Seminars and Colloquia by Series

Non-asymptotic approach in random matrix theory

Series
School of Mathematics Colloquium
Time
Friday, November 16, 2018 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Mark Rudelson – University of Michigan –

Please Note: please note special time!

Random matrices arise naturally in various contexts ranging from theoretical physics to computer science. In a large part of these problems, it is important to know the behavior of the spectral characteristics of a random matrix of a large but fixed size. We will discuss a recent progress in this area illustrating it by problems coming from combinatorics and computer science: Condition number of “full” and sparse random matrices. Consider a system of linear equations Ax = b where the right hand side is known only approximately. In the process of solving this system, the error in vector b gets magnified by the condition number of the matrix A. A conjecture of von Neumann that with high probability, the condition number of an n × n random matrix with independent entries is O(n) has been proven several years ago. We will discuss this result as well as the possibility of its extension to sparse matrices. Random matrices in combinatorics. A perfect matching in a graph with an even number of vertices is a pairing of vertices connected by edges of the graph. Evaluating or even estimating the number of perfect matchings in a given graph deterministically may be computationally expensive. We will discuss an application of the random matrix theory to estimating the number of perfect matchings in a de- terministic graph. Random matrices and traffic jams. Adding another highway to an existing highway system may lead to worse traffic jams. This phenomenon known as Braess’ paradox is still lacking a rigorous mathematical explanation. It was recently explained for a toy model, and the explanation is based on the properties of the eigenvectors of random matrices.

TRIAD Distinguished Lecture Series: Lectures on Combinatorial Statistics

Series
School of Mathematics Colloquium
Time
Thursday, October 25, 2018 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Gábor Lugosi – Pompeu Fabra University, Barcelona
In these lectures we discuss some statistical problems with an interesting combinatorial structure behind. We start by reviewing the "hidden clique" problem, a simple prototypical example with a surprisingly rich structure. We also discuss various "combinatorial" testing problems and their connections to high-dimensional random geometric graphs. Time permitting, we study the problem of estimating the mean of a random variable.

Counting integer points in polytopes

Series
School of Mathematics Colloquium
Time
Thursday, September 27, 2018 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Igor Pak – UCLA
Given a convex polytope P, what is the number of integer points in P? This problem is of great interest in combinatorics and discrete geometry, with many important applications ranging from integer programming to statistics. From computational point of view it is hopeless in any dimensions, as the knapsack problem is a special case. Perhaps surprisingly, in bounded dimension the problem becomes tractable. How far can one go? Can one count points in projections of P, finite intersections of such projections, etc? We will survey both classical and recent results on the problem, emphasizing both algorithmic and complexity aspects. Some elegant hardness results will make an appearance in dimension as small as three. If time permits, we will discuss connections to Presburger Arithmetic and decidability problems for irrational polyhedra. Joint work with Danny Nguyen.

Efficient Network Analysis: Sparsity, Algorithms, and... Colorings!

Series
School of Mathematics Colloquium
Time
Thursday, September 20, 2018 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Blair Sullivan – Department of Computer Science, NC State University
Techniques from structural graph theory hold significant promise for designing efficient algorithms for network science. However, their real-world application has been hampered by the challenges of unrealistic structural assumptions, hidden costs in big-O notation, and non-constructive proofs. In this talk, I will survey recent results which address many of these concerns through an algorithmic pipeline for structurally sparse networks, highlighting the crucial role of certain graph colorings, along with several open problems. For example, we give empirical and model-based evidence that real-world networks exhibit a form of structural sparsity known as "bounded expansion,'' and discuss properties of several low-treedepth colorings used in efficient algorithms for this class. Based on joint works with E. Demaine, J. Kun, M. O'Brien, M. Pilipczuk, F. Reidl, P. Rossmanith, F. Sanchez Villaamil, and S. Sikdar.

TRIAD Distinguished Lecture Series: Sparsity, oracles and inference in high-dimensional statistics

Series
School of Mathematics Colloquium
Time
Tuesday, September 4, 2018 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sara van de Geer – ETH Zurich
The colloquium will be the second lecture of the TRIAD Distinguished Lecture Series by Prof. Sara van de Geer. For further information please see http://math.gatech.edu/events/triad-distinguished-lecture-series-sara-van-de-geer-0.

