Asaf Katz Mathematics

I am an Assistant Professor of mathematics at the School of Mathematics, Georgia Institute of Technology (Georgia Tech).

Before that, I was 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:

Ergodic theory.
Homogeneous dynamics and relations to number theory.
Automorphic representations and spectral theory.
Number theory - analytic number theory and additive combinatorics.
Harmonic analysis.

Papers

Generalizations of Furstenberg's diophantine result - Ergodic Theory and Dynamical Systems, Vol 38, Issue 3.

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.

On mixing and sparse ergodic theorems - Journal of Modern Dynamics, Vol 17.

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.

Quantitative disjointness of nilflows from horospherical flows - accepted, Journal d'Analyse Mathematique.

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.

Applications of joinings to sparse equidistribution problems.

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.

Margulis' inequality for translates of horospherical orbits and applications.

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.

Measure rigidity for Anosov flows via the Factorization method - To appear, Geometric and Functional Analysis

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.

Density and equidistribution of intersecting geodesics

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

School of Mathematics

Georgia Institute of Technology, Skiles Classroom Building

686 Cherry Street

Atlanta, GA, 30332-0160 USA

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