# Fall 2021 Titles and Abstracts

Date: 23 September 2021
Title: Geometric Structures and Point Processes
Abstract: We’ll give several concrete examples of how to go from geometry to point processes, following work of Siegel, Veech, Masur, Eskin, Marklof, Mirzakhani, Wright, and others. We’ll discuss how this “probabilistic” perspective helps inform both the direction of questions one asks, as well as providing ideas of how to prove things. We’ll discuss some pieces of joint work with Cheung-Masur, Ghosh, Margulis, and Arana-Herrera.

Date: 30 September 2021
Title: Height Gap, an Arithmetic Margulis Lemma and Almost Laws
Abstract: We provide a new (more elementary) proof of a result of E. Breuillard, which state that a set of matrices with algebraic entries generating a non-virtually solvable group has a positive lower bound in its arithmetic height (we will explain this notion), this is a non-abelian version of Lehmer’s problem. We also show that in arithmetic locally symmetric spaces, short geodesics tend to be far from each other if the degree of the trace field is large. This lemma allows us to prove new results about growth of cohomology of sequences of locally symmetric spaces and to give a proof of a conjecture of Gelander. These results are works in progress with Joe Chen and Homin Lee, and with Mikolaj Fraczyk and Jean Raimbault.

Date: 7 October 2021
Speaker: Lei Yang
Title: Khintchine’s theorem on manifolds
Abstract: In this talk, we will prove the convergence part of Khitchine’s theorem on non-degenerate manifolds. This confirms a conjecture of Kleinbock and Margulis in 1998. Our approach uses geometric and dynamical ideas together with a new technique of major and minor arcs’. In particular, we establish sharp upper bounds for the number of rational points of bounded height lying near major arcs’ and give explicit exponentially small bounds for the measure of `minor arcs’. This is joint work with Victor Beresnevich.

Date: 14 October 2021
Speaker: Pengyu Yang
Title: Equidistribution of degenerate curves and Dirichlet improvability
Abstract: In the space of 3-lattices, we study the translates of a line segment under a diagonal flow. Sharp conditions for non-divergence and equidistribution will be given. As an application, we will show that Lebesgue-almost every point on a planar line is Dirichlet non-improvable if and only if the line is irrational. This is joint work with Kleinbock, de Saxcé and Shah. Generalizations to higher dimensions will also be discussed (work in progress with Shah).

Date: 21 October 2021
Speaker: Dubi Kelmer
Title: The light cone Siegel transform, its moment formulas, and their applications
Abstract: In this talk I will describe an analogue of the Siegel transform where the role of Euclidean space is replaced by a light cone corresponding to an indefinite quadratic form.In this case one can use results on the spectral theory of incomplete Eisenstein series to establish moment formulas analogous to the classical formulas of Siegel, Rogers, and Schmidt.I will then describe several applications of these formulas to counting lattice points on the light cone, as well as for the distribution of rational points on the sphere.   All new results are based on joint work with Shucheng Yu.

Date: 28 October 2021
Speaker: Demi Allen
Title: An inhomogeneous Khintchine-Groshev Theorem without monotonicity
Abstract: The classical (inhomogeneous) Khintchine-Groshev Theorem tells us that for a monotonic approximating function $\psi: \mathbb{N} \to [0,\infty)$ the Lebesgue measure of the set of (inhomogeneously) $\psi$-well-approximable points in $\mathbb{R}^{nm}$ is zero or full depending on, respectively, the convergence or divergence of $\sum_{q=1}^{\infty}{q^{n-1}\psi(q)^m}$. In the homogeneous case, it is now known that the monotonicity condition on $\psi$ can be removed whenever $nm>1$ and cannot be removed when $nm=1$. In this talk I will discuss recent work with Felipe A. Ramírez (Wesleyan, US) in which we show that the inhomogeneous Khintchine-Groshev Theorem is true without the monotonicity assumption on $\psi$ whenever $nm>2$. This result brings the inhomogeneous theory almost in line with the completed homogeneous theory. I will survey previous results towards removing monotonicity from the homogeneous and inhomogeneous Khintchine-Groshev Theorem before discussing the main ideas behind the proof our recent result.

