**Seminars and Colloquiums**

for the week of April 23, 2018

for the week of April 23, 2018

*SPEAKERS*

**Monday**

Pawel Grzegrzolka and Jeremy Siegert, University of Tennessee

Maggie Wieczorek, University of Tennessee

Xiaoyang Pan, University of Tennessee

**Tuesday**

Jan Rosinski, University of Tennessee

Ryan Unger, University of Tennessee

**Wednesday**

Vincent Heningburg, University of Tennessee

**Thursday**

Kody Law, University of Manchester

Adam Spannaus, University of Tennessee

Sajjad Lakzian, Fordham University

**Friday**

Gigliola Staffilani, MIT

**Monday, April 23**

TOPOLOGY/ GEOMETRY SEMINAR

TITLE: Category of Coarse Proximity Spaces and Proximity at Infinity

SPEAKER: Pawel Grzegrzolka and Jeremy Siegert, University of Tennessee

TIME: 2:30 PM-3:20 PM

ROOM: Ayres 112

Coarse topology (also known under the name of large scale geometry) emerged in the 20th century as a dual notion to uniform spaces. While uniformities capture primarily what happens "on the small scale" (e.g., limits, continuity), coarse structures focus predominantly on what happens "on the large scale" (e.g., asymptotic dimension, coarse equivalence of spaces). Large scale geometry has received worldwide attention thanks to its wide range of applications, including its connection to the Novikov conjecture and the coarse Baum-Connes conjecture.

In this talk, we will introduce coarse proximity maps and their basic properties. Consequently, we will show the existence of a category of coarse proximity spaces whose morphisms are closeness classes of coarse proximity maps. Then we will restrict ourselves to metric spaces to define a proximity on the collection of unbounded subsets of a given metric space. This will allow us to define the proximity space at infinity for any metric space X. Finally, we will prove that proximity space at infinity induces a functor from a full subcategory of coarse proximity spaces (whose objects are unbounded metric spaces and whose morphisms are closeness classes of coarse proximity maps) to the category of proximity spaces (whose objects are proximity spaces and whose morphisms are proximity maps).

Note from organizers: This could be the last talk at the seminar due to expected End of the World on April 23rd, 2018. We advise to take advantage of this unique (possibly) opportunity.

ALGEBRA SEMINAR

TITLE: The partition function and its congruences

SPEAKER: Maggie Wieczorek, University of Tennessee

TIME: 3:35 PM-4:25 PM

ROOM: Ayres 112

In this talk we will define a partition of a nonnegative integer and give congruences for the number of partitions. We will also connect this partition function to the theory of modular forms. This relationship allows us to find more congruences for the partition function, many of which are characterized by a powerful result of Ahlgren and Ono.

DOCTORAL DEFENSE

TITLE: Asymptotics for dynamical systems driven by jump noise

SPEAKER: Xiaoyang Pan, University of Tennessee

TIME: 3:45 PM

ROOM: Ayres 111

This dissertation studies asymptotic estimates for dynamical systems with jumps. We first focus on

the statistical inference problem for a linear partially observed system. A least- squares

estimator for the intensity of a Poisson process is proposed, where the signal process

is driven by the mixture of a Brownian motion and a Poisson precess and the observation is a

diffusion process. Precisely, we verify the unbiasedness, consistency for the estimator of the

intensity. Furthermore, the asymptotic distribution and convergence rate of the consistent

estimator are studied as well as a statistics for statistical inference is constructed employing

the central limit theorem, large and moderate deviation principles. The last part of this

dissertation is concerned with large deviation principles for the optimal filtering of a general

nonlinear model. First, the uniqueness of the solution of the Zakai and Kushner-Stratonovich

equations are proved, by applying a pertinent transformation of the associated equations into

SDEs in an appropriate Hilbert space. Taking into account the controlled analogue of

Zakai and Kushner-Stratonovich equations, respectively, the large deviation principle

follows by employing some qualitative properties of their solutions using

weak convergence arguments.

Committee members: Drs. Vasileios Maroulas (Chair), Xia Chen, Jan Rosinski, Kody Law, Abner Salgado

and Haileab Hilafu (STAT)

**Tuesday, April 24**

STOCHASTICS/PROBABILITY SEMINAR

TITLE: Embedding of Levy Processes in Brownian motion

SPEAKER: Jan Rosinski, University of Tennessee

TIME: 2:10 PM-3:20 PM

ROOM: Ayres 114

Embedding of Levy Processes in Brownian motion has been studied by several authors as an offspring of the famous Skorohod embedding problem. We will give a short introduction to this problem and show some consequences of this embedding for the trajectories of Levy Processes.

GENERAL RELATIVITY SEMINAR

TITLE: Dimension reduction and minimal surface singularities II

SPEAKER: Ryan Unger, University of Tennessee

TIME: 5:00 PM-6:00 PM

ROOM: Ayres 113

We will present Federer's dimension reduction argument and indicate how this can be used to control the dimension of the singular set of a minimizing current. This is important for the regularity theory of minimal surfaces and Schoen and Yau's minimal slicings.

