# Mathematics

## Limit Theorems for the Frontier of One-Dimensional Branching Diffusions

Speaker:

Thomas Sellke
Date:

Thu, Nov 24, 2016
Location:

PIMS, University of Manitoba
Conference:

PIMS-UManitoba Distinguished Lecture Abstract:

This talk will discuss results in a joint paper of mine with Steve Lalley from 1992.
Suppose a particle, starting at position 0, moves according to a diffusion process on the real line. Suppose also that this particle emits daughter particles according to a branching process whose instantaneous rate can depend on location, though not on time. The daughter particles move independently according to the same diffusion process, starting at their points of birth, and in turn emit their own daughters according to the same branching process. The simplest special case of this situation is standard branching Brownian motion, with the rate of reproduction not depending on location.
Let R_t be the position of the right-most particle at time t, and let m_t be the median of R_t. In 1937, Kolmogorov, Petrovskii, and Piskunov showed that, for standard branching Brownian motion, (m_t)/ t converges to SQRT(2) and that (R_t - m_t) converges in distribution to a nondegenerate limiting distribution.It turns out that results like those proved by Kolmogorov, et al, hold in great generality for one-dimensional branching diffusions. If the branching diffusion is "recurrent" (in the sense that the initial position is re-visited at arbitrarily large times by _some_ particle), and if space is rescaled so that m_t grows linearly, then (R_t - m_t) converges in distribution to a location-mixture of extreme value distributions. We also have the Andy Warhol Theorem, according to which every particle ever born has a descendant in the lead at some point in the future.

## Asynchronous Consensus

Speaker:

Faith Ellen
Date:

Thu, Oct 20, 2016
Location:

PIMS, University of Manitoba
Conference:

PIMS-UManitoba Distinguished Lecture Abstract:

The consensus problem plays a central role in the theory of distributed computing. I will prove that consensus is impossible to solve in some asynchronous shared memory systems and I will present some algorithms for solving it in others, together with matching lower bounds on the amount of time and space needed. Consensus is universal: using consensus and read/write registers, I will show how to implement any shared object. The consensus hierarchy is used to classify the computational power of shared objects. I will conclude by discussing some limitations of this classification that have been recently discovered.

## On De Giorgi Conjecture and Beyond

Speaker:

Jun-Cheng Wei
Date:

Thu, Sep 29, 2016
Location:

PIMS, University of Manitoba
Conference:

PIMS-UManitoba Distinguished Lecture Abstract:

Classifying solutions is one of central themes in nonlinear partial differential equations. This is the content of various Liouville type theorems and more recently De Giorgi type conjectures. I will report recent progress towards De Giorgi's conjecture for Allen-Cahn equation and free boundary problems, and related issues in nonlinear Schroedinger equations.

## Extrema of 2D Discrete Gaussian Free Field - Lecture 16

Speaker:

Marek Biskup
Date:

Sun, Jul 30, 2017
Location:

PIMS, University of British Columbia
Conference:

PIMS-CRM Summer School in Probability 2017 Abstract:

The Gaussian free field (GFF) is a fundamental model for random fluctuations of a surface. The GFF is closely related to local times of random walks via relations that originated in the study of spin systems. The continuous GFF appears as the limit law of height functions of dimer covers, uniform spanning trees and other models without apparent Gaussian correlation structure. The GFF is also a simple example of a quantum field theory. Intriguing connections to SLE, the Brownian map and other recently studied problems exist. The GFF has recently become subject of focused interest by probabilists. Through Kahane's theory of multiplicative chaos, the GFF naturally enters into models of Liouville quantum gravity. Multiplicative chaos is also central to the description of level sets where the GFF takes values proportional to its maximum, or values order-unity away from the absolute maximum. Random walks in random environments given as exponentials of the GFF show intriguing subdiffusive behavior. Universality of these conclusions for other models such as gradient systems and/or local times of random walks are within reach.

## Graphical approach to lattice spin models - Lecture 16

Speaker:

Hugo Duminil-Copin
Date:

Fri, Jun 30, 2017
Location:

PIMS, University of British Columbia
Conference:

PIMS-CRM Summer School in Probability 2017 Abstract:

Phase transitions are a central theme of statistical mechanics, and of probability more generally. Lattice spin models represent a general paradigm for phase transitions in finite dimensions, describing ferromagnets and even some fluids (lattice gases). It has been understood since the 1980s that random geometric representations, such as the random walk and random current representations, are powerful tools to understand spin models. In addition to techniques intrinsic to spin models, such representations provide access to rich ideas from percolation theory. In recent years, for two-dimensional spin models, these ideas have been further combined with ideas from discrete complex analysis. Spectacular results obtained through these connections include the proof that interfaces of the two-dimensional Ising model have conformally invariant scaling limits given by SLE curves, that the connective constant of the self-avoiding walk on the hexagonal lattice is given by \u221a 2 + \u221a 2 , and that the magnetisation of the three-dimensional Ising model vanishes at the critical point.

## Graphical approach to lattice spin models - Lecture 15

Speaker:

Hugo Duminil-Copin
Date:

Thu, Jun 29, 2017
Location:

PIMS, University of British Columbia
Conference:

PIMS-CRM Summer School in Probability 2017 Abstract:

## Extrema of 2D Discrete Gaussian Free Field - Lecture 15

Speaker:

Marek Biskup
Date:

Thu, Jun 29, 2017
Location:

PIMS, University of British Columbia
Conference:

PIMS-CRM Summer School in Probability 2017 Abstract:

## Graphical approach to lattice spin models - Lecture 14

Speaker:

Hugo Duminil-Copin
Date:

Tue, Jun 27, 2017
Location:

PIMS, University of British Columbia
Conference:

PIMS-CRM Summer School in Probability 2017 Abstract:

Phase transitions are a central theme of statistical mechanics, and of probability more generally. Lattice spin models represent a general paradigm for phase transitions in finite dimensions, describing ferromagnets and even some fluids (lattice gases). It has been understood since the 1980s that random geometric representations, such as the random walk and random current representations, are powerful tools to understand spin models. In addition to techniques intrinsic to spin models, such representations provide access to rich ideas from percolation theory. In recent years, for two-dimensional spin models, these ideas have been further combined with ideas from discrete complex analysis. Spectacular results obtained through these connections include the proof that interfaces of the two-dimensional Ising model have conformally invariant scaling limits given by SLE curves, that the connective constant of the self-avoiding walk on the hexagonal lattice is given by √ 2 + √ 2 , and that the magnetisation of the three-dimensional Ising model vanishes at the critical point.

## Extrema of 2D Discrete Gaussian Free Field - Lecture 14

Speaker:

Marek Biskup
Date:

Tue, Jun 27, 2017
Location:

PIMS, University of British Columbia
Conference:

PIMS-CRM Summer School in Probability 2017 Abstract:

## Extrema of 2D Discrete Gaussian Free Field - Lecture 13

Speaker:

Marek Biskup
Date:

Mon, Jun 26, 2017
Location:

PIMS, University of British Columbia
Conference:

PIMS-CRM Summer School in Probability 2017 Abstract: