Georgia Institute of Technology College of Sciences

The arithmetic quantum unique ergodicity (AQUE) conjecture predicts that the L^2 mass of Hecke-Maass cusp forms on an arithmetic hyperbolic manifold becomes equidistributed as the Laplace eigenvalue grows. If the underlying manifold is non-compact, mass could “escape to infinity”. This possibility was ruled out by Soundararajan for arithmetic surfaces, which when combined with celebrated work of Lindenstrauss completed the proof of AQUE for surfaces.

We establish non-escape of mass for Hecke-Maass cusp forms on a congruence quotient of hyperbolic 4-space. Unlike in the setting of hyperbolic 2- or 3-manifolds (for which AQUE has been proved), the number of terms in the Hecke relations is unbounded, which prevents us from naively applying Cauchy-Schwarz. We instead view the isometry group as a group of quaternionic matrices, and rely on non-commutative unique factorization, along with certain structural features of the Hecke action. Joint work with Zvi Shem-Tov.

Houdré and Tetali defined a class of isoperimetric constants phi_p of graphs for 1/2 <= p <= 1.  When p=1, the classical Cheeger's inequality relates phi_1 to the second smallest eigenvalue of the normalized Laplacian matrix.  Houdré and Tetali conjectured that a similar Cheeger-type inequality holds for p=1/2, which if true would be a strengthening of Cheeger's inequality.  Morris and Peres proved the Houdré-Tetali conjecture up to an additional log factor, using techniques from evolving sets.  In this talk, we discuss the following results about this conjecture:

  - There is a family of counterexamples to the conjecture of Houdré and Tetali, showing that the logarithmic factor is needed.

  - Morris and Peres' result can be recovered using standard spectral arguments. 

  - The Houdré-Tetali conjecture is true for any constant p strictly bigger than 1/2, which is also a strengthening of Cheeger's inequality.

If time permits, we also discuss other strengthenings of Cheeger's inequality.  No background is assumed from the audience.

 I will discuss recent work with Chantal David, Alexander Dunn, and Joshua Stucky, in which we prove that a positive proportion of Hecke L-functions associated to the cubic residue symbol modulo squarefree Eisenstein integers do not vanish at the central point. Our principal new contribution is the asymptotic evaluation of the mollified second moment with power saving error term. No such asymptotic formula was previously known for a cubic family (even over function fields). Our new approach makes crucial use of Patterson's evaluation of the Fourier coefficients of the cubic metaplectic theta function, Heath-Brown's cubic large sieve, and a Lindelöf-on-average upper bound for the second moment of cubic Dirichlet series that we establish. The significance of our result is that the (unitary) family considered does not satisfy a perfectly orthogonal large sieve bound. This is quite unlike other families of Dirichlet L-functions for which unconditional results are known (namely the symplectic family of quadratic characters and the unitary family of all Dirichlet characters modulo q). Consequently, our proof has fundamentally different features from the corresponding works of Soundararajan and of Iwaniec and Sarnak.

We will talk about recent work establishing a quantitative nonlinear scattering theory for asymptotically de Sitter solutions of the Einstein vacuum equations in (n+1) dimensions with n ≥ 4 even, which are determined by small scattering data at future infinity and past infinity. We will also explain why the case of even spatial dimension n poses significant challenges compared to its odd counterpart and was left open by the previous works in the literature.