Georgia Institute of Technology College of Sciences

Each point x in Gr(r, n) corresponds to an r × n matrix Ax which gives rise to a matroid Mx on its columns. Gel’fand, Goresky, MacPherson, and Serganova showed that the sets {y ∈ Gr(r, n)|My = Mx} form a stratification of Gr(r, n) with many beautiful properties. However, results of Mnëv and Sturmfels show that these strata can be quite complicated, and in particular may have arbitrary singularities. We study the ideals Ix of matroid varieties, the Zariski closures of these strata. We construct several classes of examples based on theorems from projective geometry and describe how the Grassmann-Cayley algebra may be used to derive non-trivial elements of Ix geometrically when the combinatorics of the matroid is sufficiently rich.

Gaussian processes (GPs) are widely employed as versatile modeling and predictive tools in spatial statistics, functional data analysis, computer modeling and diverse applications of machine learning. They have been widely studied over Euclidean spaces, where they are specified using covariance functions or covariograms for modelling complex dependencies. There is a growing literature on GPs over Riemannian manifolds in order to develop richer and more flexible inferential frameworks. While GPs have been extensively studied for asymptotic inference on Euclidean spaces using positive definite covariograms, such results are relatively sparse on Riemannian manifolds. We undertake analogous developments for GPs constructed over compact Riemannian manifolds. Building upon the recently introduced Matérn covariograms on a compact Riemannian manifold, we employ formal notions and conditions for the equivalence of two Matérn Gaussian random measures on compact manifolds to derive the microergodic parameters and formally establish the consistency of their maximum likelihood estimates as well as asymptotic optimality of the best linear unbiased predictor.

Cohomology groups of moduli spaces of curves are fruitfully studied from several mathematical perspectives, including geometric group theory, stably homotopy theory, and quantum algebra.  Algebraic geometry endows these cohomology groups with additional structures (Hodge structures and Galois representations), and the Langlands program makes striking predictions about which such structures can appear.  In this talk, I will present recent results inspired by, and in some cases surpassing, such predictions.  These include the vanishing of odd cohomology on moduli spaces of stable curves in degrees less than 11, generators and relations for H^11, and new constructions of unstable cohomology on M_g.  

Based on joint work with Jonas Bergström and Carel Faber; with Sam Canning and Hannah Larson; with Melody Chan and Søren Galatius; and with Thomas Willwacher. 

We introduce a large family of one-dimensional full branch maps which generalize the classical “intermittency maps” by admitting two neutral fixed points and possibly also critical points and/or singularities. We study the statistical properties of these maps for various parameter values, including the existence (and non-existence) of physical measures, and their properties such as decay of correlations and limit theorems. In particular we describe a new mechanism for relatively persistent non-statistical chaotic dynamics. This is joint work with Douglas Coates and Muhammad Mubarak.