## Project Details

### Description

Overview:

This project is about the asymptotic distribution of periodic torus orbits on arithmetic homogeneous

spaces of algebraic groups. Periodic torus orbits unify several objects in number theory and homogeneous dynamics into a single framework. These include Heegner points and closed geodesics on modular curves and their higher rank generalizations, integral points on homogeneous varieties with a torus stabilizer, Galois orbits of special points on Shimura varieties and questions about representations of integral quadratic forms. Periodic torus orbits are closely related to some automorphic L-functions by period formulae like the Waldspurger formula and Hecke's formula for Eisenstein series.

This project will study limits of period measures on periodic torus orbits. This is an exceptionally

interdisciplinary area of mathematics which combines homogeneous dynamics and ergodic theory,

automorphic forms, arithmetic geometry and multiplicative number theory. The prototypical result in this field is Duke's theorem about the equidistribution of packets of closed geodesics and Heegner points on the modular curve. Our main focus is analogous higher rank equidistribution problems. Now is an exciting time to study these questions as several long standing conjectures appear attainable.

Intellectual Merit:

The main aim of this project is to develop novel techniques to study limits of period measures invariant under the action of a torus. Linnik's "ergodic method" has been recast into a modern version by Einsiedler, Lindenstrauss, Michel and Venkatesh who demonstrated the usefulness in this area of the phenomena of measure rigidity for diagonalizable actions. Measure rigidity by itself fails to settle any single case due to several fundamental limitations: positivity of entropy, concentration on intermediate orbits and escape of mass.

The recent work by the PI has provided new methods to deal with several of these issues. The PI

suggested studying these questions by bounding correlations between periodic measures. This work has constructed a bridge between correlation quantities and arithmetic properties of toral packets. These correlations can now be bounded in terms of arithmetic invariants which appear in the geometric expansion of a relative trace. The culmination of these efforts to date is the recent progress by the PI on equidistribution of torus orbits of special points and the partial resolution of the mixing conjecture of Michel and Venkatesh. These ideas show potential for significant progress on previously unattainable higher rank questions.

Broader Impacts:

The PI has a constantly expanding portfolio of broader impact activities. These are mostly focused on mentoring of undergraduate students outside of the classroom and exposing a broad body of

undergraduate students to research mathematics -- its beauty and depth. The PI is strongly committed to junior mentoring as evident by his activity last year even when he was a member in the IAS. During that year the PI has supervised a junior thesis and a summer research project, he has given two popular talks to undergraduate students and served on the senior thesis committee of a student.

This project is about the asymptotic distribution of periodic torus orbits on arithmetic homogeneous

spaces of algebraic groups. Periodic torus orbits unify several objects in number theory and homogeneous dynamics into a single framework. These include Heegner points and closed geodesics on modular curves and their higher rank generalizations, integral points on homogeneous varieties with a torus stabilizer, Galois orbits of special points on Shimura varieties and questions about representations of integral quadratic forms. Periodic torus orbits are closely related to some automorphic L-functions by period formulae like the Waldspurger formula and Hecke's formula for Eisenstein series.

This project will study limits of period measures on periodic torus orbits. This is an exceptionally

interdisciplinary area of mathematics which combines homogeneous dynamics and ergodic theory,

automorphic forms, arithmetic geometry and multiplicative number theory. The prototypical result in this field is Duke's theorem about the equidistribution of packets of closed geodesics and Heegner points on the modular curve. Our main focus is analogous higher rank equidistribution problems. Now is an exciting time to study these questions as several long standing conjectures appear attainable.

Intellectual Merit:

The main aim of this project is to develop novel techniques to study limits of period measures invariant under the action of a torus. Linnik's "ergodic method" has been recast into a modern version by Einsiedler, Lindenstrauss, Michel and Venkatesh who demonstrated the usefulness in this area of the phenomena of measure rigidity for diagonalizable actions. Measure rigidity by itself fails to settle any single case due to several fundamental limitations: positivity of entropy, concentration on intermediate orbits and escape of mass.

The recent work by the PI has provided new methods to deal with several of these issues. The PI

suggested studying these questions by bounding correlations between periodic measures. This work has constructed a bridge between correlation quantities and arithmetic properties of toral packets. These correlations can now be bounded in terms of arithmetic invariants which appear in the geometric expansion of a relative trace. The culmination of these efforts to date is the recent progress by the PI on equidistribution of torus orbits of special points and the partial resolution of the mixing conjecture of Michel and Venkatesh. These ideas show potential for significant progress on previously unattainable higher rank questions.

Broader Impacts:

The PI has a constantly expanding portfolio of broader impact activities. These are mostly focused on mentoring of undergraduate students outside of the classroom and exposing a broad body of

undergraduate students to research mathematics -- its beauty and depth. The PI is strongly committed to junior mentoring as evident by his activity last year even when he was a member in the IAS. During that year the PI has supervised a junior thesis and a summer research project, he has given two popular talks to undergraduate students and served on the senior thesis committee of a student.

Status | Active |
---|---|

Effective start/end date | 7/1/19 → 6/30/22 |

### Funding

- National Science Foundation (DMS‐1946333)

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