## Project Details

### Description

I work in dynamics and number theory. A central theme in my research is the equidistribution of arithmetic objects. Let me present three major topics I am interested in. The first is the distribution of integral points of an algebraic variety in its real points when varying the integral model, for example, the radial projection of lattice points in Euclidean space to the unit sphere, i.e. d −1/2 x Zn x, x = d Sn−1 for large d N. The second one is the distribution of Galois orbits of algebraic points in the complex points of a variety when varying the algebraic point, e.g. the primitive roots of unity of large order in C× and Galois orbits of special points on Shimura varieties (generalizing Heegner points on the modular curve). The third topic is the asymptotic distribution of automorphic forms, such as the limit when of g ψλ 2 dVolY for ψλ : Y C an L2-normalized Laplace eigenfunction of eigenvalue λ on an arithmetic manifold Y and g : Y C a fixed nice test function. My work in these areas is distinguished by introducing ideas from the theory of automorphic forms, arithmetic geometry, and multiplicative number theory into

homogeneous dynamics.

For the duration of the Sloan Research Fellowship I plan to pursue three major research directions. The first one will focus on equidistribution of periodic torus orbits and builds upon my recent breakthrough around the Michel-Venkatesh mixing conjecture [Kha19b]1 and a conjecture of Aka-Einsiedler-Shapira [Kha18]2. A special point on a product of modular curves is a tuple of Heegner points. The action of the Galois group on special points is related, by class field theory, to the action of a non-maximal torus. The Michel-Venkatesh conjecture [MV06] is a major strengthening of the André-Oort conjecture for a product of modular curves. The latter has been established in the pioneering work of Pila [Pil11]. The André- Oort conjecture claims that the Zariski closure of a sequence of special points is a special sub-variety (a Cartesian product of special points and modular curves of different levels). The stronger mixing conjecture claims that the Galois orbits of these points equidistribute in the special sub-variety. A major challenge that appears in this problem is the possibility of accumulation in the limit on intermediate sub-varieties. The proof is a fusion of measure rigidity [EL19] and a method I have developed based on the relative trace formula. The classical trace formula, the simplest of which is the Poisson summation formula, relates spectral information to more tractable geometric information. I have discovered another side to the relative trace formula – a dynamic one. This establishes a bridge between dynamical quantities such as Kolmogorov- Sinai entropy [Kha19a]3 or correlation between measures [Kha19b, Kha18] and the geometric expansion of a relative trace. In a very recent ongoing work I made progress on the problem of equidistribution of packets of maximal tori in Γ SL3(R) for an arbitrary lattice Γ. I believe there is hope to establish effective equidistribution in this case using the uniqueness of the measure of maximal entropy. The study of periodic torus orbits is often related to special values of automorphic L-functions. Indeed, in [Kha19c]4 I am able to establish a non-vanishing result for class group Hecke L-functions of arbitrary degree. This is the first time such a result is proved in a setting more general then imaginary quadratic fields.

A second direction I pursue is closely related to the distributional properties of automorphic forms mentioned above. In a re

homogeneous dynamics.

For the duration of the Sloan Research Fellowship I plan to pursue three major research directions. The first one will focus on equidistribution of periodic torus orbits and builds upon my recent breakthrough around the Michel-Venkatesh mixing conjecture [Kha19b]1 and a conjecture of Aka-Einsiedler-Shapira [Kha18]2. A special point on a product of modular curves is a tuple of Heegner points. The action of the Galois group on special points is related, by class field theory, to the action of a non-maximal torus. The Michel-Venkatesh conjecture [MV06] is a major strengthening of the André-Oort conjecture for a product of modular curves. The latter has been established in the pioneering work of Pila [Pil11]. The André- Oort conjecture claims that the Zariski closure of a sequence of special points is a special sub-variety (a Cartesian product of special points and modular curves of different levels). The stronger mixing conjecture claims that the Galois orbits of these points equidistribute in the special sub-variety. A major challenge that appears in this problem is the possibility of accumulation in the limit on intermediate sub-varieties. The proof is a fusion of measure rigidity [EL19] and a method I have developed based on the relative trace formula. The classical trace formula, the simplest of which is the Poisson summation formula, relates spectral information to more tractable geometric information. I have discovered another side to the relative trace formula – a dynamic one. This establishes a bridge between dynamical quantities such as Kolmogorov- Sinai entropy [Kha19a]3 or correlation between measures [Kha19b, Kha18] and the geometric expansion of a relative trace. In a very recent ongoing work I made progress on the problem of equidistribution of packets of maximal tori in Γ SL3(R) for an arbitrary lattice Γ. I believe there is hope to establish effective equidistribution in this case using the uniqueness of the measure of maximal entropy. The study of periodic torus orbits is often related to special values of automorphic L-functions. Indeed, in [Kha19c]4 I am able to establish a non-vanishing result for class group Hecke L-functions of arbitrary degree. This is the first time such a result is proved in a setting more general then imaginary quadratic fields.

A second direction I pursue is closely related to the distributional properties of automorphic forms mentioned above. In a re

Status | Not started |
---|---|

Effective start/end date | 9/15/21 → 9/14/23 |

### Funding

- Alfred P. Sloan Foundation (NOT SPECIFIED)

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