We relate the McMullen polynomial of a free-by-cyclic group to its Alexander polynomial. To do so, we introduce the notion of an orientable fully irreducible outer automorphism F and use it to characterize when the homological stretch factor of F is equal to its geometric stretch factor. This is joint work with Spencer Dowdall and Radhika Gupta.

Gaussian process regression is widely used in geostatistics, time-series analysis, and machine learning. It infers an unknown continuous function in a principled fashion from noisy measurements at $N$ scattered data points. The prior on the function is Gaussian, with covariance given by some user-chosen translationally invariant kernel. Yet $N$ has been limited to about $10^6$, even with modern low-rank methods. Focusing on low spatial dimension (1--3), we present a GP regression method using kernel approximation by an equispaced quadrature grid in the Fourier domain. This enables the iterative solution of a smaller Toeplitz linear system, exploiting both the FFT and the nonuniform FFT to give ${\mathcal O}(N)$ cost. The result is often one to two orders of magnitude faster than state of the art methods, and enables cheap massive-scale regressions. For example, for a 2D Matérn-3/2 kernel and $N = 10^9$ points, the posterior mean function is found to 3-digit accuracy in two minutes on a desktop.

Joint work with Philip Greengard (Columbia) and Manas Rachh (Flatiron Institute)

The study of "small" eigenvalues of the Laplacian on hyperbolic surfaces has a long history and has recently seen many developments. In this talk I will focus on the recent work (joint with Yunhui Wu and Haohao Zhang) on the higher spectral gaps, where we study the differences of consecutive eigenvalues up to $\lambda_{2g-2}$ for genus $g$ hyperbolic surfaces. We show that the supremum of such spectral gaps over the moduli space has infimum limit at least 1/4 as genus goes to infinity. The analysis relies on previous joint works with Richard Melrose on degenerating hyperbolic surfaces