Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Number Theory - Siddhi Pathak (Queen's University)

Wednesday, March 28th, 2018

Time: 2:15 p.m.  Place: Jeffery Hall 319

Speaker: Siddhi Pathak (Queen's University)

Title: On the primitivity of Dirichlet characters.

Abstract: A Dirichlet character modulo q is said to be imprimitive if it is induced from a lower level. A characterization of the primitivity of characters is the separability of the Gauss sum ( Fourier transform of $\chi$ ), i.e., $G_q(n, \bar{chi}) = \chi(n) G_q(1, \bar{ \chi } )$ for all n. In this talk, we discuss a paper of R. Daielda and N. Jones in which they introduce another way of extending primitive Dirichlet characters so that the above separability property holds even for imprimitive characters.

Information Theory - Silas Fong (University of Toronto)

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.

Probability Seminar - Pei-Lun Tseng (Queen's University)

Tuesday, March 27th, 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

Abstract:  For given infinitesimal distribution of selfadjoint elements $X$, $Y$, and given a selfadjoint polynomial $P$ with variable $X$ and $Y$. The natural question is whether we can write down the precise formula for the infinitesimal distribution of $P$? In 2009 Belinschi and Shlyakhtenko gave a precise formula to solve for the infinitesimal distribution of $P$ for $P(X,Y)=X+Y$. In the talk, we will discuss how to find the formula for an arbitrary polynomial by using the linearization trick.

Free Probability and Random Matrices Seminar Webpage:

Department Colloquium - Mihai Nica (University of Toronto)

Mihai Nica, University of Toronto

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.

Math Club - Tianyuan Xu (Queen's University)

Thursday, March 22nd, 2018

Time: 5:30 p.m.  Place: Jeffery Hall 118

Speaker: Tianyuan Xu (Queen's University)

Title: Knot Invariants by Pulling Strings.

Abstract: A knot is a smooth closed curve in the 3-dimensional space, and two knots are equivalent if they can be distorted into each other without cutting or gluing. How can we tell whether two knots are equivalent? For example, is the trefoil equivalent to its mirror image? This turns out to be a highly non-trivial problem, and one approach to solving it is to study so-called knot invariants, quantities associated to knots that remain unchanged under equivalences.

We will discuss a knot invariant called the Kauffman bracket of knots. While knots are inherently 3-dimensional objects, in developing this invariant we will simplify their study to the study of 2-dimensional objects called non-crossing pairings. To achieve this, we will need to literally "pull some strings"!

Number Theory - Francois Seguin (Queen's University)

Wednesday, March 21st, 2018

Time: 2:15 p.m.  Place: Jeffery Hall 319

Speaker: François Séguin (Queen's University)

Title: Frequency of primes dividing n and $\Phi_n$.

Abstract: In a previous talk, we have seen that at most one prime Pn, can divide both an integer n and the nth cyclotomic polynomial evaluated at some integer a. During this talk, we will use methods in Dirichlet series to investigate how often Pn actually divides the nth cyclotomic polynomial $\Phi_n$.

Probability Seminar - Mihai Popa (University of Texas, San Antonio)

Tuesday, March 20th, 2018

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

Speaker: Mihai Popa (University of Texas, San Antonio)

Title: Permutations of Entries and Asymptotic Free Independence for Gaussian Random Matrices

Abstract:  Since the 1980's, various classes of random matrices with independent entries were used to approximate free independent random variables. But asymptotic freeness of random matrices can occur without independence of entries: in 2012, in a joint work with James Mingo, we showed the (then) surprising result that unitarily invariant random matrices are asymptotically (second order) free from their transpose. And, in a more recent work, we showed that Wishart random matrices are asymptotically free from some of their partial transposes. The lecture will present a development concerning Gaussian random matrices. More precisely, it will describe a rather large class of permutations of entries that induces asymptotic freeness, suggesting that the results mentioned above are particular cases of a more general theory.

Free Probability and Random Matrices Seminar Webpage:

Math Club - Greg Smith (Queen's University)

Thursday, March 15th, 2018

Time: 5:30 p.m.  Place: Jeffery Hall 118

Speaker: Greg Smith (Queen's University)

Title: Realistic Expectations.

Abstract: How many real roots should we expect a real polynomial to have? In this talk, we will convert this vague question into a well-posed mathematical problem. With the help of geometry, we will also provide the surprisingly beautiful solution.