Tuesday, November 19, 2019
11/19/2019 - 4:00pm
Abstract: Informed by physiology, the wave-shape oscillatory model for biomedical time series asserts that cycles lie on or near a low-dimensional manifold.
Recovering this manifold provides insight into the dynamics of the underlying system. I present a manifold learning framework for biomedical time series analysis, and I provide a guarantee on the dynamical information which can be recovered using this framework. We will discuss several applications in the analysis of electrocardiograms and blood pressure monitoring. Time-permitting, we will examine further approaches to biomedical time series analysis which fuse traditional signal processing with modern mathematics.
11/19/2019 - 4:15pm
When geometric structures on surfaces are determined by the lengths of curves, it is natural to ask which curves’ lengths do we really need to know? It is a classical result of Fricke that a hyperbolic metric on a surface is determined by its marked simple length spectrum. More recently, Duchin–Leininger–Rafi proved that a flat metric induced by a unit-norm quadratic differential is also determined by its marked simple length spectrum. In this talk, I will describe a generalization of the notion of simple curves to that of q-simple curves, for any positive integer q, and show that the lengths of q-simple curves suffice to determine a non-positively curved Euclidean cone metric induced by a q-differential metric.
11/19/2019 - 4:15pm
: We describe a Breuil–Mezard type conjecture that relates structure of mod l fibers of deformation rings of the Galois group of a local field K (which has residue characteristic p different from l) to deformations of smooth mod l representations of GL_n(K). These conjectures are centered around the compatibility between inertial local Langlands and its mod l version. We sketch a proof of a weaker version of the conjecture, when K is a local function field, using the Taylor–Wiles–Kisin patching method. If time permits, we will report on some progress on variants for the conjecture for inner forms.
11/19/2019 - 5:00pm
Abstract: A matroid is a combinatorial object generalizing the idea of “linear independence” of a set in vector spaces. One example of a matroid is the edge set of a finite graph together with all forests in the graph. Such matroids can be used to prove the correctness of various greedy algorithms for finding spanning trees of a finite graph. Besides this instance of matroids, I want to share with you some invariants of general matroids and their incarnations in geometry that connect combinatorics, graph theory and algebra.