Un of Tenn | Math Department

Seminars and Colloquiums
for the week of November 12, 2012

Speakers:

Prof. Jan Rosinski, Monday
Prof. Colin Adams, Williams College, Tuesday
Dr. Guannah Zhang, Householder Fellow, ORNL, Wednesday
Prof. Carl Sundberg, Wednesday
Mr. Josh Brewer, Thursday
Prof. Almut Burchard, University of Toronto, Thursday
Prof. Almut Burchard, University of Toronto, Friday

**Tea Time this week by the "News Guys from 109", Monday through Wednesday, 3:00 pm. Everyone welcome!**

Monday, November 12

PROBABILITY SEMINAR
TIME: 3:35 - 4:25 p.m.
ROOM: Ayres Hall 112
SPEAKER: Prof. Jan Rosinski
TITLE: Wiener chaos, Malliavin calculus, and the Central Limit Theorem, Part 1.
ABSTRACT: Introduction to Wiener chaos will be given. Fundamental notions of the Malliavin calculus, gradient and divergence operators on the Wiener chaos space, will be defined and discussed. Using Malliavin calculus, a surprisingly simple characterization of the asymptotic independence of a Wiener chaos will be obtained. As a consequence, a multidimensional version of the celebrated Central Limit Theorem for Wiener chaos, the so called fourth moment theorem of Nualart and Peccati (2005), will be derived and new bounds on the rate of convergence will be shown.

Tuesday, November 13

TOPOLOGY SEMINAR
TIME: 3:35 p.m.
ROOM: Ayres 405
SPEAKER: Prof. Colin Adams, Williams College
TITLE: Turning Knots into Flowers
ABSTRACT: Knots have traditionally been investigated by considering projections with crossings where two strands of the knot cross one another. Here, we consider multi-crossings (or n-crossings) where n strands of the knot cross at a single point. We show that for each integer n greater than or equal to 2, every knot has a projection made up entirely of n-crossings, and therefore a minimal n-crossing number c_n(K). We investigate what is known about c_n(K) and then show that for every knot there is an n such that c_n(K) = 1. In fact, every knot has a projection with a single multi-crossing that looks like a daisy. We will consider the implications of this.

Wednesday, November 14

APPLIED MATH SEMINAR
TIME: 3:35 - 4:25 p.m.
ROOM: Ayres Hall 110
SPEAKER: Guannah Zhang, Householder Fellow, ORNL
TITLE: An Adaptive Sparse-grid High-order Stochastic Collocation Method
for Bayesian Inference with Computationally Expensive Simulations
ABSTRACT: Bayesian inference has become vital to stochastic parameter identification for complex physical systems, but its application has been hindered due to the computational cost associated with numerous model executions needed for exploring the posterior probability density function (PPDF) of model parameters. This is particularly the case when the PPDF is estimated using Markov Chain Monte Carlo (MCMC) sampling. In this study, we develop a new approach that improves computational efficiency of Bayesian inference by constructing a surrogate system based on an adaptive sparse-grid high-order stochastic collocation (aSG-hSC) method. Unlike previous works using first-order hierarchical basis, we utilize a compactly supported higher-order hierarchical basis to construct the surrogate system, resulting in a significant reduction in the number of computational simulations required. In addition, we use hierarchical surplus as an error indicator to determine adaptive sparse grids. This allows local refinement in the uncertain domain and/or anisotropic detection with respect to the random model parameters, which further improves computational efficiency. Finally, we incorporate a global optimization technique and propose an iterative algorithm for building the surrogate system for the PPDF with multiple significant modes. Once the surrogate system is determined, the PPDF can be evaluated by sampling the surrogate system directly with very little computational cost. Several numerical examples demonstrate that the aSG-hSC is an effective and efficient tool for Bayesian inference in parameter identification in comparison with conventional MCMC simulations. The computational efficiency is expected to be more beneficial to more computational expensive physical problems.

ANALYSIS SEMINAR
TIME: 3:35 - 4:25 p.m.
ROOM: Ayres Hall 112
SPEAKER: Prof. Carl Sundberg
TITLE: More on the Transitive Algebra Problem

Thursday, November 15

M.S. THESIS DEFENSE
Mr. Joshua Brewer
TIME: 11:30 a.m.
ROOM: Perkins 61
TITLE: On the spherical symmetry of perfect-fluid stellar models in General Relativity
COMMITTEE: Professors Freire (chair), Simpson, Thistlethwaite.

JUNIOR COLLOQUIUM
TIME: 3:35-4:30 p.m.
ROOM: Ayres Hall 405
SPEAKER: Prof. Almut Burchard, University of Toronto
TITLE: Steiner symmetrization: Some new twists in an old story
ABSTRACT: Steiner symmetrization was invented in the 1830's as a tool for proving the isoperimetric inequality, that circles enclose the largest area among all planar curves of a given length. Since then, it has found many applications in Geometry, Mathematical Physics, and
Functional Analysis.

In this talk I will describe Steiner's original argument and two recent results. The first concerns infinite sequences of Steiner symmetrizations that fail to converge to the ball, but still converge "in shape". The second bounds the perimeter of a set in R^d that has been subjected to Steiner symmetrization along d linearly independent directions. Time permitting, I will mention some open problems.

Pizza available at 3:00 p.m.

Friday, November 16

COLLOQUIUM
TIME: 3:35 - 4:30 p.m.
ROOM: Ayres Hall 405
SPEAKER: Prof. Almut Burchard, University of Toronto
TITLE: : "Convergence to equilibrium for a thin-film equation on a horizontal cylinder"
ABSTRACT: In this talk, I will discuss recent work with Marina Chugunova and Ben Stephens on the long-term evolution of a thin liquid film on a horizontal cylinder, modeled by the degenerate parabolic equation

u_t + [u^3(u_xxx + u_x - sin x)]_x=0 .

For each given mass, we find that the unique steady state is a droplet that hangs from the bottom of the cylinder and meets the dry part of surface at zero contact angle. The steady state attracts all strong solutions, but the distance decays no faster than a power law.

One difficulty of thin-film equations is the lack of a uniqueness theorem for a good class of non-negative solutions. The key to our results is an energy (made up from the surface tension and gravitational potential energy) that decreases along solutions, and an entropy that increases at most linearly with time. Time permitting, I will discuss how the evolution can be viewed as the gradient flow of the energy, by realizing it on a space of measures with a distance function related to mass transportation. This promises to open new approaches to the long-standing uniqueness problem.

Refreshments will be available at 3:15 p.m.

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@math.utk.edu.

Past notices: