AMS Centennial Fellowship

Project: Research project

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 |d|−1/2 {x ∈ Zn | ⟨x, x⟩ = d} ⊂ Sn−1 for large d ∈ N, i.e. the radial projection of lattice points in Euclidean space to the unit sphere. 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 of quantum observables for automorphic forms, such as the asymptotic of ∫Yg|ψ|2 dVolY for ψ : Y → C
an L2-normalized Laplace eigenfunction of large eigenvalue on an arithmetic manifold Y and g : Y → C a nice test function. My research has specifically focused on problems that are connected to homogeneous dynamics, as is the case with all the examples above. In homogeneous dynamics we study the orbits of the action of a closed subgroup H < G of a locally compact group G on a space of the form Γ\G for a lattice Γ < G. An all important example is SLn(Z)\SLn(R) which parameterizes unimodular lattices in Rn. My results are distinguished by introducing ideas from the theory of automorphic forms, arithmetic geometry, and multiplicative number theory into homogeneous dynamics.
StatusActive
Effective start/end date9/1/208/31/22

Funding

  • American Mathematical Society (AGMT 8/24/20)

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