Topology and Geometry Seminar

Title: A (gentle) introduction to Poisson geometry, Part 1. 

Abstract: I will give an introduction to Poisson geometry, starting with its role in the dynamics of physical systems and its relationship to symplectic geometry. Through the use of illustrative examples, I will highlight some of the basic features of Poisson manifolds, such as their symplectic leaves. 

Title: A (gentle) introduction to Poisson geometry, Part 2. 

Abstract:  This is the second part of an introduction to Poisson geometry.  In this part, I will discuss the geometry of Poisson manifolds, such as their symplectic leaves, through the use of illustrative examples. I will also briefly discuss the quantization of Poisson manifolds. 

Title: Introduction to deformation quantization

Abstract: A deformation quantization of a manifold is a noncommutative deformation of its algebra of functions; the idea originated in physics, as a way of relating the classical and quantum descriptions of mechanical systems.  At leading order in the deformation parameter, a deformation quantization gives rise to a Poisson structure on the manifold.  A deep theorem of Kontsevich states that this gives an equivalence between Poisson structures and noncommutative deformations; it thus provides a direct link between geometry and noncommutative algebra.  In fact, he gives an explicit formula for the quantization of any Poisson bracket, via integrals of differential forms associated to graphs (a Feynman expansion).  I will give an introduction to deformation quantization, with several examples, leading to a sketch of Kontsevich's proof.

Title:  Dehn Surgery: From Torus Knots to 3-manifold.

Abstract: I will introduce Dehn surgery via torus knots and links, show how 3-manifolds arise from surgeries, note basic surgery equivalences, and sketch homology computations of the resulting manifolds. I will also include surgeries on torus knots that yield lens spaces.

Note: This talk is being held on a Thursday, rather than Tuesday as usual. 

Title: Geometric Approaches for Functional Data

Abstract: Functional data such as time series and images are ubiquitous in applications and are equipped with natural concatenation operations. For machine learning applications, it is often helpful to build structured representations of such data which preserve the underlying algebraic structure, and satisfy universality (allows us to approximate functions) and characteristicness (allows us to characterize probability measures). By viewing such data as paths and surfaces, we discuss how (higher) parallel transport can be used to provide such representations, and discuss their universal and characteristic properties. Based on joint work with Harald Oberhauser (Oxford).

Title: Functor calculus and vector bundle enumerations.

Abstract:  Pick your favorite manifold M and a positive integer r: how many rank r (topological) complex vector bundles are there over M up to isomorphism? While the question is accessible via K-theory in the stable range, unstably such bundles become much harder to compute and to detect.

In this talk, we will demonstrate how the orthogonal/unitary calculus of Weiss — a version of functor calculi — can be applied to study unstable topological vector bundles. We will present several counting results for complex vector bundles over CPn in the metastable range, and, time permitting, introduce an equivariant version of the theory along with some potential applications in equivariant geometry. This talk contains joint work with Hood Chatham and Morgan Opie, and with Prasit Bhattacharya.

Title: Orientation orders and chromatic defects.

Abstract: A ring spectrum (a.k.a. a multiplicative cohomology theory) E is complex-orientable if every complex vector bundle admits a Thom class in E. While many familiar spectra are complex-orientable, notable examples fail to be so — such as real K-theory and topological modular forms. To measure this failure precisely, two competing ideas have been proposed: the complex orientation order and the chromatic defect. A natural question is whether the two notions are related. In this talk, we will present examples that suggest a negative answer. This is joint work with Prasit Bhattacharya and Christian Carrick.

Note: This talk is being held on a Thursday, rather than Tuesday as usual. 

Title: An introduction to Toric Topology

Abstract: In this talk, I will present a crash course in Toric Topology. We will start by discussing basic combinatorial structures (such as polytopes, fans, and simplicial complexes) and their role in studying torus actions. After some examples we will move on to introducing various algebraic objects that arise from the combinatorics of these spaces and pave the way to start discussing polyhedral products if time permits.

Note: This talk is being held on a Thursday, rather than Tuesday as usual. 

Title: An introduction to Toric Topology, part 2

Abstract: We continue last week’s talk by introducing two more combinatorial structures: simplicial and cubical complexes, which we’ll introduce and present some examples and constructions that arise from them. These complexes let us construct a family of Toric spaces we call moment-angle complexes. To study the homotopical features of these spaces we’ll introduce two algebraic objects which arise from the underlying simplicial complex: The Tor algebra and the Stanley-Reisner ring. Both encode the homological features of a MAC; we’ll present this equivalence, a proof sketch and some examples. If time permits, I’ll introduce polyhedral products and talk about some of the current work being done, such as the de-suspension problem and the triviality of the product.

Note: This talk is being held on a Thursday, rather than Tuesday as usual. 

Title: Lie 2-algebras and higher quantization

Abstract: Bundle gerbes on a manifold M provide geometric realizations of degree 3 cohomology classes of M. The space of infinitesimal symmetries of a bundle gerbe naturally carries the structure of a Lie 2-algebra, a deformation of the notion of Lie algebra where the Jacobi identity only holds ‘up to homotopy.’ This Lie 2-algebra of symmetries is related to other Lie 2-algebras associated to a closed differential 3-form, when the 3-form encodes the ‘higher curvature’ of the bundle gerbe—namely, the Poisson Lie 2-algebra of observables, and the Lie 2-algebra of sections of an exact Courant algebroid. This talk reviews the notion of bundle gerbe, and discusses the relations between the aforementioned Lie 2-algebras.

Note: This talk is being held on a Wednesday.

Title: Steenrod algebra

Abstract: The Steenrod algebra acts on ordinary cohomology with coefficients in a finite field, thereby refining the invariant that is cohomology. In this talk, I discuss its formal properties and applications.