Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Probability Seminar - Jamie Mingo (Queen's University)

Tuesday, February 6th, 2018

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

Speaker: Jamie Mingo (Queen's University)

Title: The Infinitesimal Law of the GOE

Abstract:  If $X_N$ is the $N \times N$ Gaussian Orthogonal Enemble (GOE) of random matrices, we can expand $\mathrm{E}(\mathrm{tr}(X_N^n))$ as a polynomial in $1/N$, often called a genus expansion. Following the celebrated formula of Harer and Zagier for the GUE, Ledoux (2009) found a five term recurrence for the coefficients of $\mathrm{E}(\mathrm{tr}(X_N^n))$. We show that the coefficient of $1/N$ counts the number of non-crossing annular pairings of a certain type.

Our method is quite elementary. A similar formula holds for the Wishart ensemble. This identification is related to the theory of infinitesimal freeness of Belinschi and Shlyakhtenko.

Free Probability and Random Matrices Seminar Webpage:

Geometry & Representation - David Wehlau (Queen's University)

Monday, February 5th, 2018

Time: 4:30 p.m.  Place: Jeffery Hall 319

Speaker: David Wehlau (Queen's University)

Title: Khovanski Bases and Derivations

Abstract:  Let $R=K[x_1,,\dots,x_m,y_1,\dots,y_m,z_1,\dots,z_m]$ be a polynomial algebera over a field $K$ of characteristic zero, Let $\Delta$ be the locally nilpotent derivation on $R$ determined by $\Delta(z_i) = y_i$, $\Delta(y_i) = x_i$ and $\Delta(x_i)=0$ for $i=1,2,\dots,m$. This is an example of

a Weitzenb\"ock derivation. We exhibit a minimal set of generators $\mathcal G$ for the algebra of

constants $R^\Delta = \ker \Delta$. We also construct a Khovanski (or sagbi) basis for this algebra.

Even though this basis is infinite our proof yields an algorithm to express any element of $R^\Delta$ as
a polynomial in the elements of $\mathcal G$. In particular, this method shows how the classical techniques of polarization and restitution may be used in combination with Khovanski bases to yield a constructive method for expressing elements of a subalgebra as a polynomials in its generators.

Department Colloquium - Jory Griffin (Queen’s University)

Jory Griffin

Friday, February 2nd, 2018

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

Speaker: Jory Griffin, Queen’s University

Title: The Lorentz Gas – Macroscopic Transport from Microscopic Dynamics

Abstract: The Lorentz Gas is microscopic model for conductivity in which a point particle representing an electron moves through an infinite array of scatterers representing the background medium. On the macroscopic scale the dynamics can instead be modelled by the linear Boltzmann transport equation, an irreversible equation where motion of particles appears to be stochastic. How can these two pictures be reconciled? Can we 'derive' the macroscopic picture from the microscopic one? I will talk about the solution to this problem as well as its quantum mechanical analogue where much less is currently known.

Jory Griffin (Queen's University): Jory Griffin received his Ph.D. in Mathematics from the University of Bristol in 2017 under the supervision of Jens Marklof. He recently joined the Department of Mathematics and Statistics at Queen's University as a Coleman Postdoctoral Fellow. Dr. Grin's research focuses on Mathematical Physics, specifically in the quantum propagation of wave packets in the presence of scatterers.

Number Theory - Vaidehee Thatte (Queen's University)

Wednesday, January 31st, 2018

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

Speaker: Vaidehee Thatte (Queen's University)

Title: Defect Extensions – I

Abstract: Let $K$ be a valued field of characteristic $p > 0$ with henselian valuation ring $A$. Let $L$ be a non-trivial Artin-Schreier extension of $K$ with $B$ as the integral closure of $A$ in $L$. In the classical theory of complete discrete valuation rings, $B$ is generated as an $A$-algebra by a single element. This in particular, is not true in the defect case. We will discuss a result that allows us to write $B$ as a "filtered union over $A$", when there is defect. Similar results can be obtained in the mixed characteristic case.

Free Probability Seminar - Neha Prabu (Queen's University)

Tuesday, January 30th, 2018

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

Speaker: Neha Prabu (Queen's University)

Title: Semicircle distribution in number theory, Part II

Abstract:  In free probability theory, the role of the semicircle distribution is analogous to that of the normal distribution in classical probability theory. However, the semicircle distribution also shows up in number theory: it governs the distribution of eigenvalues of Hecke operators acting on spaces of modular cusp forms. In this talk, I will give a brief introduction to this theory of Hecke operators and sketch the proof of a result which is a central limit type theorem from classical probability theory, that involves the semicircle measure.

Free Probability and Random Matrices Seminar Webpage:

Department Colloquium - Milian Derpich (USM, Valparaiso, Chile)

Milan Derpich

Friday, January 26th, 2018

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

Speaker: Milian Derpich, Universidad Técnica Federico Santa Maria, Valparaiso, Chile

Title: The Differential Entropy Gain Created by Linear Time-Invariant Systems

Abstract: The differential entropy of a continuous-valued random variable quantifies the uncertainty associated with the latter, and plays a crucial role in many fundamental result of Information Theory. This talk will discuss how the differential entropy rate of a random process exciting a discrete-time linear time invariant (LTI) system relates to that of the random process coming out of it. First, an apparent contradiction between existing results characterizing the difference between these two differential entropy rates, referred to a 'differential entropy gain', will be exposed. It will then be shown how and when these results can be reconciled, presenting a geometric interpretation as well as novel results which quantify the differential entropy gain introduced by LTI systems. Finally, some of the implications of these results will be illustrated for three different problems, namely: the rate-distortion function for non stationary processes, an inequality in networked control systems, and the capacity of stationary Gaussian channels.

Milan S. Derpich (Universidad Tecnica Federico Santa Maria, Valparaiso, Chile): Milan S. Derpich received the 'Ingeniero Civil Electronico' degree from Federico Santa Maria Technical University, in Valparaso, Chile in 1999. Dr. He then worked by the electronic circuit design and manufacturing company Protonic Chile S.A. between 2000 and 2004. In 2009 he received the PhD degree in electrical engineering from the University of Newcastle, Australia. He received the Guan Zhao-Zhi Award at the Chinese Control Conference 2006, and the Research Higher Degrees Award from the Faculty of Engineering and Built Environment, University of Newcastle, Australia, for his PhD thesis. Since 2009 he has been with the Department of Electronic Engineering at UTFSM, currently as associate professor. His main research interests include rate-distortion theory, networked control systems, and signal processing. He has just started a sabbatical one-year visit to the Department of Mathematics and Statistics in Queen's University, Canada, as a visiting professor.

Number Theory - François Séguin (Queen's University)

Wednesday, January 24th, 2018

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

Speaker: François Séguin (Queen's University)

Title: Heights of elliptic curves and the elliptic analogue of the two-variable Artin conjecture

Abstract: Similar to the way Lang and Trotter adapted Artin's primitive root conjecture in the case of elliptic curves, we consider this natural adaptation for the two-variable Artin Conjecture. In light of our recent results for the two-variable setting, we present similar, unconditional lower bounds for this elliptic analogue.

Free Probability Seminar - Rob Martin (University of Cape Town)

Tuesday, January 23rd, 2018

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

Speaker: Rob Martin (University of Cape Town)

Title: Non-commutative Clark measures for Free and Abelian multi-variable Hardy space

Abstract:  In classical Hardy space theory, there is a natural bijection between the Schur class of contractive analytic functions in the complex unit disk and Aleksandrov- Clark measures on the unit circle. A canonical several-variable analogue of Hardy space is the Drury-Arveson space of analytic functions in the unit ball of d-dimensional complex space, and the canonical non-commuting or free multi- variable analogue of Hardy space is the full Fock space over d-dimensional complex space. Here, the full Fock space is naturally identified with a non- commutative reproducing kernel Hilbert space of free or non-commutative ana- lytic functions acting on a several-variable non-commutative open unit ball. We will extend the concept of Aleksandrov-Clark measure, the bijection between the Schur class and AC measures, Clark’s unitary perturbations of the shift, Lebesgue decomposition formulas and additional related results from one to several commuting and non-commuting variables.

Free Probability and Random Matrices Seminar Webpage:

Department Colloquium - Svetlana Jitomirskaya (UC Irvine)

Svetlana Jitomirskaya

Friday, January 19th, 2018

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

Speaker: Svetlana Jitomirskaya, UC Irvine

Title: Lyapunov exponents, small denominators, arithmetic spectral transitions, and universal hierarchical structure of quasiperiodic eigenfunctions

Abstract: A very captivating question in solid state physics is to determine/understand the hierarchical structure of spectral features of operators describing 2D Bloch electrons in perpendicular magnetic fields, as related to the continued fraction expansion of the magnetic flux. In particular, the hierarchical behavior of the eigenfunctions of the almost Mathieu operators, despite significant numerical studies and even a discovery of Bethe Ansatz solutions has remained an important open challenge even at the physics level. I will present a complete solution of this problem in the exponential sense throughout the entire localization regime. Namely, I will describe, with very high precision, the continued fraction driven hierarchy of local maxima, and a universal (also continued fraction expansion dependent) function that determines local behavior of all eigenfunctions around each maximum, thus giving a complete and precise description of the hierarchical structure. In the regime of Diophantine frequencies and phase resonances there is another universal function that governs the behavior around the local maxima, and a reflective-hierarchical structure of those, a phenomena not even described in the physics literature. These results lead also to the proof of sharp arithmetic transitions between pure point and singular continuous spectrum, in both frequency and phase, as conjectured since 1994. The talk is based on papers joint with W. Liu.

Svetlana Jitomirskaya, (UC Irvine): Svetlana Jitomirskaya earned her Ph.D. in Mathematics from Moscow State University in 1991 under the
supervision of Ya.G. Sinai with a thesis on Spectral and Statistical Properties of Lattice Hamiltonians.  Her awards include the A.P. Sloan Research Fellowship (1996-2000), the AMS Satter Prize (2005), the EPSRC Fellowship at Cambridge University (2008), the Simons Fellowship (2014-2015), and the Aisenstadt Chair at the CRM in Montreal (2018). Prof. Jitomirskaya was also an invited speaker at the 2002 International Congress of Mathematicians in Beijing. She solved (with Artur Avila) the famous Ten Martini Problem in 2009. Her research focuses on Mathematical Physics and Dynamical Systems.