Talks

Valentin Blomer

Variations on the mixing conjecture

Low-lying horocycles are known to equidistribute on the modular curve. In the context of the mixing conjecture of Michel and Venkatesh, we consider the joint distribution of two low-lying horocycles of different speeds in the product of two modular curves and show equidistribution under certain necessary diophantine conditions. The techniques involve a combination of analytic number theory, automorphic representation theory and ergodic theory. This is joint work with Philippe Michel. 

Yitwah Cheung

Staircases, Z-expansions and BCZ cocycles

Orbits of the geodesic flow on the modular surface are encoded by the theory of continued fractions, allowing many precise statements about the dynamics, e.g. the detailed description of Lagrange’s spectrum.  Generalizing the tight correspondence between number theory and dynamics is possible from the perspective of best approximation theory, but in more than one way.  In the first part of this talk, I will describe 3 different generalizations and some recent new results that can be obtained using them.  In the second part of this talk, I will describe a variation on the central theme where diagonal flows are replaced by a unipotent one.

Sam Chow

Dispersion and Littlewood's conjecture

I’ll discuss some problems related to Littlewood’s conjecture in diophantine approximation, and the role hitherto played by discrepancy theory. I’ll explain why our new dispersion-theoretic approach should, and does, deliver stronger results. Our dispersion estimate is proved using Poisson summation and diophantine inequalities. This is joint work with Niclas Technau.

Shreyasi Datta

Bad is null

The set of badly approximable vectors in Diophantine approximation plays a significant role. Given any C^2 manifold, we show almost every point is not badly approximable. Our methods apply in various set-ups where the previous methods do not seem to apply easily. This is a joint work with Victor Beresnevich, Anish Ghosh, and Ben Ward.

Nicolas de Saxcé

Rational approximations to linear subspaces

We shall present some questions from Schmidt's 1967 paper "On heights of subspaces and diophantine approximation", give an overview of the recent progress on the program he suggested and try to highlight the remaining challenges.

Mikolaj Fraczyk

Optimal approximation exponents on Hilbert modular surfaces

Hilbert modular surface S admits two natural foliations by copies of the hyperbolic planes (horizontal and vertical) coming from the local product structure. Given a generic pair of points x,y in S we are interested how far do we need to travel in the horizontal x-leaf to get close to y. This is measured by the diophantine approximation exponents, a classical notion going back to 60's. Using pigeonhole principle and basic volume estimates, one can easily give a lower bound on the approximation exponent, and the exponents obtained in this way are called optimal (because they are the best possible). I will explain how density estimates on the automorphic spectrum, more specifically Sarnak's density hypothesis, can be used to show that generically the optimal exponent is the correct approximation exponent. Based on a joint work with Alex Gorodnik and Amos Nevo. 

Subhajit Jana

Fourier Asymptotics and Effective Equidistribition

We talk about effective equidistribution of the expanding horocycles on the unit cotangent bundle of the modular surface with respect to various classes of Borel probability measures on the reals, depending on their Fourier asymptotics. In particular, I will talk about examples involving certain fractal measures with small Hausdorff dimensional support and measures whose Fourier transforms have arbitrary slow polynomial decay. This is a joint work with Shreyasi Datta.

Dubi Kelmer

The Light Cone Siegel Transform, its Moment formulas, and their applications

In this talk I will describe the light cone Siegel transform corresponding to an indefinite rational quadratic form, and its moment formulas (analogous to Siegel's mean value theorem and Roger's second moment formula). I will then describe various applications of such formulas for counting integral points on the light cone as well as to the distribution of rational points on the sphere (as well as other quadratic hypersurfaces). This is based on joint work with Shucheng Yu.

Dmitriy Kleinbock

A general construction of singular objects in Diophantine approximation and dynamics

A vector x = (x_1, ..., x_n) in R^n is singular if for any ε > 0 and for all large enough T there are solutions p in Z^n and q in {1, ...,T} to the inequality ∥qx − p∥ < εT^{−1/n}. This corresponds to divergence of a certain trajectory in the space of unimodular lattices in R^{n+1}. In previous work, using a variation on an old argument employed by Khintchine and Jarnik, we showed that analytic manifolds of dimension at least two contain totally irrational (those with 1, x_1, ..., x_n linearly independent over Z) singular vectors. In this new work we adapt the argument and develop a scheme for constructing singular objects of a similar type in a wide variety of general situations. Applications include: approximation with weights, approximation by algebraic vectors, inhomogeneous approximation, divergence with super-fast rates and with respect to higher minima of lattices, etc etc. The key step is verifying the so-called total density property of resonant sets. Joint with Nikolay Moshchevitin, Vasiliy Nekrasov, Jacqueline Warren and Barak Weiss.

Manuel Luethi

Almost prime times in the maximal horospherical flow on lattices in three dimensions

Following work by Taylor McAdam, it is known that orbits of the maximal horospherical group are effectively equidistributed with polynomial rate unless there is a Diophantine obstruction and that this can be used to deduce density of almost prime times along generic orbits of the full upper triangular group in the space of lattices, with the number of prime factors depending on the basepoint. In ongoing work with Taylor McAdam, we provide a finer analysis of the Diophantine obstruction. Namely, we show that for a point generic under the horospherical group but facing the Diophantine obstruction, there exists a sequence of well equidistributed closely tracked intermediate orbit closures such that pieces of the orbits of the horospherical group are effectively equidistributed inside the intermediate orbit closures. As an application, we remove the base-point dependence of the number of prime factors.

