Abstract: We give a generalized Kuznetsov formula that allows one to impose additional conditions at finitely many primes.  The formula arises from the relative trace formula. I will discuss applications to spectral large sieve inequalities and subconvexity. This is work in progress with M.P. Young and Y. Hu.

For a family of finite sets whose cardinalities are naturally called class numbers, the Gauss problem asks to determine the subfamily in which every member has class number one. We study the Siegel moduli space of abelian varieties in characteristic $p$ and solve the Gauss problem for the family of central leaves, which are the loci consisting of points whose associated $p$-divisible groups are isomorphic. Our solution involves mass formulae, computations of automorphism groups, and a careful analysis of Ekedahl-Oort strata in genus $4$. This geometric Gauss problem is closely related to an arithmetic Gauss problem for genera of positive-definite quaternion Hermitian lattices, which we also solve.

We have recently proposed a new kind of neural network, called a Legendre Memory Unit (LMU) that is provably optimal for compressing streaming time series data. In this talk, I describe this network, and a variety of state-of-the-art results that have been set using the LMU. I will include recent results on speech and language applications that demonstrate significant improvements over transformers. I will discuss variants of the original LMU that permit effective scaling on current GPUs and hold promise to provide extremely efficient edge time series processing.

Dispersion relations for S-matrices and CFT correlators translate UV consistency into bounds on IR observables. In this talk, I will begin with briefly introducing dispersionrelations in 2D flat space which will guide the analogous discussion in AdS₂/CFT₁. I will introduce a set of functionals acting on the 1D CFT. These will allow us to prove bounds on higher-derivative couplings in weakly coupled non-gravitational EFTs in AdS₂. At the leading order in the bulk-point limit, the bounds agree with the flat-space result. Furthermore we can compute the leading universal effect of finite AdS radius on the bounds.

The Chabauty-Kim method is an effective algorithm for finding the $S$-integral points of hyperbolic curves by directly using the hyperbolicity in group-cohomological arguments. Central objects in the theory are affine spaces known as a Selmer schemes. We'll introduce the CK method and Selmer schemes, and demonstrate some additional structures possessed by Selmer schemes which can aid in implementing the CK method.
 

Spectral methods offer a tractable, global framework for clustering in graphs via eigenvector computations on graph matrices. Hypergraph data, in which entities interact on edges of arbitrary size, poses challenges for matrix representations and therefore for spectral clustering. We study spectral clustering for arbitrary hypergraphs based on the hypergraph nonbacktracking operator. After reviewing the definition of this operator and its basic properties, we prove a theorem of Ihara-Bass type which allows eigenpair computations to take place on a smaller matrix, often enabling faster computation. We then propose an alternating algorithm for inference in a hypergraph stochastic blockmodel via linearized belief-propagation which involves a spectral clustering step, again using nonbacktracking operators. We provide proofs related to this algorithm that both formalize and extend several previous results. We pose several conjectures about the limits of spectral methods and detectability in hypergraph stochastic blockmodels in general, supporting these with in-expectation analysis of the eigeinpairs of our studied operators. We perform experiments with real and synthetic data that demonstrate the benefits of hypergraph methods over graph-based ones when interactions of different sizes carry different information about cluster structure.

Joint work with Nicole Eikmeier (Grinnell) and Jamie Haddock (Harvey Mudd).