Geometry and Topology Seminar 2019-2020
|September 21||Owen Sizemore (Wisconsin)||local|
|September 28||Mireille Boutin (Purdue)||Mari Beffa|
|October 5||Ben Schmidt (Michigan State)||Dymarz|
|October 12||Ian Biringer (Boston College)||Dymarz|
|October 19||Peng Gao (Simons Center for Geometry and Physics)||Wang|
|October 26||Jo Nelson (Wisconsin)||local|
|November 2||Jennifer Taback (Bowdoin)||Dymarz|
|November 9||Jenny Wilson (Chicago)||Ellenberg|
|November 16||Jonah Gaster (UIC)||Kent|
|November 30||Shinpei Baba (Caltech)||Kent|
|December 7||Kathryn Mann (Chicago)||Kent|
Owen Sizemore (Wisconsin)
Operator Algebra Techniques in Measureable Group Theory
Measurable group theory is the study of groups via their actions on measure spaces. While the classification for amenable groups was essentially complete by the early 1980's, progress for nonamenable groups has been slow to emerge. The last 15 years has seen a surge in the classification of ergodic actions of nonamenable groups, with methods coming from diverse areas. We will survey these new results, as well as, give an introduction to the operator algebra techniques that have been used.
Mireille Boutin (Purdue)
The Pascal Triangle of a discrete Image: definition, properties, and application to object segmentation
We define the Pascal Triangle of a discrete (gray scale) image as a pyramidal ar- rangement of complex-valued moments and we explore its geometric significance. In particular, we show that the entries of row k of this triangle correspond to the Fourier series coefficients of the moment of order k of the Radon transform of the image. Group actions on the plane can be naturally prolonged onto the entries of the Pascal Triangle. We study the induced action of some common group actions, such as translation, rotations, and reflections, and we propose simple tests for equivalence and self- equivalence for these group actions. The motivating application of this work is the problem of recognizing ”shapes” on images, for example characters, digits or simple graphics. Application to the MERGE project, in which we developed a fast method for segmenting hazardous material signs on a cellular phone, will be also discussed.
This is joint work with my graduate students Shanshan Huang and Andrew Haddad.
Ben Schmidt (Michigan State)
Three manifolds of constant vector curvature.
A Riemannian manifold M is said to have extremal curvature K if all sectional curvatures are bounded above by K or if all sectional curvatures are bounded below by K. A manifold with extremal curvature K has constant vector curvature K if every tangent vector to M belongs to a tangent plane of curvature K. For surfaces, having constant vector curvature is equivalent to having constant curvature. In dimension three, the eight Thurston geometries all have constant vector curvature. In this talk, I will discuss the classification of closed three manifolds with constant vector curvature. Based on joint work with Jon Wolfson.
Ian Biringer (Boston College)
Growth of Betti numbers and a probabilistic take on Gromov Hausdorff convergence
We will describe an asymptotic relationship between the volume and the Betti numbers of certain locally symmetric spaces. The proof uses an exciting new tool: a synthesis of Gromov-Hausdorff convergence of Riemannian manifolds and Benjamini-Schramm convergence from graph theory.
Peng Gao (Simons Center for Geometry and Physics)
string theory partition functions and geodesic spectrum
String theory partition functions often have nice modular properties, which is well understood within the context of representation theory of (supersymmetric extensions) of Virasoro algebra.
However, many questions of physical importance are preferrably addressed when string theory is formulated in terms of non-linear sigma model on a Riemann surface with a Riemannian manifold as target space. Traditionally, physicists have studied such sigma models within the realm of perturbation theory, overlooking a large class of very natural critical points of the path integral, namely, closed geodesics on the target space Riemannian manifold. We propose how to take into account the effect of these critical points on the path integral, and initiate its study on Ricci flat targe spaces, such as the K3 surface.
Jo Nelson (Wisconsin)
Cylindrical contact homology as a well-defined homology theory? Part I
In this talk I will define all the concepts in the title, starting with what a contact manifold is. I will also explain how the heuristic arguments sketched in the literature since 1999 fail to define a homology theory and provide a foundation for a well-defined cylindrical contact homology, while still providing an invariant of the contact structure. A later talk will provide us with a large class of examples under which one can compute a well-defined version of cylindrical contact homology via a new approach the speaker developed for her thesis that is distinct and completely independent of previous specialized attempts.
Jennifer Taback (Bowdoin)
Jenny Wilson (Chicago)
Jonah Gaster (UIC)
Shinpei Baba (Caltech)
Kathryn Mann (Chicago)
|March 22||Michelle Lee (Michigan)||Kent|
Michelle Lee (Michigan)