**Seminars and Colloquiums**

for the week of March 20, 2017

for the week of March 20, 2017

*SPEAKER:*

Jifeng Shen, Yale University, Monday

Eddie Tu, University of Tennessee, Tuesday

Michael Hartz, Washington University, Wednesday

Jiguang Shen, University of Minnesota, Wednesday

Pablo Stinga, Iowa State University, Thursday

Wojbor Woyczynski, Case Western Reserve University, Friday

*TEA TIME -
3:00 pm – 3:30 pm
Monday - Wednesday: Ayres 401
*

*Hosted by: Hannah Thompson*

**Monday, March 20th**

ALGEBRA SEMINAR

TITLE: Break divisors and compactified Jacobians

SPEAKER: Jifeng Shen, Yale University

TIME: 3:35p – 4:25p

ROOM: 113 Ayres

Let X be a genus g strictly semistable family of curves over C[[t]]. We show, using break divisors introduced by Mikhalkin and Zharkov, that all degree g Simpson compactified Jacobians of X are identical. As a consequence, we resolve a conjecture of Payne, and show that the unique degree g Simpson compactified Jacobian can be constructed from the break divisor decomposition introduced by An-Baker-Kuperberg-Shokrieh, using Mumford's non-Archimedean uniformization theory.

**Tuesday, March 21st**

STOCHASTICS/ PROBABILITY SEMINAR

TITLE: Special dependence structures in Markov processes: Part I

SPEAKER: Eddie Tu, University of Tennessee

TIME: 2:10p - 3:25p

ROOM: 113 Ayres

In probability and statistics, there are various characterizations to describe the dependence in a multivariate distribution, beyond the basic notion of positive and negative correlation. These other characterizations, which include “association,” “supermodular dependence,” and “orthant dependence,” can describe the strength of the dependence between components in a random vector, which is often more revealing and useful. In Part I, we will discuss these different forms of dependence for multivariate distributions, their importance in applications, and illustrate such notions on examples of distributions, particularly infinitely divisible distributions. In Part II, we will discuss how these different forms of dependence can help us study the evolution of stochastic processes. We will investigate and characterize these special kinds of dependence for Levy and Feller Markov processes.

**Wednesday, March 22nd**

ANALYSIS SEMINAR

TITLE: A multiplier functional calculus

SPEAKER: Michael Hartz, Washington University

TIME: 2:30p – 3:20p

ROOM: 113 Ayres

A classical result of Sz.-Nagy and Foias shows that every contraction $T$ on a Hilbert space without unitary summand admits an $H^\infty$-functional calculus, that is, one can make sense of $f(T)$ for every bounded analytic function $f$ in the unit disc. I will talk about a generalization of this result, which applies to tuples of commuting operators and multiplier algebras of a large class of Hilbert function spaces on the unit ball. In particular, this extends a recent theorem of Clou^atre and Davidson for commuting row contractions. This is joint work with Kelly Bickel and John McCarthy.

COMPUTATIONAL and APPLIED MATHEMATICS (CAM) SEMINAR

TITLE: An automated algorithm for stabilizing HDG method for nonlinear elasticity

SPEAKER: Jiguang Shen, University of Minnesota

TIME: 3:35p – 4:35p

ROOM: 113 Ayres

We address the stability issue of the hybridizable discontinuous Galerkin (HDG) method for the nonlinear elasticity raised in the paper by Kabaria et.al. (2015) such that if the inter-element jumps are not properly penalized, the method may lose the convergence to the exact solution. We show that the stabilization function which controls the magnitude of inter-element jumps must be at least C/h in the subdomain where the elastic moduli is locally indefinite to ensure the so-called linearized stability. In particular, we design an automated algorithm to explicitly compute the analytic bounds on C over the domain and update them as the deformation evolves. In the stability analysis, we extend the idea proposed by Celiker et.al.(2008) for DG scheme and improve the results by proposing a new continuous reconstruction which fits in the structure of the HDG formulation. The novel features of this reconstruction enable us to get rid of solving an global eigenvalue problem within each nonlinear iteration and to avoid modifying the bilinear form of the scheme, which are both required in the approach by Celiker et.al.(2008). Numerical experiments demonstrates that our automated algorithm is necessary and effective to ensure the convergence of the HDG scheme to the exact solution especially in the case when large compressions are presented. Finally, we recover the super-convergence of the HDG scheme over the entire domain by applying a variable degree HDG formulation which is numerically shown to be beneficial for the approximations of both the deformation mapping and the stresses. For the practically interesting linear element, third order convergence in deformation mapping and second order in stresses approximations are obtained.

Joint work with Bernardo Cockburn.

**Thursday, March 23rd**

DIFFERENTIAL EQUATIONS SEMINAR

TITLE: Extension problem for fractional heat equations and applications.

SPEAKER: Pablo Stinga, Iowa State University

TIME: 2:00p – 3:00p

ROOM: 113 Ayres

We report on recent advances in the regularity theory for fractional nonlocal space-time equations like the master equation considered by L. A. Caffarelli and L. Silvestre. We prove a parabolic extension problem that characterizes the fractional powers of the heat operator. In particular, we recover the extension problem for the Marchaud fractional derivative previously proved in a joint work with A. Bernardis, F. J. Martín-Reyes and J. L. Torrea. As a byproduct of our ideas we show novel mixed-norm Sobolev estimates with weights for some basic parabolic equations. The latter results not only recover previous works by N. V. Krylov but also extend them to their fullest extent to include weak type estimates and a.e. convergence of principal values formulas for derivatives. The results presented are based in joint works with J. L. Torrea.

**Friday, March 24th**

COLLOQUIUM

TITLE: Multiscale conservation laws driven by L\'evy stable and Linnik diffusions: asymptotics, explicit representations, shock creation, preservation and dissolution.

SPEAKER: Wojbor Woyczynski, Case Western Reserve University

TIME: 3:30p – 4:30p

ROOM: 405 Ayres

Asymptotic behavior of supercritical nonlinear multifractal conservation laws with integrable initial conditions is dictated by the linearized case. Thus obtaining explicit solutions of the latter is of interest. For $ \alpha< 1 $, conservation laws driven by L\'evy $\alpha$-stable diffusions exhibit shocks for bounded, odd, and convex on $R^+$, initial data. For L\'evy $\alpha$-Linnik diffusions, $0<\alpha\le 2$, the local behavior is strikingly different. The relevant conservation laws display shocks that do not dissipate over time while those for $\alpha$-stable diffusion ($0<\alpha\le 1$) do.

*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 *

**Past notices:**

3/13/17 - Spring Break

Winter Break