Department of Mathematics and Statistics

Thursday, April 5th, 2018

Time: 5:30 p.m.  Place: Jeffery Hall 127

Speaker: Amie Wilkinson (University of Chicago)

Title: The Mathematics of Deja Vu

Abstract: Dynamics is an area of mathematics concerned with the motion of spaces (" dynamical systems") over time. Dynamics has its roots in the late nineteenth century, when it was developed as a tool to understand physical phenomena, such as the motion of gas molecules in a box and planets around the sun. A simple and yet powerful concept in dynamics is that of recurrence. In everyday language, recurrence is the mathematical version of deja vu: a motion of a space is recurrent if, given enough time, it eventually returns to its original configuration (allowing for a small amount of error). In this talk, I will describe how mathematical results about recurrence can be used to answer surprisingly disparate questions, from the mixing and unmixing of two ideaI gases in a box, to deep properties of the prime numbers, to the discovery of exoplanets in nearby solar systems.

Amie Wilkinson (University of Chicago): Prof Wilkinson (University of Chicago) is a leading researcher in ergodic theory and dynamical systems. Among other recognitions of her many achievements, she was an invited speaker at the International Congress of Mathematicians in 2010, was awarded the Satter Prize in Mathematics in 2011, and was elected a fellow of the AMS in 2014. Prof Wilkinson has also been very active in public outreach, giving numerous public lectures, interviews, and recently publishing an article in the New York Times.

Tuesday, April 3rd, 2018

Time: 3:30-5:00 p.m.  Place: Jeffery Hall 319

Speaker: Pei-Lun Tseng (Queen's University)

Title: Linearization trick of infinitesimal freeness II

Abstract:  Last week, we introduced how to find a linearization for a given selfadjoint polynomial and showed some properties of this linearization. We will continue our discussion this week and introduce the operator-valued Cauchy transform. Then, we will show the algorithm for finding the distribution of $P$ where $P$ is a selfadjoint polynomial with selfadjoint variables $X$ and $Y$. Based on this method, we will discuss how to extend this algorithm for finding infinitesimal distribution for $P$.

Free Probability and Random Matrices Seminar Webpage:

Wednesday, March 28th, 2018

Time: 3:30 p.m.  Place: Jeffery Hall 319

Speaker: Mike Zabrocki (York University)

Title: A Multiset Partition Algebra

Abstract:  Schur-Weyl duality is a statement about the relationship between the actions of the general linear group $Gl_n$ and the symmetric group $S_k$ when these groups act on $V_n^{\otimes k}$ (here $V_n$ is an $n$ dimensional vector space). If we consider the symmetric group $S_n$ as permutation matrices embedded in $Gl_n$, then the partition algebra $P_k(n)$ (introduced by Martin in the 1990's) is the algebra which commutes with the action of $S_n$.

In this talk I will explain how an investigation of characters of the symmetric group leads us to consider analogues of the RSK algorithm involving multiset tableaux. To explain the relationship of the combinatorics to representation theory we were led to the multiset partition algebra as an analogue of the partition algebra and the dimensions of the irreducible representations are the numbers of multiset tableaux.

This is joint work with Rosa Orellana of Dartmouth College.

Tuesday, March 27th, 2018

Time: 3:00 p.m.  Place: Jeffery Hall 422

Speaker: Silas Fong (University of Toronto)

Title: Strong converse theorems for multimessage networks with tight cut-set bound

Abstract:  In Shannon’s seminal work that established the maximum coding rate of point-to-point communication, it was shown that communicating reliably over a noisy medium is possible as long as the coding rate is below the capacity, i.e., the Shannon’s limit. Conversely, no reliable communication can be supported for any coding rate above the capacity. For communication engineers, this leaves open the possibility of the following tradeoff between coding rate and error probability: Communicating at a rate above the Shannon’s limit while tolerating a non-zero error probability. In this talk, we focus on various multi-user communication systems where this tradeoff does not exist, i.e., there is a sharp phase transition of the performance of the system (quantified by the error probability) between rates below the Shannon’s limit which can be supported for reliable communication and rates above the Shannon’s limit that must lead to catastrophic failure of communication. In this case, we say that a strong converse exists for the system.

In the first part of my talk, I will briefly discuss the latest development of strong converse results for several common multi-terminal systems including the multiple access channel (MAC), the broadcast channel (BC) and the relay channel.  The second part of this talk will cover my recent result which proves a strong converse theorem for any multimessage network with tight cut-set bound. In particular, the result yields the first strong converse theorem for the degraded relay channel. A proof sketch based on the method of types will be presented. The Gaussian version of this result yields the first strong converse theorem for the Gaussian MAC with feedback.

Biography:  Silas L. Fong is currently a postdoctoral fellow with the Department of Electrical and Computer Engineering at University of Toronto. He received his B.Eng., M.Phil. and Ph.D. degrees in Information Engineering from the Chinese University of Hong Kong in 2005, 2007 and 2011 respectively. He has performed postdoctoral research at City University of Hong Kong from 2011 to 2013, at Cornell University from 2013 to 2014, and at National University of Singapore from 2014 to 2017. His research interests include information theory and its applications to communication networks such as relay networks, wireless networks, and energy-harvesting channels.

Friday, March 23rd, 2018

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Mihai Nica (University of Toronto)

Title: Phase transitions in random matrices and the spiked tensor model

Abstract: Given a matrix of noisy data, principal component analysis (PCA) can be viewed as "de-noising" technique that recovers the closest rank-one approximation. In certain matrix models, it is known that this procedure exhibits a phase transition: if the signal-to-noise ratio is below a critical value then PCA returns uninformative information. In this talk, we also consider a generalization of this problem to k-tensors (the matrix case corresponds to k=2). By studying the energy landscape of this model, we also find phase transitions akin to the matrix case. The proof of the results uses the Kac-Rice formula for the expected number of critical points of a random function and results about spiked random matrices. Based on joint work with Gerard Ben Arous, Song Mei and Andrea Montanari.