Seminars and Colloquia by Series

A numerical analysis approach to convex optimization

Series
ACO Student Seminar
Time
Friday, January 11, 2019 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Richard Peng – CS, Georgia Tech –
In current convex optimization literature, there are significant gaps between algorithms that produce high accuracy (1+1/poly(n))-approximate solutions vs. algorithms that produce O(1)-approximate solutions for symmetrized special cases. This gap is reflected in the differences between interior point methods vs. (accelerated) gradient descent for regression problems, and between exact vs. approximate undirected max-flow. In this talk, I will discuss generalizations of a fundamental building block in numerical analysis, preconditioned iterative methods, to convex functions that include p-norms. This leads to algorithms that converge to high accuracy solutions by crudely solving a sequence of symmetric residual problems. I will then briefly describe several recent and ongoing projects, including p-norm regression using m^{1/3} linear system solves, p-norm flow in undirected unweighted graphs in almost-linear time, and further improvements to the dependence on p in the runtime.

Network data: Modeling and Statistical Analysis

Series
Job Candidate Talk
Time
Thursday, January 10, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Subhabrata Sen – MIT –
Network data arises frequently in modern scientific applications. These networks often have specific characteristics such as edge sparsity, heavy-tailed degree distribution etc. Some broad challenges arising in the analysis of such datasets include (i) developing flexible, interpretable models for network datasets, (ii) testing for goodness of fit, (iii) provably recovering latent structure from such data.In this talk, we will discuss recent progress in addressing very specific instantiations of these challenges. In particular, we will1. Interpret the Caron-Fox model using notions of graph sub-sampling, 2. Study model misspecification due to rare, highly “influential” nodes, 3. Discuss recovery of community structure, given additional covariates.

A modern maximum-likelihood approach for high-dimensional logistic regression

Series
Job Candidate Talk
Time
Tuesday, January 8, 2019 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Pragya Sur – Statistics Department, Stanford University
Logistic regression is arguably the most widely used and studied non-linear model in statistics. Classical maximum-likelihood theory based statistical inference is ubiquitous in this context. This theory hinges on well-known fundamental results—(1) the maximum-likelihood-estimate (MLE) is asymptotically unbiased and normally distributed, (2) its variability can be quantified via the inverse Fisher information, and (3) the likelihood-ratio-test (LRT) is asymptotically a Chi-Squared. In this talk, I will show that in the common modern setting where the number of features and the sample size are both large and comparable, classical results are far from accurate. In fact, (1) the MLE is biased, (2) its variability is far greater than classical results, and (3) the LRT is not distributed as a Chi-Square. Consequently, p-values obtained based on classical theory are completely invalid in high dimensions. In turn, I will propose a new theory that characterizes the asymptotic behavior of both the MLE and the LRT under some assumptions on the covariate distribution, in a high-dimensional setting. Empirical evidence demonstrates that this asymptotic theory provides accurate inference in finite samples. Practical implementation of these results necessitates the estimation of a single scalar, the overall signal strength, and I will propose a procedure for estimating this parameter precisely. This is based on joint work with Emmanuel Candes and Yuxin Chen.

Finite Dimensional Balian-Low Theorems

Series
Applied and Computational Mathematics Seminar
Time
Monday, January 7, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 154
Speaker
Dr. Michael Northington – GT Math
Gabor systems, or collections of translations and modulations of a window function, are often used for time-frequency analysis of signals. The Balian-Low Theorem and its generalizations say that if a Gabor system obeys certain spanning and independence properties in L^2(R), then the window function of such a system cannot be well localized in both time and frequency. Recently, Shahaf Nitzan and Jan—Fredrik Olsen show that similar behavior extends to Gabor systems of finite length signals in l^2(Z_d). In this talk, I will discuss these finite dimensional results as well as recent extensions proven in collaboration with Josiah Park.

The Technique to Solve the Variable Coefficients Homological Equations

Series
CDSNS Colloquium
Time
Monday, December 17, 2018 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Hongyu Cheng – MSRI & Nankai University
In the infinite-dimensional KAM theory, solving the homological equations is the one of the main parts. Generally, the coefficients of the homological equations are constants, by comparing the coefficients of the functions, it is easy to solve these equations. If the coefficients of homological equations depend on the angle variables, we call these equations as the variable coefficients homological equations. In this talk we will talk about how to solve these equations.

Congruence subgroups of braid groups

Series
School of Mathematics Colloquium
Time
Friday, December 7, 2018 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tara Brendle – University of Glasgow
The Burau representation plays a key role in the classical theory of braid groups. When we let the complex parameter t take the value -1, we obtain a symplectic representation of the braid group known as the integral Burau representation. In this talk we will give a survey of results on braid congruence subgroups, that is, the preimages under the integral Burau representation of principal congruence subgroups of symplectic groups. Along the way, we will see the (perhaps surprising) appearance of braid congruence subgroups in a variety of other contexts, including knot theory, homotopy theory, number theory, and algebraic geometry.

Scattering maps and instability in Hamiltonian mechanics.

Series
Dynamical Systems Working Seminar
Time
Friday, December 7, 2018 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 170
Speaker
Rafael de la Llave – School of Mathematics

Given a Hamiltonian system, normally hyperbolic invariant manifolds and their stable and unstable manifolds are important landmarks that organize the long term behaviour.

When the stable and unstable manifolds of a normally hyperbolic invarriant manifold intersect transversaly, there are homoclinic orbits that converge to the manifold both in the future and in the past. Actually, the orbits are asymptotic both in the future and in the past.

One can construct approximate orbits of the system by chainging several of these homoclinic excursions.

A recent result with M. Gidea and T. M.-Seara shows that if we consider long enough such excursions, there is a true orbit that follows it. This can be considered as an extension of the classical shadowing theorem, that allows to handle some non-hyperbolic directions

Canonical measures on graphs and a Kazhdan’s theorem

Series
Algebra Seminar
Time
Wednesday, December 5, 2018 - 14:30 for 1 hour (actually 50 minutes)
Location
Skiles 249
Speaker
Farbod Shokrieh – University of Copenhagen –
Classical Kazhdan's theorem for Riemann surfaces describes the limiting behavior of canonical (Arakelov) measures on finite covers in relation to the hyperbolic measure. I will present a generalized version of this theorem for metric graphs. (Joint work with Chenxi Wu.)

Some well disguised ribbon knots

Series
Geometry Topology Student Seminar
Time
Wednesday, December 5, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Agniva Roy – Georgia Tech

The talk will discuss a paper by Gompf and Miyazaki of the same name. This paper introduces the notion of dualisable patterns, a technique which is widely used in knot theory to produce knots with similar properties. The primary objective of the paper is to first find a knot which is not obviously ribbon, and then show that it is. It then goes on to construct a related knot which is not ribbon. The talk will be aimed at trying to unwrap the basic definitions and techniques used in this paper, without going too much into the heavy technical details.

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