Seminars and Colloquia by Series

Estimation of Low Rank Kernels on Graphs

Series
High-Dimensional Phenomena in Statistics and Machine Learning Seminar
Time
Tuesday, February 14, 2012 - 16:00 for 1.5 hours (actually 80 minutes)
Location
Skyles 006
Speaker
Vladimir Koltchinskii – Georgia Institute of Technology, School of Mathematics
Let (V, E) be a graph with vertex set V and edge set E. Let (X, X', Y) V \in V × {-1,1} be a random triple, where X, X' are independent uniformly distributed vertices and Y is a label indicating whether X, X' are "similar", or not. Our goal is to estimate the regression function S_*(u, v) = \mathbb{E}(Y|X = u, X' = v), u, v \in V based on n i.i.d. copies (X_1, X'_1, Y_1), ... , (X_n, X'_n, Y_n) of (X, X', Y). We are interested in this problem in the case when S_*: V × V \mapsto [-1,1] is a symmetric low rank kernel (or it can be well approximated by low rank kernels). In addition to this, assume that S_* is "smooth" on the graph. We study estimators based on a modified least squares method with complexity penalization involving both the nuclear norm and Sobolev type norms of symmetric kernels on the graph (defined in terms of its Laplacian). We prove error bounds for such estimators that improve the existing bounds in low rank matrix recovery in the cases when the target kernel is not only low rank, but also sufficiently smooth. The talk is based in part on a joint project with Pedro Rangel.

Viscoelastic Navier-Stokes equations with damping

Series
PDE Seminar
Time
Tuesday, February 14, 2012 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ryan Hynd – Courant Institute of Mathematical Sciences, New York University
We prove an analog of the Caffarelli-Kohn-Nirenberg theorem for weak solutions of a system of PDE that model a viscoelastic fluid in the presence of an energy damping mechanism. The system was recently introduced in a method of establishing the global in time existence of weak solutions of the well known Oldroyd model, which remains an open problem.

Complex Geometry and Operator Theory

Series
School of Mathematics Colloquium
Time
Tuesday, February 14, 2012 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ron Douglas – Texas A&M University
An intesting class of bounded operators or algebras of bounded operators on Hilbert spaces, particularly on Hilbert spaces of holomorphic functions, have a natural interpretation in terms of concepts from complex geometry. In particular, there is an intrinsic hermitian holomorphic vector bundle and many questions can be answered in terms of the Chern connection and the associated curvature. In this talk we describe this setup and some of the results obtained in recent years using this approach. The emphasis will be on concrete examples, particularly in the case of Hilbert spaces of holomorphic functions such as the Hardy and Bergman spaces on the unit sphere in C^n.

How Advances in Science are Made

Series
Other Talks
Time
Monday, February 13, 2012 - 18:00 for 1 hour (actually 50 minutes)
Location
CULC Room 152
Speaker
Douglas Osheroff – Nobel Laureate, Stanford University

Please Note: Host: Carlos Sa de Melo, School of Physics

How advances in science are made, and how they may come to benefit mankind at large are complex issues. The discoveries that most infuence the way we think about nature seldom can be anticipated, and frequently the applications for new technologies developed to probe a specific characteristic of nature are also seldom clear, even to the inventors of these technologies. One thing is most clear: seldom do individuals make such advances alone. Rather, they result from the progress of the scientific community, asking questions, developing new technologies to answer those questions, and sharing their results and their ideas with others. However, there are indeed research strategies that can substantially increase the probability of one's making a discovery, and the speaker will illustrate some of these strategies in the context of a number of well known discoveries, including the work he did as a graduate student, for which he shared the Nobel Prize for Physics in 1996.

Parameterization of Invariant Manifolds for Lagrangian Systems with Long-range Interactions

Series
CDSNS Colloquium
Time
Monday, February 13, 2012 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hector Lomeli – Univ. of Texas at Austin/ITAM
We generalize some notions that have played an important role in dynamics, namely invariant manifolds, to the more general context of difference equations. In particular, we study Lagrangian systems in discrete time. We define invariant manifolds, even if the corresponding difference equations can not be transformed in a dynamical system. The results apply to several examples in the Physics literature: the Frenkel-Kontorova model with long-range interactions and the Heisenberg model of spin chains with a perturbation. We use a modification of the parametrization method to show the existence of Lagrangian stable manifolds. This method also leads to efficient algorithms that we present with their implementations. (Joint work with Rafael de la Llave.)

Discrete Mathematical Biology Working Seminar

Series
Other Talks
Time
Monday, February 13, 2012 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Svetlana Poznanovik – Georgia Tech
A discussion of the paper "Linear trees and RNA secondary structure" by Schmitt and Waterman (1994) and, as time permits, "Combinatorics of RNA secondary structures" by Hofacker, Schuster, and Stadler (1998).

Oral Comprehensive Exam: Low rank estimation of similarities on graphs

Series
Other Talks
Time
Monday, February 13, 2012 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 170
Speaker
Pedro Rangel – School of Mathematics, Georgia Tech
The goal in matrix recovery problems is to estimate an unknown rank-r matrix S of size m based on a set of n observations. It is easy to see that even in the case where the observations are not contaminated with noise, there exist low rank matrices that cannot be recovered based on n observations unless n is very large. In order to deal with these cases, Candes and Tao introduced the called low-coherence assumptions and a parameter \nu measuring how low-coherent the objective matrix S is. Using the low-coherence assumptions, Gross proved that S can be recovered with high probability if n>O(\nu r m \log^2(m)) by an estimator based on nuclear norm penalization. Let's consider the generalization of the matrix recovery problem where the matrix S is not only low-rank but also "smooth" with respect to the geometry given by a graph G. In this 40 minutes long talk, the speaker will present an approximation error bound for a proposed estimator in this generalization of the matrix recovery problem.

An Overview of Lattice Cryptography

Series
ACO Student Seminar
Time
Friday, February 10, 2012 - 13:00 for 1 hour (actually 50 minutes)
Location
TBD
Speaker
Christopher Peikert – School of Computer Science
I'll give a high-level tour of how lattices are providing a powerful new mathematical foundation for cryptography. Lattices provide simple, fast, and highly parallel cryptoschemes that, in contrast with many of today's popular methods (like RSA and elliptic curves), even appear to remain secure against quantum computers. No background in lattices, cryptography, or quantum computers will be necessary -- you only need to know how to add and multiply vectors and matrices.

Log-Sobolev inequalities for subelliptic operators satisfying a generalized curvature dimension inequality

Series
Stochastics Seminar
Time
Thursday, February 9, 2012 - 15:05 for 1 hour (actually 50 minutes)
Location
Skyles 006
Speaker
Fabrice Baudoin – University of Purdue
Let $\M$ be a smooth connected manifold endowed with a smooth measure $\mu$ and a smooth locally subelliptic diffusion operator $L$ which is symmetric with respect to $\mu$. We assume that $L$ satisfies a generalized curvature dimension inequality as introduced by Baudoin-Garofalo \cite{BG1}. Our goal is to discuss functional inequalities for $\mu$ like the Poincar\'e inequality, the log-Sobolev inequality or the Gaussian logarithmic isoperimetric inequality.

Gel'fand Duality

Series
Geometry Topology Student Seminar
Time
Wednesday, February 8, 2012 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 169
Speaker
Jonathan Paprocki – Georgia Tech
We will prove a duality between locally compact Hausdorff spaces and the C*-algebra of continuous complex-valued functions on that space. Formally, this is the equivalence of the opposite category of commutative C*-algebras and the category of locally compact Hausdorff spaces.

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