# PIMS lectures

Title: Discrete Gaussian Free Field and Invariant Measures for the Non-Linear Schrödinger Equation

Speaker: Kesav Krishnan, University of Victoria

Date and time:
20 Mar 2024,
9:30am -
10:30am

Location: via Zoom registration required

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For more information and registration: https://www.pims.math.ca/seminars/PIMSPDF.

ABSTRACT:

In this talk, I will describe how certain non-linear
exponential tilts of the Discrete Gaussian Free Field yield
invariant measures for a discretized PDE, the focusing Non
Linear Schrödinger equation. I will briefly talk about how this
tilted measure relates to self-intersections of the random
walk on $\mathbb{Z}^{d}$. I will then describe joint work
with Partha Dey and Kay Kirkpatrick on how this measure
undergoes a phase transition in the thermodynamic limit
with general $d\geq 3$ and non-linearity $p$. I will conclude
with some new results with Gourab Ray, on the weak
convergence of the random field sampled with respect to
this measure to the massive discrete Gaussian Free Field.

SPEAKER BIO:

Kesav is a PIMS postdoctoral fellow at the University
of Victoria who works with Gourab Ray. Kesav’s
academic trajectory began as an undergraduate
student in physics at the University of Delhi. After
realizing that mathematical methods (specifically
probability) excited him more, Kesav switched
track and completed his masters in mathematics
at the Indian Statistical Institute, Bangalore.
As a PhD student at the University of Illinois,
Urbana-Champaign, Kesav focused on problems
in probability that have origins in statistical and
mathematical physics. He continues to work in
this field, specifically on phase transitions in
lattice field theories and disordered models in
statistical mechanics. He is thrilled to have gotten
the opportunity to spend two years in the Pacific
Northwest and explore its spectacular natural
beauty.

Download poster (PDF).

Title: PIMS Network-Wide Colloquium Series - Complex dynamics: the real story

Speaker: Sarah Koch, University of Michigan

Date and time:
29 Feb 2024,
1:30pm -
2:30pm

Location: via Zoom Registration required

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Abstract:

We will introduce the research area of complex dynamics by studying the family of (complex) quadratic polynomials, $z->z^2+c$. Historically, mathematicians first studied the dynamics of {\em real} quadratic polynomials and then moved to the complex realm. Recently, there has been a resurgence of mathematical activity surrounding real quadratic polynomials. We will explore some well known features and some rather new features of real polynomials in contrast with those enjoyed by their complex counterparts.

Speaker Biography:

Sarah Koch is a an Arthur F. Thurnau Professor in the Department of Mathematics at the University of Michigan. Her research incorporates topology, algebraic geometry, complex analysis, and Teichmueller theory to better understand complex dynamical systems (in one and several variables) and their associated dynamical moduli spaces from both analytic and algebraic points of view.

Click here to Register for the Series.

Title: PIMS Network-wide Colloquium: Hamilton-Jacobi equations on the Wasserstein space on graphs

Speaker: Wilfrid Gangbo, UCLA

Date and time:
25 Jan 2024,
1:30pm -
2:30pm

Location: via Zoom registration required

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Description:
We consider metric tensors on undirected weighted graphs G, which allows us to treat P(G), the set of probability vectors on G, as a length space. On defines a divergence operator div_\mu(G) for mu in P(G), in such a way that we can use control vectors m to define paths s:[0,T] \to P(G), satisfying the system of ODEs: d\sigma/dt + div_G(m) + \hbar div_\sigma(\nabla_G log \sigma)=0. These paths serve as characteristics for Hamilton-Jacobi equations involving graph-individual noise operators. We propose a well posedness theory on P(G). (This talk is based on a joint work with C. Mou and A. Swiech)

Other Information:
Time:
All network wide colloquia take place at 1:30pm Pacific Time with a few exceptions.

Registration:

Participants register once on Zoom and can attend any of the Colloquium talks. Please remember to download the calendar information to save the dates on your calendar. PIMS will resend the confirmation from Zoom prior to the event date.

Title: PIMS Emergent Research PDF Seminar: Spaces of geodesic triangulations of surfaces

Speaker: Yanwen Luo, SFU and UVic

Date and time:
29 Nov 2023,
9:30am -
10:30am

Location: via Zoom registration required

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Register for Zoom link.

In 1962, Tutte proposed a simple method to produce a straight-line embedding of a planar graph in the plane, known as Tutte’s spring theorem. It leads to a surprisingly simple proof of a classical theorem proved by Bloch, Connelly, and Henderson in 1984, which states that the space of geodesic triangulations of a convex polygon is contractible. In this talk, I will introduce spaces of geodesic triangulations of surfaces, review Tutte’s spring theorem, and present this short proof. It time permits, I will briefly report the recent progress in identifying the homotopy types of spaces of geodesic triangulations of general surfaces.

