Seminars and Colloquia by Series

Convergence of the extremal eigenvalues of empirical covariance matrices with dependence

Series
Stochastics Seminar
Time
Thursday, November 19, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Konstantin Tikhomirov – University of Alberta
Consider a sample of a centered random vector with unit covariance matrix. We show that under certain regularity assumptions, and up to a natural scaling, the smallest and the largest eigenvalues of the empirical covariance matrix converge, when the dimension and the sample size both tend to infinity, to the left and right edges of the Marchenko-Pastur distribution. The assumptions are related to tails of norms of orthogonal projections. They cover isotropic log-concave random vectors as well as random vectors with i.i.d. coordinates with almost optimal moment conditions. The method is a refinement of the rank one update approach used by Srivastava and Vershynin to produce non-asymptotic quantitative estimates. In other words we provide a new proof of the Bai and Yin theorem using basic tools from probability theory and linear algebra, together with a new extension of this theorem to random matrices with dependent entries. Based on joint work with Djalil Chafai.

Thin Position for Knots and Topological Data Analysis

Series
School of Mathematics Colloquium
Time
Thursday, November 19, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jesse Johnson – Google
Topological data analysis is the study of Machine Learning/Data Mining problems using techniques from geometry and topology. In this talk, I will discuss how the scale of modern data analysis has made the geometric/topological perspective particularly relevant for these subjects. I'll then introduce an approach to the clustering problem inspired by a tool from knot theory called thin position.

Random matrix, concentration and almost sure convergence of the distribution of eigenvalues

Series
Regular Seminars
Time
Wednesday, November 18, 2015 - 17:00 for 1 hour (actually 50 minutes)
Location
Skies 169
Speaker
Inoel Popescu – Georgia Tech
We will summarize what we did so far in this sequence of seminars, among other things, the convergence of eigenvalues of Wigner random matrices and also GUE in expectation. This time we will explore concentration inequalities and use these to go from the convergence in expectation to convergence almost surely.

Fourier restriction to degenerate manifolds

Series
Analysis Seminar
Time
Wednesday, November 18, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Betsy Stovall – UW-Madison
We will discuss the problem of restricting the Fourier transform to manifolds for which the curvature vanishes on some nonempty set. We will give background and discuss the problem in general terms, and then outline a proof of an essentially optimal (albeit conditional) result for a special class of hypersurfaces.

How Geometry plays a role in Industry

Series
Research Horizons Seminar
Time
Wednesday, November 18, 2015 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dr. Jesse Johnson – Google Company

Please Note: Food and Drinks will be provided before the seminar.

In this talk, we will discuss: (1) How geometry plays a role in machine learning/data science? (2) What it's like being a mathematician at a software company.

Almost-reducibility for fibered holomorphic dynamics

Series
Dynamical Systems Working Seminar
Time
Tuesday, November 17, 2015 - 17:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Mikel Viana – Georgia Tech (Math)
In previous talks, we discussed an algorithm (Nash-Moser iteration) to compute invariant whiskered tori for fibered holomorphic maps. Several geometric and number-theoretic conditions are necessary to carry out each step of the iteration. Recently, there has been interest in studying what happens if some of the conditions are removed. In particular, the second Melnikov condition we found can be hard to verify in higher dimensional problems. In this talk, we will use a method due to Eliasson, Moser and Poschel to obtain quasi-periodic solutions which, however, lose an important geometric property relative to the solutions previously constructed.

Uniqueness and Finsler type optimal transport metric for nonlinear wave equations

Series
PDE Seminar
Time
Tuesday, November 17, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Geng Chen – School of Mathematics, Georgia Tech
In this talk, we will discuss a sequence of recent progresses on the global well-posedness of energy conservative Holder continuous weak solutions for a class of nonlinear variational wave equations and the Camassa-Holm equation, etc. A typical feature of solutions in these equations is the formation of cusp singularity and peaked soliton waves (peakons), even when initial data are smooth. The lack of Lipschitz continuity of solutions gives the major difficulty in studying the well-posedness and behaviors of solutions. Several collaboration works with Alberto Bressan will be discussed, including the uniqueness by characteristic method, Lipschitz continuous dependence on a Finsler type optimal transport metric and a generic regularity result using Thom's transversality theorem.

Random graph processes with dependencies

Series
Job Candidate Talk
Time
Tuesday, November 17, 2015 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Lutz Warnke – University of Cambridge
Random graphs are the basic mathematical models for large-scale disordered networks in many different fields (e.g., physics, biology, sociology). Their systematic study was pioneered by Erdoes and Renyi around 1960, and one key feature of many classical models is that the edges appear independently. While this makes them amenable to a rigorous analysis, it is desirable (both mathematically and in terms of applications) to understand more complicated situations. In this talk I will discuss some of my work on so-called Achlioptas processes, which (i) are evolving random graph models with dependencies between the edges and (ii) give rise to more interesting percolation phase transition phenomena than the classical Erdoes-Renyi model.

Topological full groups

Series
Colloquia
Time
Tuesday, November 17, 2015 - 10:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Professor Volodymyr Nekrashevych – Department of Mathematics, Texas A&M –

Please Note: This talk should interest people in Algebra, Dynamical Systems and Mathematical Physics in addition to Geometry and Topology. Volodia Nekrashevych will visit Atlanta from Sunday November 15th evening until Tuesday November 17th afternoon. He will be available for private talks on Monday November 14th after noon or on Tueasday morning before 10AM. Contact him directly by email or contact jeanbel@math.gatech.edu to schedule a meeting or to have a dinner with him.

Topological full groups are naturally associated with semigroups of local homeomorphisms: iterations of a single homeomorphism, holonomy groupoids of laminations, groupoids of local isomorphisms of quasiperiodic sets (for example Penrose tilings), etc. Some of these groups have interesting properties from the point of view of group theory. For instance, they provide first examples of amenable infinite simple finitely generated groups (by a result of K. Juschenko and N. Monod) and first examples of simple amenable groups of Burnside type. The full group of the Penrose tiling is another interesting example from the point of view of amenability.

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