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Seminars and Colloquiums
for the week of September 14, 2015


Andrew Geng (PhD student at University of Chicago), Monday
Anna Kazanova, UGA, Monday
Liguo Wang, UTK, Tuesday
Joseph Daws, UTK, Wednesday
Yanzhi Zhang, MS&T, Wednesday
Dat Cao, UTK, Thursday
Dustin Cartwright, UTK, Thursday
Brittany Stephenson & Lindsay Fox, UTK, Friday
David Horton, UTK, Friday
Yanzhao Cao, Auburn University, Friday

Monday September 14

TITLE: Classification and examples of 5-dimensional geometries
TIME: 2:30-3:20
ROOM: Ayres 114
SPEAKER: Andrew Geng (PhD student at University of Chicago)
ABSTRACT: Thurston's eight homogeneous geometries formed the building blocks of 3-manifolds in the Geometrization Conjecture. Filipkiewicz classified the 4-dimensional geometries in 1983, finding 18 and one countably infinite family. I have recently classified the 5-dimensional geometries. I will review what a geometry in the sense of Thurston is, survey related ideas, and outline the classification in 5 dimensions. Salient features, especially those first occurring in dimension 5, will be illustrated using particular geometries from the list. The classification touches a number of topics including foliations, fiber bundles, representations of compact Lie groups, Lie algebra cohomology, Galois theory in algebraic number fields, and conformal transformation groups. I hope to give some indication of how all of these come into play.

TITLE: Vector bundles on moduli space of stable curves with marked points
TIME: 3:35 – 4:25pm
ROOM: Ayres 114
SPEAKER: Anna Kazanova, UGA
ABSTRACT: Conformal block vector bundles are vector bundles on the moduli space of stable curves with marked points defined using certain Lie theoretic data. Over smooth curves, these vector bundles can be identified with generalized theta functions. In this talk we discuss extension of this identification over the stable curves. This is joint work with P. Belkale and A. Gibney

Tuesday September 15

TITLE:  Weak Convergence of Numerical Solutions of SDEs via Malliavin Calculus
TIME: 2:10pm -3:25
ROOM: Ayres 114
SPEAKER: Liguo Wang, UTK
ABSTRACT: The classical time discretization numerical methods for stochastic differential equations rely on the Ito-Taylor expansion, and hence they highly depend on the smoothness of coefficients of the equation and the test function. In this talk we will show that using Malliavin Calculus one can lower the smoothness requirements, and therefore justify the rate of convergence for the numerical schemes under weaker assumptions. We consider the Euler scheme as an example. Preliminary material from Malliavin Calculus needed to understand our method will be provided. 

Wednesday September 16

TIME: 9:05 - 9:55
ROOM: Ayres 405
SPEAKER: Joseph Daws, UTK
TITLE: At the first ever meeting of the Graduate Students Seminar, we will make introductions and briefly discus plans for the semester. Then, Joseph will discuss some of his research interests, including parallel computing problems. All mathematics graduate students are welcome and encouraged to attend.

TITLE: Eigenvalues and eigenfunctions the fractional Laplacian and their applications to the fractional quantum mechanics
TIME: 3:35 -4:35pm
ROOM: Ayres 112
SPEAKER: Yanzhi Zhang, MS&T
ABSTRACT: Recently, the fractional Laplacian has received great attention in modeling complex phenomena that involve long-range interactions. However, the nonlocality of the fractional Laplacian introduces considerable challenges in its analysis and simulations. So far, the eigenvalues and eigenfunctions of the fractional Laplacian still remain an open problem. In this talk, I present a novel numerical method to compute the eigenvalues and eigenfucntions of the fractional Laplacian. The errors of our method in discretizing the fractional Laplacian are evaluated both analytically and numerically. As an application, we study the fractional Schrodinger equation in an infinite potential well, one fundamental problem in the fractional quantum mechanics. We find that the eigenfunctions of the fractional Schrodinger equation in an infinite potential are different from those of the standard Schrodinger equation, and this provides insights to one open problem in this field.

Thursday September 17

TITLE: Pointwise estimates of solutions of Brezis-Kamin type to sublinear elliptic equations 
TIME: 2:10 -3:25Pm
ROOM: Ayres 114
ABSTRACT : We study equation ??p u = ?uq in Rn in the case 0 < q < p?1, where ?p u = ? · (?u|?u|p?2 ) is the p-Laplacian and ? ? 0 is an arbitrary locally integrable function or measure. Necessary and sufficient conditions for the existence of a certain class of solutions are given. We prove the existence of W 1,p -solutions and provide sharp global pointwise estimates of solutions in terms of Wolff potentials. The equations with the fractional Laplacian (??)? are treated as well. These results extend those of Brezis- Kamin to the case p j= 2. This is joint work with Igor Verbitsky.

TITLE: Better communication through linear algebra
TIME: 3:40 – 4:35pm
ROOM: Ayres 405
SPEAKER: Dustin Cartwright, UTK
ABSTRACT: From cell phones to laptops, wireless networking is everywhere around us. How does this work? To a physicist or engineer, the signal your cell phone sends is an electromagnetic wave, but to a mathematician like me, this wave is just a complex number. We can work with these numbers using linear algebra and, in some cases, it allows us to devise more efficient ways of getting information to and from our devices.

Friday September 18

TITLE: Traveling Waves in heterogeneous environments
TIME: 10:10 -11:00am
ROOM: Ayres 405
SPEAKER: Brittany Stephenson & Lindsay Fox, UTK

TITLE: Continuous Nowhere Differentiable Functions and the Loewner Equation
TIME: 2:30 -3:20
ROOM: BU 476
SPEAKER: David Horton, UTK
ABSTRACT: We'll begin with a brief overview of continuous nowhere differentiable functions and see some examples. Then continuing on from previous talks on the Loewner equation, we'll see some results when our driving term is a scaled version of the Weierstrass function.

TITLE: Backward SDE methods for nonlinear filtering problems
TIME: 3:30 -4:30pm
ROOM: Ayres 405
SPEAKER: Yanzhao Cao, Auburn University
ABSTRACT: A nonlinear filtering problem can be classified as an inverse problem of identifying the state of a system with a noise perturbation given noisy observations of the system. Well known numerical simulation methods include unscented Karlman filters and particle filters. In this talk, we attempt to construct efficient numerical methods using forward backward stochastic differential equations. The backward SDEs for the nonlinear filtering problems are the counter parts of Fokker-Planck equations for SDEs. In this talk we will present the derivation of such backward SDEs as well as the resulting high order numerical algorithms for nonlinear filtering problems.

If you are interested in giving or arranging a talk for one of our seminars or colloquiums, please review our calendar.

If you have questions, or a date you would like to confirm, please contact colloquium AT math DOT utk DOT edu



last updated: February 2016

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