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As of September 2020, I am a Hill Assistant Prof. (post-doc) at Rutgers university, my mentor is Alex Kontorovich. My research interests lie at the intersection of several topics including number theory, dynamical systems, and mathematical physics. Before this I received my PhD at the University of Bristol under the supervision of Jens Marklof and Bálint Tóth.

I am currently co-organizing the Rutgers number theory seminar. Please feel free to contact me about the possibility of giving a talk.

# Research Interests

Asymptotic Distribution of Thin Group Orbits [1], [2]

The central image in the above picture, is a so-called Apollonian circle packing (named after Apollonius of Perga circa 200 B.C). To generate such a packing we start with an initial configuration of three mutually tangent circles, like the one drawn in solid lines on the left. Now notice that there are two circles tangent to the three in the initial packing (in dotted lines on the left). We add these to the configuration, and then continue to add circles in all of the resulting gaps. This generates an infinite packing, sketched in the middle. These packings have fascinated the likes of Descartes, the Princess Elizabeth of Bohemia, and the Nobel prize winning chemist Frederick Soddy (who even eulogized them in a poem). While these are geometrically very interesting objects, there is a simple relationship between the radii of the circles discovered by Descartes. This quadratic form, relating the radii of four mutually tangent circles in the packing, allows one to make a connection between Apollonian circle packings, and number theory.

In fact, the packing can also be generated by reflecting the circles in the initial packing, by the circles drawn in red on the right (for example, to generate the circle “a” we reflect the circle “b” through the circle labelled “d”). Thus, the packing can be thought of as the orbit of the initial configuration by a group generated by four reflections. It turns out this group, is a “thin group” (meaning the associated hyperbolic manifold has infinite volume - see this review paper for more information). One can think of thin groups as an extension of a (hyperbolic) lattice, except with sparser points (while a lattice has constant density, a thin group has bigger and bigger gaps as you move out). The study of thin groups has become a bit of a hot topic at the moment owing to the ubiquity of these groups, their applications to number theory and geometry, and the development of new tools in this setting.

One of the projects from my thesis was to study the fine-scale local properties of these thin groups (see this paper), and certain number theoretic sequences derived from thin groups (Farey sequences for thin groups – see this paper). In both of these papers we used techniques from the world of homogeneous dynamics (namely the equidsitribution of expanding horospheres) to understand fine-scale statistics of these objects.

Pseudo-Random Properties of Deterministic Sequences [1], [2]

In the $20^{th}$ century, starting with the work of Weyl, mathematicians became interested in the statistical properties of monomial sequences of the form $(\alpha n^\theta \mod 1 )_{n>0}$ where $\alpha >0$ and $\theta >0$. From a number theoretic point of view there is a natural question one can ask: in the limit as we consider more and more sequence elements, what can be said about the asymptotic distribution of these points? In other words, can we see random behavior emerging from these sequences?

While this question has a natural motivation from a number theory perspective, there is also a deep connection to quantum chaos. Namely, the Berry-Tabor conjecture (See these notes by J. Marklof) hypothesizes a connection between the the chaotic properties of a given system with the pseudo-random properties of the eigenvalues of corresponding quantom analog. These monomial sequences can appear as the eigenvalues of a quantum system, and thus determining the pseudo-random properties can shed light on this deep hypothesized connection in quantum chaos. The above plot shows the cumulative gap distribution (probability $P(x)$ that the gap between neighboring points divided by the average gap size is larger than $x$) of the sequence $(\pi \sqrt{n} \mod 1 )_{n>0}$ compared to the function $f(x)=e^{-x}$, which is the expected value for uniformly distributed random variables.

One very strong measure of pseudo-randomness, is the so-called ‘pair correlation function’. In this paper, together with A. Sourmelidis and N. Technau, we showed that a wide class of these monomial sequences exhibit Poissonian pair correlation. In fact, this is one of the only examples where the pair correlation function has been shown to converge to the Poissonian limit for explicit values of $\alpha$ and $\theta$. The paper builds on a methodology outlined in this paper which proved convergence of (so-called) ‘long-range correlations’ (a weaker measure of pseudo-randomness) for some other monomial sequences.

Lorentz Gas and Non-Interacting Particle Systems [1], [2]

In the $19^{th}$ Centruy physicists started to ask a simple question, which gave rise to an entire field of mathematical physics – statistical mechanics. The question was the following: given a simple model for the movement of atoms, can we see the laws of physics which goven large scale objects by studying large ensembles of these atoms. For example, can we use particle dynamics to derive the heat equation (the equation governing how heat diffuses outwards, through a fluid)? Since we observe the heat equation in everyday life, any reasonable model of atomistic behavior should have the heat equation as a natural limit. From a probabilistic point of view, one can rephrase this question as follows: given a model for particle dynamics, do the trajectories of the particles converge (in distribution) to Brownian motions (random walks)? This is known as an invariance principle.

One example of a very simple (although highly nontrivial) model for particle dynamics, is the Lorentz gas (introduced in 1908 by physicist Henrik Lorentz). In this model we consider an array of spherical scatterers (think of these like nuclei in a metal) which do not move. Then we consider the motion of a point particle which moves in straight lines, bouncing off of the scatterers (see the left most above image) – if the scatterers model the heavy nuclei in a metal, then the point particle models the motion of a light electron scattering off the nuclei. Again, one of the central questions in the study of the Lorentz gas is: when observed for a long time and scaled appropriately, does the particle trajectory (on average) converge to that of a Brownian motion? In this paper together with B. Tóth, and by introducing techniques from classical probabiltiy theory we were able to show that the Lorentz gas, in a particular intermediate scaling regime, does indeed satisfy this invariance principle. That is, the trajectory of the point particle converges to that of a Brownian motion when scaled correctly and observed over long times. In our case we arranged the scatterers randomly through space (the random Lorentz gas) and worked in 3 dimensions. This was the first result in this direction for the random Lorentz gas since the 1980s, and the first result which took into account the fact that the dynamics are not independent (that is the first result which handled recollisions with the same scatterer).

In this paper we were able to apply the techniques developed in the previous paper to the so-called Ehrenfest wind-tree model (which dates back to 1912 and Paul and Tatiana Ehrenfest’s monograph on statistical mechanics) where the scatterers are replaced by cubes (the middle image in the above). That is, we again proved an invariance principle in a particular scaling limit.