Title: PIMS Emergent Research PDF Seminar: Around Artin's primitive root conjecture

Speaker: Paul Péringuey, University of British Columbia

Date and time:
22 Nov 2023,
9:30am -
10:30am

Location: via Zoom registration required

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Abstract: In this talk we will first discuss this soon to be 100 years old conjecture, which states that the set of primes for which an integer $a$ different from $-1$ or a perfect square is a primitive root admits an asymptotic density among all primes. In 1967 Hooley proved this conjecture under the Generalized Riemann Hypothesis.

After that, we will look into a generalization of this conjecture, where we don't restrain ourselves to look for primes for which $a$ is a primitive root but instead elements of an infinite subset of $\N$ for which $a$ is a generalized primitive root. In particular, we will take this infinite subset to be either $\N$ itself or integers with few prime factors.

Speaker Biography: Paul Péringuey is a PIMS Postdoctoral fellow at the University of British Columbia where he is working with Prof. Greg Martin, under the sponsorship of the PIMS Collaborative Research Group "L-Functions in Analytic Number Theory". He obtained his PhD in 2022 at Université de Lorraine under the supervision of Cécile Dartyge. His area of research is in Analytic Number Theory, Comparative Prime Number Theory, as well as Additive Combinatorics.

Poster: The poster for the talk is available.

Title: PIMS Network-wide Colloquium: AI4Crypto

Speaker: Kristin Lauter, Meta

Date and time:
16 Nov 2023,
1:30pm -
2:30pm

Location: via Zoom registration required

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Speaker Bio:

Kristin Lauter is an American mathematician and cryptographer whose research interest is broadly in application of number theory and algebraic geometry in cryptography. She is particularly known for her work in the area of elliptic curve cryptography. She was a researcher at Microsoft Research in Redmond, Washington, from 1999 - 2021, and the head of the Cryptography Group from 2008 - 2021; her group developed Microsoft SEAL. In April 2021, Lauter joined Facebook AI Research (FAIR) as the West Coast Head of Research Science. She became the President-Elect of the Association for Women in Mathematics in February 2014 and served as President from 2015 - 2017.

Registration:

Participants register once on Zoom and can attend any of the Colloquium talks. Please remember to download the calendar information to save the dates on your calendar. PIMS will resend the confirmation from Zoom prior to the event date.

Title: PIMS Network-wide Colloquium: Surface sums and Yang-Mills gauge theory

Speaker: Scott Sheffield, MIT

Date and time:
19 Oct 2023,
1:30pm -
2:30pm

Location: via Zoom registration required

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Constructing and understanding the basic properties of Euclidean Yang-Mills theory is a fundamental problem in physics. It is also one of the Clay Institute's famous Millennium Prize problems in mathematics. The basic problem is not hard to understand. You can begin by describing a simple random function from a set of lattice edges to a group of matrices. Then you ask whether you can construct/understand a continuum analog of this object in one way or another. In addition to a truly enormous physics literature, this topic has inspired research within many major areas of mathematics: representation theory, random matrix theory, probability theory, differential geometry, stochastic partial differential equations, low-dimensional topology, graph theory and planar-map combinatorics.

Attempts to understand this problem in the 1970's and 1980's helped inspire the study of "random surfaces" including Liouville quantum gravity surfaces. Various relationships between Yang-Mills theory and random surface theory have been obtained over the years, but many of the most basic questions have remained out of reach. I will discuss our own recent work in this direction, as contained in two long recent papers relating "Wilson loop expectations" (the fundamental objects in Yang-Mills gauge theory) to "sums over spanning surfaces."

1. Wilson loop expectations as sums over surfaces on the plane (joint with Minjae Park, Joshua Pfeffer, Pu Yu)

2. Random surfaces and lattice Yang-Mills (joint with Sky Cao, Minjae Park)

The first paper explains how in 2D (where Yang-Mills theory is more tractable) one can interpret continuum Wilson loop expectations purely in terms of flat surfaces. The second explains a general-dimensional interpretation of the Wilson loop expectations in lattice Yang-Mills theory in terms of discrete-and not-necessarily-flat surfaces, a.k.a. embedded planar maps.

Speaker Biography:
Scott Sheffield is the Leighton Family Professor of Mathematics at MIT. He is a leading figure at the interface of mathematical physics and probability. He has held positions at Microsoft Research, Berkeley, the Institute for Advanced Study and New York University. He received the Rollo Davidson and Loève prizes in probability and has twice spoken at the International Congress of Mathematicians