentz Gas – Going Beyond Boltzmann-Grad: **(joint with Bálint Tóth)

Consider an array of spherical scatterers distributed throughout the plane on the points of a Poisson process. Consider a point particle moving in straight lines and colliding elastically with the scatterers. This is the so-called *Lorentz Gas*, introduced by Henrik Lorentz in 1905. Gallavotti (1970) and later Spohn (1978) proved that under the correct scaling (Boltzmann-Grad limit) as the radius of the scatterers goes to zero, for finite time, the Lorentz gas converges in distribution to a random flight process (Gallavotti showed that the limiting process obeys a linear Boltzmann equation while Sphon proved the convergence and more).

Our goal in this project is to extend this picture. By using probabilistic coupling, one can jointly realize both the Lorentz gas and the Markovian flight process for times proportional to the radius of the scatterers to the power -2. Hence we are looking at infinite times. We can prove the necessary convergence which allows us to prove an invariance principle and central limit theorem.

It has long been of great interest to study this problem for infinite times. As we need to take the joint limit of radius to zero and time to infinity, this is a big step in that direction.

[See for example this paper]

¹ The image on the left is taken from the fantastic and beautiful book **Indra’s Pearls: The Vision of Felix Klein** – Mumford, Series and Wright. I highly recommend checking it out, it is written for any mathematical background and gives fantastic background to the theory of Schottky groups, circle packings and, in general, symmetry.