Mating habits of polynomials

Series
School of Mathematics Colloquium
Time
Wednesday, June 6, 2018 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sarah Koch – U Michigan
Given two complex polynomials, we can try to mathematically paste them together to obtain a rational function through a procedure known as mating the polynomials. In this talk, we will begin by trying to understand the "shape" of complex polynomials in general. We will then discuss the mating of two quadratic polynomials: we explore examples where the mating does exist, and examples where it does not. There will be lots of movies and exploration in this talk.

The weak Pinsker property

Series
School of Mathematics Colloquium
Time
Thursday, April 19, 2018 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tim Austin – UCLA Mathematics Department
This talk is about the structure theory of measure-preserving systems: transformations of a finite measure space that preserve the measure. Many important examples arise from stationary processes in probability, and simplest among these are the i.i.d. processes. In ergodic theory, i.i.d. processes are called Bernoulli shifts. Some of the main results of ergodic theory concern an invariant of systems called their entropy, which turns out to be intimately related to the existence of `structure preserving' maps from a general system to Bernoulli shifts. I will give an overview of this area and its history, ending with a recent advance in this direction. A measure-preserving system has the weak Pinsker property if it can be split, in a natural sense, into a direct product of a Bernoulli shift and a system of arbitrarily low entropy. The recent result is that all ergodic measure-preserving systems have this property. This talk will assume graduate-level real analysis and measure theory, and familiarity with the basic language of random variables. Past exposure to entropy, measure-theoretic probability or ergodic theory will be helpful, but not essential.

The IBM Ponder This monthly challenge

Series
School of Mathematics Colloquium
Time
Tuesday, April 10, 2018 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Prof. Oded Margalit – CTO, IBM Cybersecurity Center of Excellence, Beer Sheva, Israel

Please Note: [CV: Prof. Oded Margalit has a PhD in computer science from Tel Aviv University under the supervision of Prof. Zvi Galil. He has worked at IBM Research – Haifa in the areas of machine learning, constraint satisfaction, verification, and more. Currently, he is the CTO of the IBM Cybersecurity Center of Excellence in Beer Sheva, Israel. Oded helps organize several computer science competitions, like the international IEEEXtreme and the Israeli national CodeGuru competition. He loves riddles and authors the IBM Research monthly challenge corner Ponder This.]

For the sake of puzzle-lovers worldwide, IBM Research offers a monthly mathematical challenge known as Ponder This. Every month, a new challenge is posted together with the solution for the previous month's riddle. Prof. Oded Margalit has served as the Ponder This puzzlemaster for the last decade. In this talk, he’ll survey some of most interesting riddles posted over the years, and tell some anecdotes about various challenges and regular solvers, such as one person who sent in his solution from an intensive care unit. Several challenges have led to conference and journal papers, such as a PRL paper born from a riddle on random walks, and an ITA 2014 paper on a water hose model (using quantum entanglement to break location-based encryption). Other monthly challenges have riffed on games such as 2048, Kakuro, an infinite chess game, the probability of backgammon ending with a double, Fischer Random Chess, and more. Other challenges have been more purely mathematic, focusing on minimal hash functions, combinatorial test design, or finding a natural number n such that round ((1+2 cos(20))^n) is divisible by 10^9. The talk will present a still-open question about a permutation-firing cannon. The talk will be self contained.

The Kannan-Lovasz-Simonovits Conjecture

Series
School of Mathematics Colloquium
Time
Thursday, March 8, 2018 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Santosh Vempala – Georgia Institute of Technology, College of Computing, ISYE, Math
The KLS conjecture says that the Cheeger constant of any logconcave density is achieved to within a universal, dimension-independent constant factor by a hyperplane-induced subset. Here we survey the origin and consequences of the conjecture (in geometry, probability, information theory and algorithms) and present recent progress resulting in the current best bound, as well as a tight bound for the log-Sobolev constant (both with Yin Tat Lee). The conjecture has led to several techniques of general interest.

Non-smooth boundary value problems

Series
School of Mathematics Colloquium
Time
Friday, March 2, 2018 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jill Pipher – Brown University
The regularity properties of solutions to linear partial differential equations in domains depend on the structure of the equation, the degree of smoothness of the coefficients of the equation, and of the boundary of the domain. Quantifying this dependence is a classical problem, and modern techniques can answer some of these questions with remarkable precision. For both physical and theoretical reasons, it is important to consider partial differential equations with non-smooth coefficients. We’ll discuss how some classical tools in harmonic and complex analysis have played a central role in answering questions in this subject at the interface of harmonic analysis and PDE.

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