**Seminars and Colloquiums**

for the week of October 9, 2017

for the week of October 9, 2017

*SPEAKERS*

Prakash Chakraborty, Purdue, Tuesday

Juan Pablo Borthagaray, University of Maryland, Wednesday **
**Thomas Weighill, UTK, Wednesday

Michael Malisoff, LSU, Thursday

Dr. Erick Smith, US Naval Research Laboratory, Thursday

Kevin Sonnanburg, UTK, Thursday

Dr. Erick Smith, US Naval Research Laboratory, Friday

*TEA TIME* - cancelled

*3:00 pm – 3:30 pm
Monday, Tuesday,
Wednesday
Ayres 401
Hosted By: *

**Tuesday, October 10th**

**STOCHASTICS/ PROBABILITY SEMINAR**

TITLE: Rough Differential Equations with Power Type Coefficients

SPEAKER: Prakash Chakraborty, Purdue

TIME: 2:10-3:25pm

ROOM: Ayres 113

In this talk we will consider noisy differential equations driven by a signal which is γ-Hölder continuous with γ > 1/3. We will focus on obtaining existence of solution under the special case where the coefficients of the equation behave like power functions of the form |ξ|κ, κ (0,1). We will see some motivation for considering this problem and also some necessary notions from rough path theory.

**Wednesday, October 11th**

**COMPUTATIONAL and APPLIED MATHEMATICS (CAM) SEMINAR**

TITLE: Finite Element Approximations of the Nonhomogeneous Fractional Dirichlet Problem

SPEAKER: Juan Pablo Borthagaray, University of Maryland

TIME: 3:35-4:25pm

ROOM: Ayres 113

We study finite element approximations of the following non-homogeneous Dirichlet problem for the integral fractional Laplacian on a bounded domain $\Omega \subset \mathbb{R}^n$. We analyze two different approaches: the first one consists on the direct imposition of the Dirichlet condition; the second one is based on weak imposition of the Dirichlet condition and incorporating a nonlocal analogous of the normal derivative as a Lagrange multiplier in the formulation of the problem. In order to obtain convergence orders for our scheme, regularity estimates are developed, both for the solution and its nonlocal derivative. The method we propose requires that, as meshes are refined, the discrete problems be solved in a family of domains of growing diameter. This is a joint work with Gabriel Acosta (UBA, Argentina) and Norbert Heuer (PUC, Chile).

**TOPOLOGY/ GEOMETRY SEMINAR
**TITLE: Group actions and the maximal Roe algebra III

SPEAKER: Thomas Weighill, UTK

TIME: 3:35-4:25pm

ROOM: Ayres 405

The (maximal) Roe algebra is an important C*-algebra which appears in the index theory of non-compact complete Riemannian manifolds. It is also a coarse invariant, and so is naturally an object of study in coarse geometry. In a previous talk, we saw what it means for a group to act on a metric space by coarse equivalences and introduced a kind of "coarse quotient" which we call X_G. In this talk, we will begin to establish a relationship between the maximal Roe algebras of X and X_G for certain kinds of group action. This correspondence will involve the (full) crossed product of the Roe algebra with the group G. We will introduce all the necessary concepts from the theory of C* algebras, and recall the definition of the Roe algebra and its maximal version. This is joint work with Logan Higginbotham.

** Thursday, October 12th**

**DIFFERENTIAL EQUATIONS SEMINAR
**TITLE: Stabilization in a Chemostat with Sampled and Delayed Measurements

SPEAKER: Michael Malisoff, LSU

TIME: 2:10-3:10pm

ROOM: Ayres 114

The chemostat is a laboratory device and a mathematical model for the continuous culture of microorganisms. Chemostat models have been studied extensively, because of their importance in biotechnology and ecology.

They are often expressed as systems of ordinary differential equations, whose states are the concentrations of the substrate and the species. One important problem for chemostat models entails finding sufficient conditions that ensure that desired equilibria are globally asymptotically stable. This can often be done using feedback control, where one specifies parameters in the model that can depend on one or more of the states of the dynamics. The parameters can be chosen to be the dilution rate, or the substrate input concentrations. In this talk, we discuss chemostat models consisting of a system of ordinary differential equations, with constant substrate input concentrations. We allow growth functions that are not necessarily monotone. The measurement to be used in the feedback control is the substrate concentration, which is piecewise constant with a delay, so only time lagged sampled substrate observations are available to use in the feedback control. Under conditions on the size of the delay and on the largest sampling interval, we solve the problem of asymptotically stabilizing componentwise positive equilibria with the dilution rate as the feedback control. This talk is based on the speaker's joint work with Jerome Harmand and Frederic Mazenc.