Date: 4 November 2021
Speaker: Alexander Gorodnik
Title: Quantitative equidistribution and Randomness
Abstract: We discuss some results on quantitative equidistribution on homogeneous spaces and related problems about behaviour of arithmetic counting functions. This is a joint work with Björklund and Fregoli.

Date: 11 November 2021
Speaker: Chris Lutsko
Title: Pair correlation of monomial sequences modulo 1
Abstract: Fix $\alpha, \theta > 0$, and consider the sequence $(\alpha n^\theta \mod 1)_{n>0}$. Since the seminal work of Rudnick-Sarnak (1998), and due to the Berry-Tabor conjecture in quantum chaos, the fine-scale properties of these dilated mononomial sequences have been intensively studied. In this talk, I will briefly survey what is known about these sequences and present a recent result (joint with Sourmelidis and Technau) showing that for $\theta \le 1/3$, and $\alpha > 0$, the pair correlation function is Poissonian. While the techniques we use are derived from analytic number theory, the problem is rooted in dynamics and relates to dynamical proofs for related problems.

Date: 18 November 2021
Speaker: Anurag Rao
Title: On a theorem of Davenport-Schmidt on Dirichlet improvable pairs
Abstract: In 1970, Davenport and Schmidt studied a Diophantine property of pairs of real numbers; it concerned those pairs for which the classical Dirichlet theorem can be improved. They showed that the set of Dirichlet-improvable pairs, while small in the sense of having zero Lebesgue measure, has full Hausdorff dimension. We study a similar Dirichlet-improvable property, where the approximations are made using an arbitrary norm rather than the supremum norm, and show the same result. To this end this, we recast the Dirichlet-improvable property into a dynamical property of certain orbits in the space of unimodular lattices, and prove a Hajos-Minkowski type result in the geometry of numbers. This is joint work with Dmitry Kleinbock.

Date: 25 November 2021
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Date: 2 December 2021
Speaker: Ilya Khayutin
Title: Two-step equidistribution for bi-quadratic torus packets
Abstract: A major challenge to the asymptotic analysis of a sequence of probability measures on a homogeneous space, invariant under diagonalizable groups, is the possibility of accumulation on intermediate homogeneous subspaces. In this aspect higher rank homogeneous flows cannot be expected to share the rigidity properties of unipotent ones. In particular, the linearization technique fails for diagonalizable flows.

In a joint work in progress with A. Wieser we show how in favorable situations one can actually use the existence of intermediate homogeneous spaces in our benefit. We show that periodic measures on some packets of periodic torus orbits on PGL4(Z)\PGL4(R) converge in the limit to a measure with a non-trivial Haar component. The proof goes by establishing high entropy for the limit measure. The method utilizes the intermediate homogeneous space to split the analysis into two more tractable steps.

Date: 9 December 2021
Speaker: Michael Bersudsky
Title: On the image in the torus of sparse points on expanding analytic curves
Abstract: It is known that the projection to the 2-torus of the normalised parameter measure on a circle of radius $R$ in the plane becomes uniformly distributed as $R$ grows to infinity. I will discuss the following natural discrete analogue for this problem. Starting from an angle and a sequence of radii {$R_n$} which diverges to infinity, I will consider the projection to the 2-torus of the n’th roots of unity rotated by this angle and dilated by a factor of $R_n$. The interesting regime in this problem is when $R_n$ is much larger than n so that the dilated roots of unity appear sparsely on the dilated circle.I will discuss 3 types of results:

1. Validity of equidistribution for all angles when the sparsity is polynomial.
2. Failure of equidistribution for some super polynomial dilations.
3. Equidistribution for almost all angles for arbitrary dilations.

I will discuss the above type of results in greater generality and I will try to explain how the theory of o-minimal structures is related to the proof.