Random Matrix & Probability Theory Seminar, Wednesdays

The Random Matrix and Probability Theory Seminar will be every Wednesday from 3pm-4pm in CMSA Building, 20 Garden Street, Room G10.

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The schedule will be updated as details are confirmed.

Date Name Title/Abstract
9-27-17 Herbert Spohn, Technische Universität München Hydrodynamics of integrable classical and quantum systems

Abstract:  In the cold atoms community there is great interest in developing Euler-type hydrodynamics for one-dimensional integrable quantum systems, in particular with application to domain wall initial states.  I provide some mathematical physics background and also compare with integrable classical systems.

10-23-17

*12:00-1:00pm, Science Center 232*

 Madhu Sudan, Harvard SEAS

 

General Strong Polarization

A recent discovery (circa 2008) in information theory called Polar Coding has led to a remarkable construction of error-correcting codes and decoding algorithms, resolving one of the fundamental algorithmic challenges in the field. The underlying phenomenon studies the “polarization” of a “bounded” martingale. A bounded martingale, X_0,…,X_t,…  is one where X_t in [0,1]. This martingale is said to polarize if Pr[lim_{t\to infty} X_t \in {0,1}] = 1. The questions of interest to the results in coding are the rate of convergence and proximity: Specifically, given epsilon and tau > 0 what is the smallest t after which it is the case that Pr[X_t in (tau,1-tau)] < epsilon? For the main theorem, it was crucial that t <= min{O(log(1/epsilon)), o(log(1/tau))}. We say that a martingale polarizes strongly if it satisfies this requirement. We give a simple local criterion on the evolution of the martingale that suffices for strong polarization. A consequence to coding theory is that a broad class of constructions of polar codes can be used to resolve the afore-mentioned algorithmic challenge.

In this talk I will introduce the concepts of polarization and strong polarization.  Depending on the audience interest I can explain why this concept is useful to construct codes and decoding algorithms, or explain the local criteria that help establish strong polarization (and the proof of why it does so).

Based on joint work with Jaroslaw Blasiok, Venkatesan Guruswami, Preetum Nakkiran, and Atri Rudra.

10-25-17

*2:00-4:00pm*

 

Subhabrata Sen (Microsoft and MIT)

Noga Alon,(Tel Aviv University)

 

 

Subhabrata Sen, “Partitioning sparse random graphs: connections with mean-field spin glasses”

Abstract: The study of graph-partition problems such as Maxcut, max-bisection and min-bisection have a long and rich history in combinatorics and theoretical computer science. A recent line of work studies these problems on sparse random graphs, via a connection with mean field spin glasses. In this talk, we will look at this general direction, and derive sharp comparison inequalities between cut-sizes on sparse Erd\ ̋{o}s-R\'{e}nyi and random regular graphs.

Based on joint work with Aukosh Jagannath.

Noga Alon, “Random Cayley Graphs”

Abstract: The study of random Cayley graphs of finite groups is related to the  investigation of Expanders and to problems in Combinatorial Number Theory and in Information Theory. I will discuss this topic, describing the motivation and focusing on the question of estimating the chromatic number of a random Cayley graph of a given  group with a prescribed number of generators.  Several intriguing questions that remain open will be mentioned as well.

11-1-17

*2:00-4:00pm*

 

Kay Kirkpatrick (Illinois)

and

Wei-Ming Wang (CNRS)

 

Kay Kirkpatrick, Quantum groups, Free Araki-Woods Factors, and a Calculus for Moments

 Abstract: We will discuss a central limit theorem for quantum groups: that the joint distributions with respect to the Haar state of the generators of free orthogonal quantum groups converge to free families of generalized circular elements in the large (quantum) dimension limit. We also discuss a connection to free Araki-Woods factors, and cases where we have surprisingly good rates of convergence. This is joint work with Michael Brannan. Time permitting, we’ll mention another quantum central limit theorem for Bose-Einstein condensation and work in progress.

Wei-Min Wang, Quasi-periodic solutions to nonlinear PDE’s

Abstract: We present a new approach to the existence of time quasi-periodic solutions to nonlinear PDE’s. It is based on the method of Anderson localization, harmonic analysis and algebraic analysis. This can be viewed as an infinite dimensional analogue of a Lagrangian approach to KAM theory, as suggested by J. Moser.

11-8-17 Elchanan Mossel Optimal Gaussian Partitions.

Abstract: How should we partition the Gaussian space into k parts in a way that minimizes Gaussian surface area, maximize correlation or simulate a specific distribution.

The problem of Gaussian partitions was studied since the 70s first as a generalization of the isoperimetric problem in the context of the heat equation. It found a renewed interest in context of the double bubble theorem proven in geometric measure theory and due to connection to problems in theoretical computer science and social choice theory.