**Wednesday, April 25**

COMPUTATIONAL and APPLIED MATHEMATICS (CAM) SEMINAR

TITLE: Importance of the Diffusion Limit of the Transport Equation in Optically Thick, Diffusive Regimes

SPEAKER: Vincent Heningburg, University of Tennessee

TIME: 3:35 PM-4:35 PM

ROOM: Ayres 112

With few exceptions, the spatial differencing schemes that have been widely used produce solutions whose errors tend to zero when the optical thickness of spatial cells tends to zero. In recent decades, however, neutronics methods have been applied to electron and thermal radiation transport problems, which are optically much thicker than neutron transport problems. We discuss the relevance of the diffusion limit in assessing the accuracy of transport differencing schemes for such optically thick meshes.

**Thursday, April 26**

MATHEMATICAL DATA SCIENCE SEMINAR

TITLE: Strategies for multilevel Monte Carlo

SPEAKER: Kody Law, University of Manchester

TIME: 3:30 PM-4:30 PM

ROOM: Ayres 405

This talk will concern the problem of inference when the posterior measure involves continuous models which require approximation before inference can be performed. Typically one cannot sample from the posterior distribution directly, but can at best only evaluate it, up to a normalizing constant. Therfore one must resort to computationally-intensive inference algorithms in order to construct estimators. These algorithms are typically of Monte Carlo type, and include for example Markov chain Monte Carlo, importance samplers, and sequential Monte Carlo samplers. The multilevel Monte Carlo method provides a way of optimally balancing discretization and sampling error on a hierarchy of approximation levels, such that cost is optimized. Recently this method has been applied to computationally intensive inference. This highly non-trivial task can be achieved in a variety of ways. This talk will review 3 primary strategies which have been successfully employed to achieve optimal convergence rates -- in other words faster convergence than i.i.d. sampling at the finest discretization level. Some of the specific resulting algorithms, and applications, will also be presented.

ORAL SPECIALTY EXAM

TITLE: Advanced Monte Carlo Markov Chain Techniques for New Materials Discovery

SPEAKER: Adam Spannaus, University of Tennessee

TIME: 5:00 PM-7:00 PM

ROOM: Ayres 111

High Entropy Alloys are a new class of materials with remarkable physical properties. Materials Science researchers are unable to predict these properties before synthesizing an alloy, due a lack of knowledge about their chemical ordering and atomic structure. The state of the art technique, Atomic Probe Tomography, for discovering the chemical ordering and atomic structure is only able to recover approximately 1/3 of the data, and what is recovered is corrupted by noise. Consequently, these alloys are unable to be tailored to specific applications, and their potential remains untapped.

In this talk, we propose an approach to uncover the true atomic structure and chemical ordering of these novel alloys simultaneously via a Bayesian perspective. Viewed through the lens of Bayesian inference, we are able to find an unambiguous representation of the crystal structure and chemical ordering. Posing the solution as not just a point, but as a distribution allows for ancillary quantities of interest to be easily computed, and the uncertainty is readily quantified. Our novel formulation of the problem results in a marginal posterior distribution on the transformation, which is explored within a Pseudo-Marginal Markov chain Monte Carlo scheme, in which the marginal density is approximated via Sequential Monte Carlo with bridging densities.

We will present numerical results from our most recent paper with synthetic Atomic Probe Tomography datasets, and extensions for further research will be discussed.

Committee members: Drs. Vasileios Maroulas (Chair), Xiaobing Feng, David Keffer (MSE), Kody Law and Tim Schulze.

GEOMETRIC ANALYSIS SEMINAR

TITLE: Compactness theory for harmonic maps from Riemann surfaces into compact locally CAT(1) spaces.

SPEAKER: Sajjad Lakzian, Fordham University

TIME: 5:00 PM-6:00 PM

ROOM: Ayres 113

In this talk, we will show the bubble tree convergence for a sequence of harmonic maps, with uniform energy bounds, from a compact Riemann surface into a compact locally CAT(1) space. In particular, we demonstrate energy quantization and the no-neck property for such a sequence. In the smooth setting, Jost and Parker respectively established these results by exploiting now classical arguments for harmonic maps. Our work demonstrates that these results can be reinterpreted geometrically. In the absence of a PDE, we take advantage of the local convexity properties of the target space. Included in this paper are an $\epsilon$-regularity theorem, an energy gap theorem, and a removable singularity theorem for harmonic maps for harmonic maps into metric spaces with upper curvature bounds. We also prove an isoperimetric inequality for conformal harmonic maps with small image. If time allows, we will also touch upon other ideas and developments regarding the existence of such maps in free homotopy classes. This is a joint work with C. Breiner.

**Friday, April 27**

COLLOQUIUM

TITLE: The many faces of dispersive and equations

SPEAKER: Gigliola Staffilani, MIT

TIME: 3:35 PM-4:35 PM

ROOM: Ayres 405

In recent years great progress has been made in the study of dispersive and wave equations. Over the years the toolbox used in order to attack highly nontrivial problems related to these equations has developed to include a variety of techniques from Fourier and harmonic analysis, analytic number theory, math physics, dynamical systems, probability and symplectic geometry. In this talk I will introduce a variety of problems connected with dispersive and wave equations, such as the derivation of a certain nonlinear Schrodinger equations from a quantum many-particles system, periodic Strichartz estimates, the concept of energy transfer, the invariance of a Gibbs measure associated to an infinite dimension Hamiltonian system and non-squeezing theorems for such systems when they also enjoy a symplectic structure.

(Bose Einstein Condensate)

*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_12_18 (spring break)