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Today, Friday, January 19, 2018

Colloquium  Questions or comments?

Posted January 12, 2018
Last modified January 17, 2018

3:30 pm - 4:20 pm Lockett 241

Jake Fillman, Virginia Tech
Spectral properties of quasicrystals

Abstract: Discovered in the early 1980s by Dan Shechtman, quasicrystals are solids that simultaneously exhibit aperiodicity (a lack of translation symmetries) and long-range order (quantified by the presence of Bragg peaks in their diffraction patterns). We will discuss almost-periodic Schroedinger operators, which supply a rich family of operator-theoretic models of quasicrystals. Our discussion will center around the spectral properties of the underlying operator and transport properties of the associated quantum dynamics. We will discuss how some of our results may be viewed as an inverse spectral theoretic obstruction to solving Deift's conjecture for the KdV equation with current technology. We will conclude with a discussion of results in higher dimensions that are motivated by the Bethe--Sommerfeld conjecture.

Monday, January 22, 2018

Colloquium  Questions or comments?

Posted January 13, 2018
Last modified January 19, 2018

3:30 pm - 4:20 pm Lockett 241

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)

Tuesday, January 23, 2018

Algebra and Number Theory Seminar  Questions or comments?

Posted November 30, 2017

3:10 pm - 4:00 pm

William Casper, Louisiana State University
TBA


Computational Mathematics Seminar  

Posted January 16, 2018

3:30 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.

Wednesday, January 24, 2018

Colloquium  Questions or comments?

Posted January 12, 2018
Last modified January 17, 2018

3:30 pm - 4:20 pm Lockett 241

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.