I will survey the little we know about this problem and the major open problems in the area.

11-10-17

*12pm SC 232*

 

Zhe Wang (NYU)

 

A Driven Tagged Particle in One-dimensional Simple Exclusion Process

Abstract: We study the long-time behavior of a driven tagged particle in a one-dimensional non-nearest- neighbor simple exclusion process.  We will discuss two scenarios when the tagged particle has a speed. Particularly, for the ASEP, the tagged particle can have a positive speed even when it has a drift with negative mean; for the SSEP with removals, we can compute the speed explicitly. We will characterize some nontrivial invariant measures of the environment process by using coupling arguments and color schemes.

11-15-17

*4:00-5:00pm*

*G02*

 

Daniel Sussman (BU)

 

Multiple Network Inference: From Joint Embeddings to Graph Matching

Abstract: Statistical theory, computational methods, and empirical evidence abound for the study of individual networks. However, extending these ideas to the multiple-network framework remains a relatively under-explored area. Individuals today interact with each other through numerous modalities including online social networks, telecommunications, face-to-face interactions, financial transactions, and the sharing and distribution of goods and services. Individually these networks may hide important activities that are only revealed when the networks are studied jointly. In this talk, we’ll explore statistical and computational methods to study multiple networks, including a tool to borrow strength across networks via joint embeddings and a tool to confront the challenges of entity resolution across networks via graph matching.

 

11-20-17

*Monday

12:00-1:00pm*

 Yue M. Lu

(Harvard)

Asymptotic Methods for High-Dimensional Inference: Precise Analysis, Fundamental Limits, and Optimal Designs
Abstract: Extracting meaningful information from the large datasets being compiled by our society presents challenges and opportunities to signal and information processing research. On the one hand, many classical methods, and the assumptions they are based on, are simply not designed to handle the explosive growth of the dimensionality of the modern datasets. On the other hand, the increasing dimensionality offers many benefits: in particular, the very high-dimensional settings allow one to apply powerful asymptotic methods from probability theory and statistical physics to obtain precise characterizations that would otherwise be too complicated in moderate dimensions. I will mention recent work on exploiting such blessings of dimensionality via sharp asymptotic methods. In particular, I will show (1) the exact characterization of a widely-used spectral method for nonconvex signal recoveries; (2) the fundamental limits of solving the phase retrieval problem via linear programming; and (3) how to use scaling and mean-field limits to analyze nonconvex optimization algorithms for high-dimensional inference and learning. In these problems, asymptotic methods not only clarify some of the fascinating phenomena that emerge with high-dimensional data, they also lead to optimal designs that significantly outperform commonly used heuristic choices.
11-29-17 David Gamarink (MIT) (Arguably) Hard on Average Constraint Satisfaction Problems

Abstract: Many combinatorial optimization problems defined on random instances such as random graphs, exhibit an apparent gap between the optimal value, which can be estimated by non-constructive means, and the best values achievable by fast (polynomial time) algorithms. Through a combined effort of mathematicians, computer scientists and statistical physicists, it became apparent that a potential and in some cases a provable obstruction for designing algorithms bridging this gap is an intricate geometry of nearly optimal solutions, in particular the presence of chaos and a certain Overlap Gap Property (OGP), which we will introduce in this talk. We will demonstrate how for many such problems, the onset of the OGP phase transition indeed nearly coincides with algorithmically hard regimes. Our examples will include the problem of finding a largest independent set of a graph, finding a largest cut in a random hypergrah, random NAE-K-SAT problem, the problem of finding a largest submatrix of a random matrix, and a high-dimensional sparse linear regression problem in statistics.

 

Joint work with Wei-Kuo Chen, Quan Li, Dmitry Panchenko,  Mustazee Rahman, Madhu Sudan and Ilias Zadik.

12-6-17

*2:00-4:00pm*

Philippe Rigollet (MIT)

&

Ankur Moitra (MIT)

Ankur Moitra, A New Approach to Approximate Counting and Sampling 

Abstract: Over the past sixty years, many remarkable connections have been made between statistical physics, probability, analysis and theoretical computer science through the study of approximate counting. While tight phase transitions are known for many problems with pairwise constraints, much less is known about problems with higher-order constraints.
Here we introduce a new approach for approximately counting and sampling in bounded degree systems. Our main result is an algorithm to approximately count the number of solutions to a CNF formula where the degree is exponential in the number of variables per clause. Our algorithm extends straightforwardly to approximate sampling, which shows that under Lovasz Local Lemma-like conditions, it is possible to generate a satisfying assignment approximately uniformly at random. In our setting, the solution space is not even connected and we introduce alternatives to the usual Markov chain paradigm.

12-14-17 TBD

 

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