Seminars and Colloquia Schedule

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.

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

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.

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.

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.

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.

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.

Recovery of High-Dimensional Low-Rank Matrices

Series
Stochastics Seminar
Time
Thursday, November 12, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tony Cai – Wharton School, University of Pennsylvania
Low-rank structure commonly arises in many applications including genomics, signal processing, and portfolio allocation. It is also used in many statistical inference methodologies such as principal component analysis. In this talk, I will present some recent results on recovery of a high-dimensional low-rank matrix with rank-one measurements and related problems including phase retrieval and optimal estimation of a spiked covariance matrix based on one-dimensional projections. I will also discuss structured matrix completion which aims to recover a low rank matrix based on incomplete, but structured observations.

On the linear span of lattice points in a parallelepiped

Series
ACO Student Seminar
Time
Friday, November 13, 2015 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Marcel Celaya – Georgia Tech
We find a good characterization for the following problem: Given a rational row vector c and a lattice L in R^n which contains the integer lattice Z^n, do all lattice points of L in the half-open unit cube [0,1)^n lie on the hyperplane {x in R^n : cx = 0}? This work generalizes a theorem due to G. K. White, which provides sufficient and necessary conditions for a tetrahedron in R^3 with integral vertices to have no other integral points. Our approach is based on a novel proof of White's result using number-theoretic techniques due to Morrison and Stevens. In this talk, we illustrate some of the ideas and describe some applications of this problem.

Signrank and its applications in combinatorics and complexity

Series
ACO Colloquium
Time
Friday, November 13, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Noga Alon – Tel Aviv University and IAS, Princeton

Refreshments will be served in the atrium after the talk.

The sign-rank of a real matrix A with no 0 entries is the minimum rank of a matrix B so that A_{ij}B_{ij} >0 for all i,j. The study of this notion combines combinatorial, algebraic, geometric and probabilistic techniques with tools from real algebraic geometry, and is related to questions in Communication Complexity, Computational Learning and Asymptotic Enumeration. I will discuss the topic and describe its background, several recent results from joint work with Moran and Yehudayoff, and some intriguing open problems.

ALS - Non-constructive combinatorics

Series
Other Talks
Time
Saturday, November 14, 2015 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Noga Alon – Tel Aviv University and IAS, Princeton

First featured lecture in the Atlanta Lecture Series in Combinatorics and Graph Theory mini-conference

I will describe several old and new applications of topological and algebraic methods in the derivation of combinatorial results. In all of them the proofs provide no efficient solutions for the corresponding algorithmic problems.

ALS - Coloring and girth

Series
Other Talks
Time
Sunday, November 15, 2015 - 09:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Noga Alon – Tel Aviv University and IAS, Princeton

Second featured lecture in the Atlanta Lecture Series in Combinatorics and Graph Theory mini-conference

The study of graphs with high girth and high chromatic number had a profound influence on the history of Combinatrics and Graph Theory, and led to the development of sophisticated methods involving tools from probability, topology, number theory, algebra and combinatorics. I will discuss the topic focusing on a recent new explicit construction of graphs (and hypergraphs) of high girth and high chromatic number, in joint work with Kostochka, Reiniger, West and Zhu.