**Seminars and Colloquiums**

for the week of February 15, 2016

for the week of February 15, 2016

*SPEAKER:*

Eleanor Abernethy, UTK, Monday

Kei Kobayashi, UTK, Tuesday

Abner Salgado, UTK, Wednesday

Brittany Stephenson, UTK, Thursday

Mark Bly, Grace McClurkin, Darrin Weber, UTK, Thursday

Dr. Junping Wang, NSFK, Friday

Stefan Richter, UTK, Friday

*TEA TIME
3:00 – 3:30 pm, Ayres 401
Monday, Tuesday, & Wednesday
Hosted by Mustafa and Mahir, UTK *

**Monday, February 15th**

GEOMETRY AND TOPOLOGY SEMINAR - cancelled

TITLE: Critical Spectra

SPEAKER: Eleanor Abernethy, UTK

TIME: 2:30pm – 3:20pm

ROOM: Ayres 113

Spectra are a classical way to understand the geometry of compact Riemannian manifolds in Differential Geometry. Two well-known spectra are the Laplace spectrum and the length spectrum. A relatively new spectrum is the covering spectrum, developed by Sormani and Wei (2003), which utilizes delta-covers of a compact geodesic space and singles out values of ?? where the covering spaces ??_?? ? ??_??? for all ???>??. More recently, Plaut and Wilkins (2012) developed the homotopy critical spectrum which arises from a discrete analog of the fundamental group construction for a compact metric space. It is already known that the covering and homotopy critical spectra are essentially the same on compact geodesic spaces. However, the homotopy critical spectrum is defined in the more general setting of metric spaces. de Smit, Garnet and Sutton (2010) extended the notion of the covering spectrum to any metric space. I will present results of comparing the definitions of the homotopy critical spectrum and the de Smit/Garnet/Sutton formulation of the covering spectrum on general metric spaces. I will also outline a strategy for extending this classification to Uniform Topological Spaces in which this will be the Entourage Critical Spectrum.

**Tuesday, February 16th **

STOCHASTICS

TITLE: Small ball probabilities for a class of time-changed self-similar processes

SPEAKER: Kei Kobayashi, UTK

TIME: 2:10pm – 3:25pm

ROOM: Ayres 114

A standard Brownian motion composed with the so-called "inverse stable subordinator" is used to model subdiffusion, where particles spread more slowly than the classical Brownian particles. This new stochastic process is significantly different from the Brownian motion; indeed, it is neither Markovian nor Gaussian and has transition probabilities satisfying a fractional-order heat equation.

Now, consider the probability that the process does not exit a small ball during the unit time interval. As the radius of the ball approaches zero, the probability converges to zero, but what is the rate of convergence? A special case of our results (originally proved by Nane in 2009) states that the probability decays polynomially. This is interesting since, for the Brownian motion, the corresponding probability decays exponentially. We extended this result to a large class of self-similar processes composed with general inverse subordinators.

**Wednesday, February 17th**

COMPUTATIONAL AND APPLIED MATHEMATICS SEMINAR

TITLE: Adaptive Finite Element Methods for an Optimal Control Problem Involving Dirac Measures

SPEAKER: Abner Salgado, UTK

TIME: 3:35pm – 4:35pm

ROOM: Ayres 114

The purpose of this work is the design and analysis of a reliable and efficient a posteriori error estimator for the so-called pointwise tracking optimal control problem. This linear-quadratic optimal control problem entails the minimization of a cost functional that involves point evaluations of the state, thus leading to an adjoint problem with Dirac measures on the right hand side; control constraints are also considered. The proposed error estimator relies on a posteriori error estimates in the maximum norm for the state and in Muckenhoupt weighted Sobolev spaces for the adjoint state. We present an analysis that is valid for two and three-dimensional domains. We conclude by presenting several numerical experiments which reveal the competitive performance of adaptive methods based on the devised error estimator.

**Thursday, February 18th**

DIFFERENTIAL EQUATIONS SEMINAR

TITLE: An Epidemiological Model of Clostridium difficile Transmission with Optimal Control of Vaccination

SPEAKER: Brittany Stephenson, UTK

TIME: 2:10pm – 3:25pm

ROOM: Ayres 113

A spore-forming, gram-negative bacterium, Clostridium difficile can survive for extended periods of time, even years, on inorganic surfaces. A striking increase in the number of cases of C. difficile infection (CDI) among hospitals has highlighted the need to better understand CDI in order to prevent its spread. A compartmental model of nosocomial C. difficile transmission is modified and updated to include vaccination. I then apply optimal control on the vaccination rate to determine the time-varying optimal rate that will both minimize the disease prevalence and spread in the hospital population while also minimizing the cost, both in terms of time and money, associated with vaccination.

GRADUATE STUDENT SEMINAR

TITLE: GS Seminar Spotlight on Algebra and Topology

SPEAKER: Mark Bly, Grace McClurkin, Darrin Weber, UTK

TIME: 3:40pm – 4:30pm

ROOM: Ayres G004

We start off the GS Seminar Spotlight Series by shining some light on Algebra and Topology at UTK. We have an outstanding panel who will share their experiences picking out classes; taking prelims; choosing an advisor; and much more! Plus, they will field any questions that you may have about moving into one of these areas during your time as a graduate student at UTK.

** Friday, February 19th**

COMPUTATIONAL AND APPLIED MATHEMATICS SEMINAR

TITLE: A Primal-Dual Weak Galerkin Finite Element Method for PDEs

SPEAKER: Dr. Junping Wang, NSFK

TIME: 10:30am – 11:30am

ROOM: TBA

In the talk, the speaker shall first introduce the weak Galerkin (WG) finite element method for partial differential equations. Weak Galerkin is a finite element method for PDEs where the differential operators (e.g., gradient, divergence, curl, Laplacian etc.) in the weak forms are approximated by discrete generalized distributions. The WG discretization procedure often involves the solution of inexpensive problems defined locally on each element. The solution from the local problems can be regarded as a reconstruction of the corresponding differential operators. The fundamental difference between the weak Galerkin finite element method and other existing methods is the use of weak functions and weak derivatives (i.e., locally reconstructed differential operators) in the design of numerical schemes based on existing weak forms for the underlying PDEs. Weak Galerkin is a natural extension of the classical Galerkin finite element method with advantages in many aspects. Due to its great structural flexibility, the weak Galerkin finite element method is well suited to most partial differential equations by providing the needed stability and accuracy in approximation.

The talk will start with the second order elliptic equation, for which WG shall be applied and explained in detail. In particular, the concept of weak gradient will be introduced and discussed for its role in the design of weak Galerkin finite element schemes. The speaker will then introduce a general notion of weak differential operators, such as weak Hessian, weak divergence, and weak curl etc. These weak differential operators shall serve as building blocks for WG finite element methods for other class of partial differential equations, such as the Stokes equation, the biharmonic equation, the Maxwell equations in electron magnetics theory, div-curl systems, and PDEs in non-divergence form (such as the Fokker-Planck equation). In particular, the speaker will introduce a primal-dual formulation for second order elliptic PDEs in non-divergence form. Numerical results and error estimates shall be discussed. The talk should be accessible to graduate students with adequate training in computational methods.

ANALYSIS SEMINAR

TITLE: Functions as quotients of multipliers, IV

SPEAKER: Stefan Richter, UTK

TIME: 2:30pm - 3:20pm

ROOM: Ayres 121

*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:**