Seminars and Colloquia by Series

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.

Folkman Numbers

Series
ACO Student Seminar
Time
Friday, November 6, 2015 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Troy Retter – Emory University
For an integer k, the Folkman number f(k) is the least integer n for which there exists a graph G on n vertices that does not contain a clique of size k and has the property that every two coloring of E(G) yields a monochromatic clique of size of size k. That is, it is the least number of vertices in a K_{k+1}-free graph that is Ramsey to K_k. A recent result of Rodl, Rucinski, and Schacht gives an upper bound on the Folkman numbers f(k) which is exponential in k. A fundamental tool in their proof is a theorem of Saxton and Thomason on hypergraph containers. This talk will give a brief history of the Folkman numbers, introduce the hypergraph container theorem, and sketch the proof of the Rodl, Rucinski, and Schacht result. Recent work with Hiep Han, Vojtech Rodl, and Mathias Schacht on two related problems concerning cycles in graphs and arithmetic progressions in subset of the integers will also be presented.

Ergodic Measures for shifts with eventually constant complexity growth

Series
CDSNS Colloquium
Time
Friday, November 6, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jon Fickenscher – Princeton University –
We will consider (sub)shifts with complexity such that the difference from n to n+1 is constant for all large n. The shifts that arise naturally from interval exchange transformations belong to this class. An interval exchange transformation on d intervals has at most d/2 ergodic probability measures. We look to establish the correct bound for shifts with constant complexity growth. To this end, we give our current bound and discuss further improvements when more assumptions are allowed. This is ongoing work with Michael Damron.

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