Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Department Colloquium - Qiang Zeng (Northwestern University)

Qiang Zeng, Northwestern University

Wednesday, February 14th, 2018

Time: 3:30 p.m.  Place: Jeffery Hall 234

Speaker: Qiang Zeng, Northwestern University

Title: Replica Symmetry Breaking for Mean Field Spin Glass Models

Abstract: In statistical physics, the study of spin glasses was initialized to describe the low temperature state of a class of magnetic alloys in the 1960s. Since then spin glasses have become a paradigm for highly complex disordered systems. Mean eld spin glass models were introduced as an approximation of the physical short range models in the 1970s. The typical mean eld models include the Sherrington- Kirkpatrick (SK) model, the (Ising) mix p-spin model and the spherical mixed p-spin model. Starting in 1979, the physicist Giorgio Parisi wrote a series of ground breaking papers introducing the idea of replica symmetry breaking (RSB), which allowed him to predict a solution for the SK model by breaking the symmetry of replicas innitely many times at low temperature. This is known as full-step replica symmetry breaking (FRSB). In this talk, we will show that Parisi's FRSB prediction holds at zero temperature for the more general mixed p-spin model. As a consequence, at positive temperature the level of RSB will diverge as the temperature goes to zero. On the other hand, we will show that there exist two-step RSB spherical mixed spin glass models at zero temperature, which are the rst examples beyond the replica symmetric, one-step RSB and FRSB phases. This talk is based on joint works with Antonio Aunger (Northwestern University) and Wei-Kuo Chen (University of Minnesota).

Qiang Zeng (Northwestern University): Qiang Zeng obtained his Ph.D. in Mathematics from the University of Illinois at Urbana-Champaign in 2014 under the supervision of Marius Junge and Renming Song. From 2014 to 2015 he was a Postdoctoral Fellow at Harvard University. In 2015, Dr. Song was a Postdoctoral Fellow at the Mathematical Sciences Research Institute in Berkeley, California. Since 2016, he is Boas Assistant Professor at Northwestern University in Evanston, Illinois. Qiang Zeng works at the interfaces of probability, functional analysis and mathematical physics. His main topic of study is noncommutative probability and spin glasses.

Probability Seminar - Jamie Mingo (Queen's University)

Tuesday, February 13th, 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, Part II

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:

Department Colloquium - Brad Rodgers (University of Michigan)

Brad Rodgers, University of Michigan

Monday, February 12th, 2018

Time: 4:30 p.m.  Place: Jeffery Hall 234

Speaker: Brad Rodgers, University of Michigan

Title: Some Applications of Random Matrix Theory to Analytic Number Theory

Abstract: In this talk I'll survey some of the ways that ideas originating from the study of random matrices have had an impact on analytic number theory. I hope to discuss in particular: 1) the statistical spacing of zeros of the Riemann zeta function, and what this spacing has to say about arithmetic, 2) a resolution of conjectures of Saari and Montgomery about the distribution of Rudin-Shapiro polynomials, using a connection to random walks on compact groups, and 3) recent work on the de Bruijn-Newman constant; de Bruijn showed that the Riemann hypothesis is equivalent to the claim that this constant is less than or equal to 0, and I will describe recent work showing the constant is greater than or equal to 0, conrming a conjecture of Newman. This includes joint work with J. Keating, E. Roditty-Gershon, and Z. Rudnick; and with T. Tao.

Brad Rodgers (University of Michigan): Brad Rodgers obtained his Ph.D. in Mathematics from the University of California, Los Angeles in 2013 under the supervision of Terence Tao. From 2013 to 2015 he held a postdoctoral position at the Institut fur Mathematik at the Universitat Zurich. Since 2015, he is a Postdoc Assistant Professor at the University of Michigan. Dr. Rodgers's awards include the AMS-Simons Travel Grant (2013-2016) and a NSF research grant (2017-2020). His research interests include random matrix theory, analytic number theory. In particular, he focuses on the interaction of these disciplines with analysis, probability, and combinatorics.

Department Colloquium - Daniel Le (University of Toronto)

Daniel Le, University of Toronto

Friday, February 9th, 2018

Time: 3:30 p.m.  Place: Jeffery Hall 234

Speaker: Daniel Le, University of Toronto

Title: The geometry of Galois representations

Abstract: The arithmetic of number fields can be profitably studied through the representation theory of their absolute Galois groups. These representations exhibit a number of elegant and surprising phenomena, most famously the quadratic reciprocity law. Many of these phenomena are explained by the modularity conjecture of Langlands that all Galois representations come from modular forms. Startling progress towards this conjecture began with Taylor and Wiles's study of Galois deformation spaces. We give a construction of local models for some Galois deformation spaces coming from geometric representation theory, and describe some applications to modularity conjectures and congruences between modular forms. Much of what we discuss is joint work with Bao Le Hung, Brandon Levin, and Stefano Morra.

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.

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