**
JR. COLLOQUIUM
**TITLE: Transionospheric Synthetic Aperture Imaging

SPEAKER: Dr. Erick Smith, US Naval Research Laboratory

TIME: 3:40-4:35pm

ROOM: Ayres 405

* Note: Math majors who would like to join the speaker for a lunch or dinner (to be scheduled) should get in touch with Ken Stephenson, kstephe2@utk.edu.

Synthetic Aperture Radars (SAR) use microwaves to image the surface of the Earth (or other planets) from above, usually by airplanes or satellites. From the standpoint of mathematics, SAR imaging is an inverse problem of reconstructing certain characteristics of the target from the information contained in the radio waves reflected off this target. It can be done at nighttime and even through cloud cover. Radars that operate in the UHF and VHF bands, with frequencies of hundreds of MHz, also can penetrate foliage and the ground to see more, but if the radar is located on a satellite, images in these frequencies are also susceptible to distortions caused by the ionosphere, a dynamic region of the atmosphere containing a high concentration of ions and free electrons.

In this talk, we will look at problems associated with spaceborne synthetic aperture imaging in these frequencies and the mathematical ways to correct for the distortions, with the primary methodology stemming from asymptotic analysis and perturbation theory. This is done without prior knowledge of the current state of the ionosphere, which is both inhomogeneous and anisotropic as well as constantly changing in time and having both deterministic and stochastic components. We also look at anisotropy in the target and how certain material parameters can be determined from the phenomenologically constructed scattering matrix. Besides mathematicians, the presentation should also be of interest to those with physics and electrical engineering backgrounds.

Published earlier this year and the basis for this talk, Transionospheric Synthetic Aperture Imaging is a follow-up to the speaker’s 2013 dissertation, SAR Imaging Through the Earth’s Ionosphere, incorporating the contents of the dissertation as well as work that came before it and afterwards.

**
ANALYSIS SEMINAR
**TITLE: Blow-up Continuity of Mean-Convex, Type-I Mean Curvature Flow

SPEAKER: Kevin Sonnanburg (UTK)

TIME: 5-6pm

ROOM: Ayres 113

Under mean curvature flow, each point of a hypersurface moves with velocity equal to its mean curvature vector, shrinking the hypersurface's area as rapidly as possible. A closed, embedded hypersurface M(t) shrinks and becomes singular in finite time. One of the most basic questions about a PDE that develops singularities is the relationship between the occurrence of its singularities and its initial data. For certain classes of mean-convex mean curvature flows, we show the first singular time T and the limit set “M(T)” are continuous with respect to the initial hypersurface.

** Friday, October 13th**

**COLLOQUIUM
**TITLE: Transionospheric Synthetic Aperture Imaging

SPEAKER: Dr. Erick Smith, US Naval Research Laboratory

TIME: 3:30-4:30pm

ROOM: Ayres 405

Synthetic Aperture Radars (SAR) use microwaves to image the surface of the Earth (or other planets) from above, usually by airplanes or satellites. From the standpoint of mathematics, SAR imaging is an inverse problem of reconstructing certain characteristics of the target from the information contained in the radio waves reflected off this target. It can be done at nighttime and even through cloud cover. Radars that operate in the UHF and VHF bands, with frequencies of hundreds of MHz, also can penetrate foliage and the ground to see more, but if the radar is located on a satellite, images in these frequencies are also susceptible to distortions caused by the ionosphere, a dynamic region of the atmosphere containing a high concentration of ions and free electrons.

In this talk, we will look at problems associated with spaceborne synthetic aperture imaging in these frequencies and the mathematical ways to correct for the distortions, with the primary methodology stemming from asymptotic analysis and perturbation theory. This is done without prior knowledge of the current state of the ionosphere, which is both inhomogeneous and anisotropic as well as constantly changing in time and having both deterministic and stochastic components. We also look at anisotropy in the target and how certain material parameters can be determined from the phenomenologically constructed scattering matrix. Besides mathematicians, the presentation should also be of interest to those with physics and electrical engineering backgrounds.

Published earlier this year and the basis for this talk, Transionospheric Synthetic Aperture Imaging is a follow-up to the speaker’s 2013 dissertation, SAR Imaging Through the Earth’s Ionosphere, incorporating the contents of the dissertation as well as work that came before it and afterwards.

*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:**