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Seminars and Colloquiums
for the week of March 5, 2018


Yu Gu, Carnegie Mellon University
Logan Higginbotham, University of Tennessee
Carl Sundberg, University of Tennessee
Michael Frazier, University of Tennessee
Brian Wetton, University of British Columbia
Shelby Scott (EEB) and Lindsey Fox, University of Tennessee
Tadele Mengesha, University of Tennessee
Brian Wetton, University of British Columbia
Daniel Ardnt, University of Heidelberg, Germany
Brian Allen, United States Military Academy

Tea Time
3:00 pm – 3:30 pm
Monday, Tuesday, and Wednesday: Ayres 401
Hosted by: Kelly Buch

Tuesday, March 6

TITLE: The Anderson model in d=2
SPEAKER: Yu Gu, Carnegie Mellon University
TIME: 2:10 PM-3:20 PM
ROOM: Ayres 114
We consider the stochastic heat/Schrodinger equation with a spatial white noise in d=2, and discuss its connection to the renormalized self-intersection local time of planar Brownian motions.

TITLE: Coarse Constructions
SPEAKER: Logan Higginbotham, University of Tennessee
TIME: 4:00 PM
ROOM: Ayres 406
His committee consists of: Dydak (Chair), Brodskiy, Thistlethwaite, and Berry (EECS).

Title: Steifel-Whitney classes, spin structures and the Lichnerowiz formula
SPEAKER: Carl Sundberg, University of Tennessee
TIME: 5:05 PM-6:05 PM
ROOM: Ayres 113
Stiefel-Whitney classes are certain $\mathbb{Z}_2$-cohomology classes of a base space $B$ associated to a real vector bundle $E$ cover $B$. I will state their basis properties and go through the standard basic examples, then indicate how they provide criteria for the tangent bundle of a compact Riemannian manifold to admit a spin structure. If time permits, I will also sketch a proof of the Lichnerowicz formula for squares of Dirac operators on spin bundles.

Wednesday, March 7

TITLE: Weighted Littlewood-Paley Theory
SPEAKER: Michael Frazier, University of Tennessee
TIME: 2:30 PM-3:20 PM
ROOM: Ayres 112
We discuss how the Littlewood-Paley characterization of L^p that we stated last week is a consequence of Calderon-Zygmund theory with kernels that take values in the bounded linear operators from one Hilbert space to another. Then we discuss the theory of A_p weights, and the associated weighted Calderon-Zygmund and Littlewood-Paley theory.

TITLE: Time Stepping for Energy Gradient Flows
SPEAKER: Brian Wetton, University of British Columbia
TIME: 3:35 PM-4:35 PM
ROOM: Ayres 112
Time stepping methods are considered for materials science models that come from energy gradient flows that lead to the evolution of structure involving two or more phases (Allen-Cahn, Cahn Hilliard and higher order relatives) . The models are considered in periodic cells and standard Fourier spectral discretization in space is used. There is ``wisdomî in the literature which states that fully implicit time stepping time stepping is not viable for these systems. This wisdom should be questioned. Using asymptotic analysis and computational comparisons we show that fully implicit time stepping can be more efficient than so-called energy-stable schemes when accurately computing metastable dynamics. In this work, the resulting implicit systems are solved iteratively with a preconditioned conjugate gradient method. Other researchers have shown that nonlinear multi-grid methods can be also used effectively. Preliminary work on benchmark computations for these systems, joint with Steven Wise and others, is shown.

Thursday, March 8

TITLE: Optimization in two agent-based model examples
SPEAKER: Shelby Scott (EEB) and Lindsey Fox, University of Tennessee
TIME: 11:10 AM-12:00 PM
ROOM: Hesler 427

TITLE: Sobolev regularity estimates for solutions to spectral fractional elliptic equations
SPEAKER: Tadele Mengesha, University of Tennessee
TIME: 2:10 PM-3:10 PM
ROOM: Ayres 114
Global Calderon-Zygmund type estimates are obtained for solutions to fractional elliptic problems over smooth domain. Our approach is based on the ìextension problemî where the fractional elliptic operator is realized as a Dirichlet-to-Neumann map corresponding to a degenerate elliptic PDE. The reformulation allows the possibility of deriving estimates for solution to the fractional problem from that of degenerate elliptic equations. We will confirm this first by obtaining weighted estimates for the gradient of solutions to a class of linear degenerate/singular elliptic equation over a bounded domain. The weighted estimates are obtained under a smallness condition on the mean oscillation of the coefficients with a weight. We then use appropriate trace theorems to demonstrate that the solution to the fractional equation not only has the desired integrability, but also, unlike solutions to local equations, has improved fractional differentiability. This is a joint work with T. Phan.

TITLE: Asymptotic error analysis
SPEAKER: Brian Wetton, University of British Columbia
TIME: 3:40 PM-4:35 PM
ROOM: Ayres 405
When computing numerical approximations to problems with smooth solutions using regular grids, the error can have additional structure. The historical example of the Euler-McLaurin formula for the approximation of integrals with the trapezoidal rule is shown. This expansion can be used to justify Richardson extrapolation of the approximations leading to the Romberg integration formula. For approximation of differential equations, similar error expansions can be derived. For standard methods on uniform grids, an expansion for the error can be constructed that is regular in the grid spacing. For some other methods, numerical artifacts (boundary layers and errors that alternate in sign between adjacent grid points) can also be present. Identifying the types of errors that are generated by a given scheme and the order at which they occur is called Asymptotic Error Analysis. Several examples are shown, including the error analysis of cubic spline interpolation which is shown to have numerical boundary layers. A numerical artifact from an idealized adaptive grid with hanging nodes used to approximate a simple elliptic problem is presented.

TITLE: Stabilized Finite Element Methods for Coupled Incompressible Flow Problems
SPEAKER: Daniel Ardnt, University of Heidelberg, Germany
TIME: 4:35 PM-5:25 PM
ROOM: Ayres 405
In this talk, a finite element discretization of the incompressible Navier-Stokes equations for a (possibly) non-isothermal fluid is considered. In order to account for instabilities and to diminish unphysical oscillations a stabilization for the incompressibility constraint as well as a local projection approach for various terms is considered. Within this model, we derive parameter bounds and suitable ansatz spaces such that quasi-optimality and semi-robustness of the resulting method can be shown both analytically and numerically. For solving the resulting linear equations efficiently, multigrid methods appear promising and we examine the suitability of Schwarz smoothers for H^1 conforming elements.

TITLE: Contrasting Notions of Convergence in Geometric Analysis
SPEAKER: Brian Allen, United States Military Academy
TIME: 5:05 PM-6:05 PM
ROOM: Ayres 113
Often times when one studies sequences of Riemannian manifolds, which arise naturally when studying stability questions, one arrives at convergence in an $L^p$ space on the way to achieving Gromov-Hausdorff (GH) or Intrinsic Flat (IF) convergence. This motivates the desire to find conditions which when combined with $L^p$ convergence imply GH and/or IF convergence. In this talk we will discuss a Theorem which identifies such conditions and look at some recent applications to stability questions from geometric analysis which take advantage of these insights. This is joint work with Christina Sormani.

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

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last updated: May 2018

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