Posted January 13, 2018
Last modified January 19, 2018
Galyna Dobrovolska, Columbia University
A geometric Fourier transform, noncommutative resolutions, and Hilbert schemes
Abstract: I will start by defining and computing an example of a geometric Fourier transform for constructible functions, and more generally for constructible sheaves. Next I will explain how geometric representation theory can be used to study categories of modules over Lie algebras and more general algebras which quantize symplectic resolutions. Lastly I will apply the above techniques in the case of the Hilbert scheme of points in the plane. (This talk is based on a joint work in progress with R. Bezrukavnikov and I. Loseu and on my Ph.D. thesis)
Posted November 30, 2017
Last modified January 22, 2018
William Casper, Louisiana State University
Algebras of Differential Operators and Algebraic Geometry with Applications
Posted January 16, 20183:30 pm - 4:30 pm 1034 Digital Media Center
Amanda Diegel, Louisiana State University
The Cahn-Hilliard Equation, a Robust Solver, and a Phase Field Model for Liquid Crystal Droplets
Abstract: We begin with an introduction to the Cahn-Hilliard equation and some motivations for the use of phase field models. We will then go on to describe a first order finite element method for the Cahn-Hilliard equation and the development of a robust solver for that method. The key ingredient of the solver is a preconditioned minimal residual algorithm (with a multigrid preconditioner) whose performance is independent of the spatial mesh size and the time step size for a given interfacial width parameter. In the second part of the talk, we present a novel finite element method for a phase field model of nematic liquid crystal droplets. The model considers a free energy comprised of three components: the Ericksen''s energy for liquid crystals, the Cahn-Hilliard energy for phase separation, and an anisotropic weak anchoring energy that enforces a boundary condition along the interface between the droplet and surrounding substance. We present the key properties of the finite element method for this model including energy stability and convergence and conclude with a few numerical experiments.
Posted January 12, 2018
Last modified January 17, 2018
Christine Lee, University of Texas at Austin
Understanding quantum link invariants via surfaces in 3-manifolds
Abstract: Quantum link invariants lie at the intersection of hyperbolic geometry, 3-dimensional manifolds, quantum physics, and representation theory, where a central goal is to understand its connection to other invariants of links and 3-manifolds. In this talk, we will introduce the colored Jones polynomial, an important example of quantum link invariants. We will discuss how studying properly embedded surfaces in a 3-manifold provides insight into the topological and geometric content of the polynomial. In particular, we will describe how relating the definition of the polynomial to surfaces in the complement of a link shows that it determines boundary slopes and bounds the hyperbolic volume of many links, and we will explore the implication of this approach on these classical invariants.
Posted January 19, 2018
Last modified January 21, 2018
Shawn X. Cui, Stanford, Institute for Theoretical Physics
Four Dimensional Topological Quantum Field Theories
Abstract: We give an introduction to topological quantum field theories (TQFTs), which have wide applications in low dimensional topology, representation theory, and topological quantum computing. In particular, TQFTs provide invariants of smooth manifolds. We give an explicit construction of a family of four dimensional TQFTs. The input to the construction is a class of tensor categories called $G$-crossed braided fusion categories where $G$ is any finite group. We show that our TQFTs generalize most known examples such as Yetter's TQFT and the Crane-Yetter TQFT. It remains to check if the resulting invariant of 4-manifolds is sensitive to smooth structures. It is expected that the most general four dimensional TQFTs should arise from spherical fusion 2-categories, the proper definition of which has not been universally agreed upon. Indeed, we prove that a $G$-crossed braided fusion category corresponds to a 2-category which does not satisfy the criteria to be a spherical fusion 2-category as defined by Mackaay. Thus the question of what axioms properly define a spherical fusion 2-category is open.