Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Curves Seminar - Daniel Erman (Wisconsin-Madison)

Wednesday, February 7th, 2018

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

Speaker: Daniel Erman (Wisconsin-Madison)

Title: Big polynomial rings and Stillman’s conjecture

Abstract: Ananyan–Hochster's recent proof of Stillman's conjecture is based on a key principle: if f_1,.., f_r are sufficiently general forms in a polynomial ring, then as the number of variables tends to infinity, they will behave increasingly like independent variables. We show that this principle becomes a theorem if ones passes to a limit of polynomial rings, using either the inverse limit or the ultraproduct. This yields the surprising fact that these limiting rings are themselves polynomial rings (in uncountably many variables). It also yields two new proofs of Stillman's conjecture. This is joint work with Steven Sam and Andrew Snowden.

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.