Seminars and Colloquia by Series

Recent progress in stochastic topology

Series
School of Mathematics Colloquium
Time
Thursday, November 12, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Prof. Dr. Matthew Kahle – Ohio State University –
The study of random topological spaces: manifolds, simplicial complexes, knots, and groups, has received a lot of attention in recent years. This talk will focus on random simplicial complexes, and especially on a certain kind of topological phase transition, where the probability that that a certain homology group is trivial passes from 0 to 1 within a narrow window. The archetypal result in this area is the Erdős–Rényi theorem, which characterizes the threshold edge probability where the random graph becomes connected. One recent breakthrough has been in the application of Garland’s method, which allows one to prove homology-vanishing theorems by showing that certain Laplacians have large spectral gaps. This reduces problems in random topology to understanding eigenvalues of certain random matrices, and the method has been surprisingly successful. This is joint work with Christopher Hoffman and Elliot Paquette.

Sharp Uncertainty Principles for Shift-Invariant Spaces

Series
Analysis Seminar
Time
Wednesday, November 11, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Michael Northington – Vanderbilt University –
Uncertainty principles are results which restrict the localization of a function and its Fourier transform. One class of uncertainty principles studies generators of structured systems of functions, such as wavelets or Gabor systems, under the assumption that these systems form a basis or some generalization of a basis. An example is the Balian-Low Theorem for Gabor systems. In this talk, I will discuss sharp, Balian-Low type, uncertainty principles for finitely generated shift-invariant subspaces of $L^2(\R^d)$. In particular, we give conditions on the localization of the generators which prevent these spaces from being invariant under any non-integer shifts.

Sums involving Diophantine numbers and applications to differential equations.

Series
Dynamical Systems Working Seminar
Time
Tuesday, November 10, 2015 - 17:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Rafael de la Llave – Georgia Tech
In the study of perturbation theories in Dynamical systems one is often interested in solving differential equations involving frequencies satisfying number theoretic properties. We will present some estimates ofsums involving Diophantine frequencies leading to sharp estimates on the differential equations.

The Fokker-Planck equation in bounded domains

Series
PDE Seminar
Time
Tuesday, November 10, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hyung Ju Hwang – POSTECH, Korea
In this talk, we consider the initial-boundary value problem for the Fokker-Planck equation in an interval or in a bounded domain with absorbing boundary conditions. We discuss a theory of well-posedness of classical solutions for the problem as well as the exponential decay in time, hypoellipticity away from the singular set, and the Holder continuity of the solutions up to the singular set. This is a joint work with J. Jang,J. Jung, and J. Velazquez.

Optimal Estimation of Low Rank Density Matrices

Series
High-Dimensional Phenomena in Statistics and Machine Learning Seminar
Time
Tuesday, November 10, 2015 - 15:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 005
Speaker
Dong Xia – Georgia Inst. of Technology, School of Mathematics

Please Note: Joint work with Vladimir Koltchinskii.

The density matrices are positively semi-definite Hermitian matrices of unit trace that describe the state of a quantum system. We develop minimax lower bounds on error rates of estimation of low rank density matrices in trace regression models used in quantum state tomography (in particular, in the case of Pauli measurements) with explicit dependence of the bounds on the rank and other complexity parameters.Such bounds are established for several statistically relevant distances, including quantum versions of Kullback-Leibler divergence (relative entropy distance) and of Hellinger distance (so called Bures distance), and Schatten p-norm distances. Sharp upper bounds and oracle inequalities for least squares estimator with von Neumann entropy penalization are obtained showing that minimax lower bounds are attained (up to logarithmic factors) for these distances.

Exponential varieties

Series
Algebra Seminar
Time
Monday, November 9, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005 or 006
Speaker
Caroline Uhler – MIT
Exponential varieties arise from exponential families in statistics. These real algebraic varieties have strong positivity and convexity properties, generalizing those of toric varieties and their moment maps. Another special class, including Gaussian graphical models, are varieties of inverses of symmetric matrices satisfying linear constraints. We develop a general theory of exponential varieties, with focus on those defined by hyperbolic polynomials. Joint work with Mateusz Michałek, Bernd Sturmfels, and Piotr Zwiernik.

Analytic methods in graph theory

Series
Joint School of Mathematics and ACO Colloquium
Time
Friday, November 6, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Daniel Kral – University of Warwick

Please Note: Refreshments will be served in the atrium after the talk.

The theory of combinatorial limits provides analytic ways of representing large discrete objects. The theory has opened new links between analysis, combinatorics, computer science, group theory and probability theory. In this talk, we will focus on limits of dense graphs and their applications in extremal combinatorics. We will present a general framework for constructing graph limits corresponding to solutions of extremal graph theory problems, which led to constructing counterexamples to several conjectures concerning graph limits. At the end, we will discuss limits of sparse graphs and possible directions to unify the existing approaches related to dense and sparse graphs.

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