Scientists / Open questions

Elon Lindenstrauss

Ergodic theory; dynamical systems; number theory

2010 Fields Medal

15 questions

Quantitative equidistribution of affine random walks on the torus beyond finite support and strong irreducibility
Shift embeddability of topological dynamical systems: completing the Lindenstrauss embedding and factor-separation program beyond the marker-property class
Joining classification for higher-rank diagonalizable actions beyond the no-local-rank-one-factor hypothesis, and the equivalent measure classification for commuting toral endomorphism actions without virtually cyclic factors
Quantum Unique Ergodicity: equidistribution of all high-energy Laplace eigenfunctions on negatively curved manifolds, and the removal of the Hecke eigenfunction assumption on arithmetic hyperbolic surfaces
Universal metric selection and the exchange of supremum and scale limit in the variational principle for metric mean dimension
Scope of Lindenstrauss's measurable-to-continuous cocycle regularization and the characterization of strictly ergodic distal models
Metric realization of mean dimension, the small boundary property at mean dimension zero, and cubical-shift embedding: the open problems of Lindenstrauss–Weiss (2000)
Does a dense rotation subgroup force a spectral gap for the rotation-averaging operator on $L^2(\mathrm{SO}(d))$ ?
Quantum unique ergodicity for Hecke eigenfunctions on the sphere (Böcherer-Sarnak-Schulze-Pillot conjecture)
Arithmetic quantum unique ergodicity under reduced Hecke input: escape of mass at a single Hecke place, and Laplace spectral multiplicities, on arithmetic hyperbolic surfaces
Optimal dependence on the ambient ternary form of the count of embeddings of a binary quadratic form
Orbit closures and closed invariant sets for reducible algebraic $\mathbb{Z}^d$ -actions on tori and solenoids
Marker-free extremal metrics and the double variational principle for mean dimension
Can one always lower topological entropy? The factor-entropy gap problem for finite-entropy systems
Bounded orbits and closed invariant sets for the diagonal group action on $\mathrm{SL}(n,\mathbb{R})/\mathrm{SL}(n,\mathbb{Z})$ and related arithmetic quotients

Back to open problems

Quantitative equidistribution of affine random walks on the torus beyond finite support and strong irreducibility

Shift embeddability of topological dynamical systems: completing the Lindenstrauss embedding and factor-separation program beyond the marker-property class

Joining classification for higher-rank diagonalizable actions beyond the no-local-rank-one-factor hypothesis, and the equivalent measure classification for commuting toral endomorphism actions without virtually cyclic factors

Quantum Unique Ergodicity: equidistribution of all high-energy Laplace eigenfunctions on negatively curved manifolds, and the removal of the Hecke eigenfunction assumption on arithmetic hyperbolic surfaces

Universal metric selection and the exchange of supremum and scale limit in the variational principle for metric mean dimension

Scope of Lindenstrauss's measurable-to-continuous cocycle regularization and the characterization of strictly ergodic distal models

Metric realization of mean dimension, the small boundary property at mean dimension zero, and cubical-shift embedding: the open problems of Lindenstrauss–Weiss (2000)

Does a dense rotation subgroup force a spectral gap for the rotation-averaging operator on L2(SO(d))L^2(\mathrm{SO}(d))L2(SO(d))?

Quantum unique ergodicity for Hecke eigenfunctions on the sphere (Böcherer-Sarnak-Schulze-Pillot conjecture)

Arithmetic quantum unique ergodicity under reduced Hecke input: escape of mass at a single Hecke place, and Laplace spectral multiplicities, on arithmetic hyperbolic surfaces

Optimal dependence on the ambient ternary form of the count of embeddings of a binary quadratic form

Orbit closures and closed invariant sets for reducible algebraic Zd\mathbb{Z}^dZd-actions on tori and solenoids

Marker-free extremal metrics and the double variational principle for mean dimension

Can one always lower topological entropy? The factor-entropy gap problem for finite-entropy systems

Bounded orbits and closed invariant sets for the diagonal group action on SL(n,R)/SL(n,Z)\mathrm{SL}(n,\mathbb{R})/\mathrm{SL}(n,\mathbb{Z})SL(n,R)/SL(n,Z) and related arithmetic quotients

Does a dense rotation subgroup force a spectral gap for the rotation-averaging operator on $L^2(\mathrm{SO}(d))$ ?

Orbit closures and closed invariant sets for reducible algebraic $\mathbb{Z}^d$ -actions on tori and solenoids

Bounded orbits and closed invariant sets for the diagonal group action on $\mathrm{SL}(n,\mathbb{R})/\mathrm{SL}(n,\mathbb{Z})$ and related arithmetic quotients