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Some Gregory Lawler Lectures
In his CV Gregory Lawler lists 111 invited lectures or mini-courses from March 2010 to February 2023 which he delivered in 22 countries: Austria, Brazil, Canada, China, Costa Rica, Czech Republic, England, Finland, France, Germany, Israel, Italy, Japan, Korea, Mexico, Peru, Romania, Russia, USA, Singapore, Sweden, and Switzerland. We give seven examples below.
2013 Conformal Geometry Program Lecture Series by Greg Lawler.
Venue: Simons Center for Mathematics and Physics, Stony Brook University.
Title: Minkowski Content and the Schramm-Loewner Evolution.
Abstract: The d-dimensional Minkowski content of a bounded set in the complex plane is defined to be the the limit as r goes to zero of r^{d-2} times the area of points within distance r of the set. We show that the Minkowski content of a Schramm-Loewner evolution curve with kappa < 8 exists and coincides with the natural parametrisation of SLE curves as defined previously. We will sketch the tools of the proof and then we will make the case for why this is the "correct" way to parametrise the curves. This is joint work with Mohammad Rezaei.
2014 13th Annual Thomas Wolff Memorial Lecture in Mathematics by Gregory F Lawler.
Title: Minkowski Content and the Schramm-Loewner Evolution.
Abstract: The d-dimensional Minkowski content of a bounded set in the complex plane is defined to be the the limit as r goes to zero of r^{d-2} times the area of points within distance r of the set. We show that the Minkowski content of a Schramm-Loewner evolution curve with kappa < 8 exists and coincides with the natural parametrisation of SLE curves as defined previously. We will sketch the tools of the proof and then we will make the case for why this is the "correct" way to parametrise the curves. This is joint work with Mohammad Rezaei.
Venue: California Institute of Technology.
Title (3 lectures): Random Walks and their Scaling Limits.
Abstract: The simple random walk on the integer lattice is well understood as well as its scaling limit, Brownian motion. However, there are a number of models of random walks with strong interactions for which we are still trying to determine the behaviour. My first talk will be a survey of the state of knowledge for three problems: intersections of random walk, loop-erased random walk, and self-avoiding random walk.
There has been considerable work in the last twenty years on the planar case and I will spend the second two lectures discussing aspect of this. In the second lecture, I will consider a particular case, the loop-erased random walk, and describe its relationship to other models in particular spanning trees, determinants of the Laplacian and the random walk loop measures. If time allows. I will discuss a recent result with C Benes and F Viklund about the probability that a loop-erased walk goes though a point.
The third talk will focus on the continuum limit of many of these walks, the Schramm-Loewner evolution (SLE). Many of the properties SLE can be seen as continuous analogues of properties of the loop-erased walk and I will discuss some of them including recent work on fractal properties with a number of coauthors.
2015 Stockholms Matematikcentrum Colloquium by Gregory F Lawler.
Title (3 lectures): Random Walks and their Scaling Limits.
Abstract: The simple random walk on the integer lattice is well understood as well as its scaling limit, Brownian motion. However, there are a number of models of random walks with strong interactions for which we are still trying to determine the behaviour. My first talk will be a survey of the state of knowledge for three problems: intersections of random walk, loop-erased random walk, and self-avoiding random walk.
There has been considerable work in the last twenty years on the planar case and I will spend the second two lectures discussing aspect of this. In the second lecture, I will consider a particular case, the loop-erased random walk, and describe its relationship to other models in particular spanning trees, determinants of the Laplacian and the random walk loop measures. If time allows. I will discuss a recent result with C Benes and F Viklund about the probability that a loop-erased walk goes though a point.
The third talk will focus on the continuum limit of many of these walks, the Schramm-Loewner evolution (SLE). Many of the properties SLE can be seen as continuous analogues of properties of the loop-erased walk and I will discuss some of them including recent work on fractal properties with a number of coauthors.
Venue: Oskar Klein lecture hall at Albanova.
Title: Self-avoiding motion.