Amir Mohammadi

Dynamics on homogeneous spaces: a quantitative viewpoint

Rigidity phenomena in homogeneous spaces have been extensively studied over the past few decades with several striking results and applications. We will give an overview of activities pertaining to the quantitative aspect of the analysis in this context with an emphasis on recent developments.

Erez Nesharim

The left shifts of the Thue-Morse sequence have partial escape of mass over F_2((1/t))

Every Laurent series in F_q((1/t)) has a continued fraction expansion whose partial quotients are polynomials. De Mathan and Teulie proved that the degrees of the partial quotients of the left shifts of every quadratic Laurent series are unbounded. Shapira and Paulin improved this by showing that, in fact, a positive proportion of the degrees are bigger than any bound. We show that their result is best possible in the following sense: For the Laurent series over F_2((1/t)) whose sequence of coefficients is the Thue-Morse sequence, this proportion is strictly less than 1. This talk is based on a work in progress with Uri Shapira and Noy Soffer-Aranov.

Nimish Shah

Equidistribution of non-contracting polynomially bounded o-minimal curves

We show that on a finite volume homogeneous space of a linear algebraic group G, any trajectory of a "non-contracting" curve in G definable in a polynomially bounded o-minimal structure gets equidistributed with respect to a homogeneous measure. This a joint work with Michael Bersudsky and Hao Xing. 

Anders Södergren

Mean value formulas in the space of lattices

In this talk I will discuss Rogers' mean value formula in the space of unimodular lattices as well as a recent generalization of Rogers' formula. In particular, I will describe a formula for mean values of products of Siegel transforms with arguments taken from both a lattice and its dual lattice, and outline a few applications of this formula. This is joint work with Andreas Strömbergsson.

Omri Solan

Gap in critical exponents of SL2(R) orbits in nonarithmetic quotients of SL2(C)

We will discuss the following result.  For every nonarithmetic lattice Γ < SL2(C) there is ε_Γ such that for every g ∈ SL2(C) the intersection gΓg^{-1} ∩ SL2(R) is either a lattice or a has critical exponent δ(gΓg^{-1} ∩ SL2(R)) ≤ 1-ε_Γ. An ergodic component of the proof is an effectivization of high directional entropy. We will describe the effectivization in details.

Andreas Strömbergsson

An effective equidistribution result in the space of 2-dimensional tori with k marked points

Let X be the homogeneous space Gamma \ G, where G is the semidirect product of SL(2,R) and a direct sum of k copies of R^2, and where Gamma is the subgroup of integer elements in G. I will present a result giving effective equidistribution of 1-dimensional unipotent orbits in the space X. The proof makes use of the delta method in the form developed by Heath-Brown. Joint work with Anders Södergren and Pankaj Vishe.

Andreas Wieser

Effective equidistribution of orbits of semisimple groups and an effective closing lemma

In recent years, quantitative equidistribution results have been a major theme in homogeneous dynamics. In the first half of this talk, we discuss a polynomially effective equidistribution result for orbits of semisimple groups on congruence quotients. This extends work of Einsiedler, Margulis, and Venkatesh by removing the assumed triviality of the centralizer in the ambient group.  An essential ingredient of the proof is an effective closing lemma for unipotent flows from work with Lindenstrauss, Margulis, Mohammadi, and Shah. We discuss this result in the second half of the talk.

Pengyu Yang

Dirichlet improvable and singular vectors on manifolds

Dirichlet improvability and singularity are Diophantine properties of real vectors (or more generally real matrices). These properties are related to the limiting behavior of flow trajectories in homogeneous spaces. We will present some recent results concerning Dirichlet improvable and singular vectors on manifolds, as well as the relevant equidistribution results in homogeneous dynamics.

Yuval Yifrach

Rigidity Properties of Tori Measures in a Fixed Number Field

We consider compact orbits for the action of the full diagonal group A on the space of 3-dimensional lattices X_3 which come from a fixed quintic number field K.

Let p be a non-split prime in K and let (M_k)_k be a sequence of full modules in K such that M_k is of index p^k in the ring of integers \Oo_K. We prove that any weak limit of the Haar measures associated to (M_k)_k contains the Haar measure m_{X_3} as an ergodic factor. This result generalizes certain aspects of a work of Aka and Shapira to dimension n=3. In addition this result is an extension of certain aspects of a joint work of Shapira and Zheng.

As a corollary of this result, we prove that for numbers $\alpha,\beta$ such that $1,\alpha,\beta$ span a full module in a cubic number field, the vector (\alpha,\beta) satisfies the p-adic Littlewood Conjecture. Namely, we prove that

\begin{equation}

\lim_{m\rightarrow \infty}\liminf_{n\rightarrow \infty}n^{1/2}\inn{np^m(\alpha,\beta)}=0.

\end{equation}

Our proof relies on estimates on the asymptotic orders of units in fixed number fields modulo families of natural numbers and on rigidity results from Einsiedler Lindenstrauss Michel and Venkatesh