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Seminars and Colloquiums
for the week of November 13, 2017


Jan Rosinski, UTK, Tuesday
Ken Stephenson, UTK, Wednesday
Joseph Daws/Chris Oballe, UTK, Wednesday
Professor Hongjie Dong, Brown University, Thursday
Soledad Villar, Courant Institute, Thursday
Brett Kotschwar, Arizona State University, Thursday
Shiferaw Berhanu, Temple University, Friday

3:00 pm – 3:30 pm
Monday, Tuesday, Wednesday
Ayres 401
Hosted By: Mahir Demir and Mustafa Elmas

Tuesday, November 14th

TITLE: Strong path-wise approximation of Gaussian and Levy processes, extension of Ito-Nisio theorem, and continuity of Ito map
SPEAKER: Jan Rosinski, UTK
TIME: 2:10 pm – 3:25 pm
ROOM: Ayres 113
Abstract: N. Wiener (1923) gave two representations of Brownian motion as random trigonometric series, and showed that these series converge path-wise uniformly. This made Brownian motion a well-defined mathematical object and explained the nature of white noise. Nowadays, such types of series representations are also used to simulate stochastic processes. The mode and type of convergence in the series is crucial when we want to approximate output of a stochastic system (such as the Ito map) by the simulated input. The uniform convergence of the input processes is often insufficient for the convergence of the outputs, which triggered the development of rough path theory.

This talk is based on a joint work with Andreas Basse-O’Connor and Jorgen Hoffmann-Jorgensen of Aarhus University.

Wednesday, November 15th

TITLE: Emergent Conformal Structure and Computational Extremal Length
SPEAKER: Ken Stephenson, University of Tennessee
TIME: 2:30pm-3:20pm
ROOM: Ayres 113
Abstract: I believe that I may have talked about circle packing in the past.  I may have mentioned that circle packings impose geometry on abstract combinatorial patterns. I may even have talked about the conformal nature of that geometry. Wouldn't surprise me. It appears, in fact, that that geometry is profoundly "conformal" --- for geometrically random triangulations, conformality seems to emerge spontaneously under the mechanics of circle packing. Humm..., seems I may have talked about that before, too. In any case, now I plan to explain how one can exploit this emergent behavior in practice. The setting is computation of extremal length for data in atmospheric science. The talk will be image-driven and so should be accessible with little background. In addition, it may suggest uses in other areas, whether pure or applied, where geometric surfaces are involved.

TITLE: The Graph Laplacian, Spectral Clustering, and Reduced Order Modeling
SPEAKER 1: Joseph Daws
TIME: 3:35 pm – 4:25 pm
ROOM: Ayres 113
Abstract: Clustering analysis on data points sampled from high-dimensional spaces suffers from the curse of dimensionality. However, with appropriate assumptions on the systems that generate these high-dimensional samples, a low dimensional representation of the data can be used for the purpose of clustering. We will discuss how associating high-dimensional data with a graph can yield more satisfactory clustering results by finding a lower dimensional representation of the data generated from the so called graph Laplacian matrix. We will then give an example of how this ``spectral clustering" technique can be used to generate reduced-order bases for models of large, parameterized systems.

CAM - Pt. 2
TITLE: Signal classification with a point process distance on the space of Persistence Diagrams
SPEAKER 2: Chris Oballe
TIME: 3:35 pm – 4:25 pm
ROOM: Ayres 113
Abstract: In this talk, I’ll present results from a recently published paper by Andrew Marchese and Vasileios Maroulas. A new metric that accounts for cardinality differences is introduced on the space of persistence diagrams to more accurately classify signals. We’ll discuss properties of the resulting metric space, introduce a classification algorithm, then examine its performance on several data sets.

Thursday, November 16th

SPEAKER: Professor Hongjie Dong, Brown University
TIME: 2:10 pm – 3:10 pm
ROOM: Ayres 114
Abstract: TBA

TITLE: K-means clustering and semidefinite programming
SPEAKER: Soledad Villar, Courant Institute
TIME: 3:35 pm – 4:35 pm
ROOM: Ayres 405
Abstract: K-means clustering aims to partition a set of n points into k clusters in such a way that each observation belongs to the cluster with the nearest mean, and such that the sum of squared distances from each point to its nearest mean is minimal. In the worst case, this is a hard optimization problem, requiring an exhaustive search over all possible partitions of the data into k clusters in order to find the optimal clustering. At the same time, fast heuristic algorithms for k-means are often applied, despite only being guaranteed to converge to local minimizers of the k-means objective.

In this talk, we consider a semidefinite programming relaxation of the k-means optimization problem. We discuss two regimes where the SDP provides an algorithm with improved clustering guarantees compared to previous results in the literature: (a) for points drawn from isotropic distributions supported in separated balls, the SDP recovers the globally optimal k-means clustering under mild separation conditions; (b) for points drawn from mixtures of distributions with bounded variance, the SDP solution can be rounded to a clustering which is guaranteed to classify all but a small fraction of the points correctly. 

TITLE: A uniqueness theorem for asymptotically cylindrical shrinking Ricci solitons.
SPEAKER: Brett Kotschwar, Arizona State University
TIME: 5:00 pm – 6:00 pm
ROOM: Ayres 113
Abstract: Shrinking Ricci solitons are generalized fixed points of the Ricci flow equation and models for the geometry of solutions to the flow in the neighborhood of a developing singularity. It is conjectured that every end of a four-dimensional complete noncompact shrinking soliton is smoothly asymptotic to either a cone or a standard cylinder at infinity. I will discuss recent joint work with Lu Wang related to this conjecture in which we prove that a shrinking Ricci soliton which is asymptotic to infinite order along some end to one of the standard cylinders $S^k\times {\mathbb{R}}^{n-k}$ for $k\geq 2$ must actually be isometric to the cylinder on that end.

Friday, November 17th

TITLE: A local Hopf lemma for holomorphic functions and the Helmholtz equation
SPEAKER: Shiferaw Berhanu, Temple University
TIME: 3:30 pm – 4:30 pm
ROOM: Ayres 405
Abstract: Abstract: We will discuss results on local unique continuation at the boundary for holomorphic functions of one variable and for the solutions of the Helmholtz equation $L_cu = ?\Delta u + cu = 0$, $c \in R$ in an open set of the half space $R^n_+$ generalizing the theorems proved by Baouendi and Rothschild for harmonic functions. The results involve a local boundary sign condition on the product of the solution and a monomial. Applications to unique continuation for CR mappings will also be discussed.

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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