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Effective equidistribution on homogeneous spaces

Applicant Einsiedler Manfred
Number 127145
Funding scheme Project funding (Div. I-III)
Research institution Departement Mathematik ETH Zürich
Institution of higher education ETH Zurich - ETHZ
Main discipline Mathematics
Start/End 01.10.2009 - 31.03.2013
Approved amount 580'500.00
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Keywords (9)

Dynamical systems; Homogeneous spaces; Equidistribution; Effectiveness; Number theory; Dynamics; Entropy; Spectral gap; Effective bounds

Lay Summary (English)

Lead
Lay summary
The aim of the project is to better understand the distribution of orbits (trajectories in certain generalized dynamical system). The questions posed have applications to number theory, in particular to the theory of Diophantine approximations. Various number theoretic problems are related to equidistribution problems of orbits on homogeneous spaces -- the latter are quotients of Lie groups and give examples of spaces with non-euclidean geometries. There are a number of different methods to study such problems. However, the most general method, which is a combination of the work of Dani, Margulis, Mozes, Ratner, and Shah has the drawback of not providing an error rate. In other words, the theorems show that the orbit is uniformly dense in the space, but do not answer the question of how long a piece of an orbit has to be in order to visit epsilon-neighborhoods of all points. An explicit dependence between the length of the orbit and the radius epsilon of the neighborhood is refered to as effectivity.The project aims to obtain more general cases of effective equidistribution and further applications in number theory. In particular, the relationship between the notion of spectral gap (property (tau)) and effective equidistribution problems will be studied. Applications could also include the classical problem (already studied by Gauss) of when a given quadratic form can represent another.
Direct link to Lay Summary Last update: 21.02.2013

Responsible applicant and co-applicants

Employees

Publications

Publication
Period functions for Hecke triangle groups, and the Selberg zeta function as a Fredholm determinant
Möller M., Pohl A. D. (2013), Period functions for Hecke triangle groups, and the Selberg zeta function as a Fredholm determinant, in Ergodic Theory Dynam. Systems, 33(1), 247-283.
A dynamical approach to Maass cusp forms
Pohl Anke D. (2012), A dynamical approach to Maass cusp forms, in J. Mod. Dyn., 6(4), 563-596.
A joinings classification and a special case of Raghunathan's conjecture in positive characteristic (with an appendix by Kevin Wortman)
Einsiedler Manfred, Mohammadi Amir (2012), A joinings classification and a special case of Raghunathan's conjecture in positive characteristic (with an appendix by Kevin Wortman), in J. Anal. Math., 116, 299-334.
Entropy and escape of mass for Hilbert modular spaces
Kadyrov Shirali (2012), Entropy and escape of mass for Hilbert modular spaces, in J. Lie Theory, 22(3), 701-722.
Entropy and escape of mass for SL(3,Z)\SL(3,R)
Einsiedler Manfred, Kadyrov Shirali (2012), Entropy and escape of mass for SL(3,Z)\SL(3,R), in Israel J. Math., 190, 253-288.
Positive entropy invariant measures on the space of lattices with escape of mass
Kadyrov Shirali (2012), Positive entropy invariant measures on the space of lattices with escape of mass, in Ergodic Theory Dynam. Systems, 32(1), 141-157.
Diophantine approximations on fractals
Einsiedler Manfred, Fishman Lior, Shapira Uri (2011), Diophantine approximations on fractals, in Geom. Funct. Anal., 21(1), 14-35.
Distribution of periodic torus orbits and Duke's theorem for cubic fields
Einsiedler Manfred, Lindenstrauss Elon, Michel Philippe, Venkatesh Akshay (2011), Distribution of periodic torus orbits and Duke's theorem for cubic fields, in Ann. of Math. (2), 173(2), 815-885.
Equidistribution of singular measures on nilmanifolds and skew products
Polo Fabrizio (2011), Equidistribution of singular measures on nilmanifolds and skew products, in Ergodic Theory Dynam. Systems, 31(6), 1785-1817.
Ergodic theory with a view towards number theory
Einsiedler Manfred, Ward Thomas (2011), Ergodic theory with a view towards number theory, Springer-Verlag, London 259.
Diagonal actions on locally homogeneous spaces
Einsiedler M., Lindenstrauss E. (2010), Diagonal actions on locally homogeneous spaces, Amer. Math. Soc., Providence, 10, 155-241.
Effective equidistribution and spectral gap
Einsiedler Manfred (2010), Effective equidistribution and spectral gap, in European Congress of Mathematics, ZürichEur. Math. Soc.,, Zurich.

Scientific events

Active participation

Title Type of contribution Title of article or contribution Date Place Persons involved
Dynamics on parameter spaces 2013 27.01.2013 Sde Boker, Zurich


Self-organised

Title Date Place

Associated projects

Number Title Start Funding scheme
152819 Equidistribution and dynamics on homogeneous spaces 01.05.2014 Project funding (Div. I-III)

Abstract

The project concerns equidistribution problems for homogeneous spaces, an area of research linking the theory of Lie groups, algebraic groups, unitary representation, and dynamical systems. Moreover, these problems are closely related to number theory, especially those involving Diophantine solutions to equations and inequalities -- where the solution set and their equations both admit large symmetry groups.The theory of unipotent flows (as developed by Margulis, Ratner, and others) is a very powerful technique to establish equidistribution but has the disadvantage of being non-effective, i.e. there is no error provided for the equidistribution statements. On the other hand representation theoretic or number theoretic arguments are more limited in scope but do provide error rates where they apply. Recently it has become clear that combined arguments can lead to proofs of theorems that are otherwise out of reach. The applicant has obtained in a joint work with Margulis and Venkatesh an effective distribution theorem for closed orbits of semisimple subgroups that currently is still limited to real Lie groups and to semisimple subgroups satisfying a technical assumption (finite centralizer). One part of the project is to generalize this method to the adelic setting and to avoid technical assumptions. This will, in particular, lead to an independent proof of property (tau) for congruence subgroups of semisimple algebraic groups defined over the rational numbers, and to a refinement of a recent joint work of Ellenberg and Venkatesh. Another case of an equidistribution problem is the study of the images of long curves in homogeneous spaces as studied in the recent work of Shah -- it is proposed to study the error rate for instances where the equidistribution has already been established.
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