Asaf Katz Mathematics

I am currently a postdoctoral assistant professor at the mathematics department of the University of Michigan, working with Prof. Ralf Spatzier.

Before that, I was an L.E. Dickson instructor at the mathematics department of the University of Chicago, working with Prof. Alex Eskin.

Before that, I completed my Ph.D. studies at the Hebrew University of Jerusalem under the guidance of Prof. Elon Lindenstrauss.

My CV (October 2023).

Research Statement (October 2023).

Research

I work in dynamics, primarily in ergodic theory and homogeneous dynamics, in particular towards applications to number theory and geometry.

Among my interests:

Papers

This paper generalizes H. Furstenberg's result about density of an irrational point in the one-torus under the x2 x3 multiplicative semigroup to a much sparser class of sequences. The proof uses ideas from the effective proof of Furstenberg's theorem by Bourgain-Lindenstrauss-Michel-Venkatesh and p-adic analysis.This paper deals with Bourgain's sparse ergodic theorem for the case of the horocyclic flow (and general one-parameter unipotent flows on homogenuous spaces). In particular, we show that the exceptional set of the point which their average along polynomial sample times do not equidistribute is a lower dimensional subset. Moreover, we show that such an estimate is free of the actual spectral gap of the homogenuous space. The proof uses ideas from homogenuous dynamics (primarily Ratner's theorems and quantitative mixing), automorphic representation theory, harmonic analysis and number theory. The paper shows a quantitative disjointness statement of unipotent flows from nilflows, based on the proof of the uniform Wiener-Wintner theorem and the results of Green,Tao and Ziegler regarding nilcharacters. Submitted.This paper deals with applications of Venkatesh's disjointness technique towards sparse equidistribution problems. We show (by Ratner's techniques) that large spheres in horospheres equidistribute and then we give some quantitative analogues like equidistribution of large annuli and some singular Bochner-Riesz means. Moreover we show by using techniques from higher Fourier analysis, quantative equidistribution of large spheres in nilmanifolds.Submitted.This paper proves a Margulis' type inequality to translates of horospherical orbits. As an application of the proof, we can deduce quantitative horospherical equidistribution result for any lattice defined over a number field, strengthening the currently known results.This paper deals with measure classification results for Anosov flows, satisfying quantitative non-integrability condition, by utilizing the Factorization technique, developed by A. Eskin and M. Mirzakhani in the context of dynamics over the moduli space of translation surfaces. As a result we get pointwise equidistribution result for systems which are highly quantitatively non-integrable, using idea developed by Chaika-Eskin and Eskin-Mirzakhani-Mohammadi.  Preprint This paper proves an effective equidistribution estimate regarding spherical averages in horospheres, proving a conjecture of Margulis and Lindenstrauss. It generalizes classical equidistribution theorems of spheres in tori. The proof is using effective equidistribution and Littlewood-Paley g-function estimates. Submitted We prove some sparse equidistribution theorems in groups of the form $SL_{2}(\mathbb{R}) \times SL_{2}(\mathbb{R})$ and $SL_{2}(\mathbb{C})$ based on recent equidistribution results of Lindenstrauss-Mohammadi-Wang and certain measure rigidty results of Marina Ratner. In particular we prove Margulis' conjecture in the case of arithmetic quotients. Preprint We prove quantitative effective density of the intersection points of the geodesics of the Bowen packet in closed hyperbolic surfaces. We also prove effective equidistribution of the emphirical measures supported over the density points in the case of surfaces of constant curvature. We provide certain results about surfaces of non-constant curvature as well. 

Contact Details

email: first name (dot) last name @ gmail.com

Department of Mathematics  , Room 1840

2074 East Hall

530 Church Street

Ann Arbor, MI 48109-1043