Abstract: The self-avoiding walk (SAW) is a model for polymers that assigns equal probability to all paths that do not return to places they have already been. The lattice version of this problem, while elementary to define, has proved to be notoriously difficult and is still open. It is initially more challenging to construct a continuous limit of the lattice model which is a random fractal. However, in two dimensions this has been done and the continuous model (Schram-Loewner evolution) can be analysed rigorously and used to understand the non-rigorous predictions about SAWs. I will survey some results in this area and then discuss some recent work on this "continuous SAW".
2020 Virginia Mathematics Lectures by Greg Lawler.
Title: Self-avoiding motion.
Abstract: The self-avoiding walk (SAW) is a model for polymers that assigns equal probability to all paths that do not return to places they have already been. The lattice version of this problem, while elementary to define, has proved to be notoriously difficult and is still open. It is initially more challenging to construct a continuous limit of the lattice model which is a random fractal. However, in two dimensions this has been done and the continuous model (Schram-Loewner evolution) can be analysed rigorously and used to understand the non-rigorous predictions about SAWs. I will survey some results in this area and then discuss some recent work on this "continuous SAW".
Venue: University of Virginia.
Title 1: Random walks: Simple and Self-avoiding.
Abstract 1: The most common model for random behaviour is the "drunkard's walk" where at each time an individual chooses their step from some probability distribution. I will review this and then discuss what happens when one puts some constraints on the walker to try to avoid places already visited. We will see the relationship between the "fractal dimension" of the random path and the ambient dimension in which it lives.
Title 2: Conformal Invariance and the Two-Dimensional Critical Phenomenon.
Abstract 2: It was predicted by theoretical physicists that lattice models from equilibrium statistical physics "at criticality" in two dimensions have limits that are conformally invariant. There has been an incredible amount of work in the last twenty years making these ideas precise and rigorous and I will survey this work. The starting point was the development of the Schramm-Loewner evolution (SLE) which I will define.
Title 3: Loop Measures and the Loop-Erased Random Walk.
Abstract 3: This talk will focus on two related models: loop measures and the loop-erased random walk which are closely related to uniform spanning trees and describe some relatively recent work in this area in dimensions two, three, and four.
2020 Master Lecture by Greg Lawler.
Title 1: Random walks: Simple and Self-avoiding.
Abstract 1: The most common model for random behaviour is the "drunkard's walk" where at each time an individual chooses their step from some probability distribution. I will review this and then discuss what happens when one puts some constraints on the walker to try to avoid places already visited. We will see the relationship between the "fractal dimension" of the random path and the ambient dimension in which it lives.
Title 2: Conformal Invariance and the Two-Dimensional Critical Phenomenon.
Abstract 2: It was predicted by theoretical physicists that lattice models from equilibrium statistical physics "at criticality" in two dimensions have limits that are conformally invariant. There has been an incredible amount of work in the last twenty years making these ideas precise and rigorous and I will survey this work. The starting point was the development of the Schramm-Loewner evolution (SLE) which I will define.
Title 3: Loop Measures and the Loop-Erased Random Walk.
Abstract 3: This talk will focus on two related models: loop measures and the loop-erased random walk which are closely related to uniform spanning trees and describe some relatively recent work in this area in dimensions two, three, and four.
Venue: Beijing Institute of Mathematical Sciences and Applications.
Title: Random walks arising in statistical physics.
Abstract: I have been studying random paths with strong interactions arising in statistical physics for over forty years. I will survey of the program and how much we have learned in this time. I will discuss the idea of scaling limits (which leads, say, to the construction of the Schramm-Loewner evolution), the interplay between the discrete and the continuous, and the relationship between the spatial dimension of the ambient space and the fractal dimension of the paths. The talk will also discuss questions that we do not know how to answer and are topics for future research.
2022 Seminar on Stochastic Processes by Greg Lawler.
Title: Random walks arising in statistical physics.
Abstract: I have been studying random paths with strong interactions arising in statistical physics for over forty years. I will survey of the program and how much we have learned in this time. I will discuss the idea of scaling limits (which leads, say, to the construction of the Schramm-Loewner evolution), the interplay between the discrete and the continuous, and the relationship between the spatial dimension of the ambient space and the fractal dimension of the paths. The talk will also discuss questions that we do not know how to answer and are topics for future research.
Venue: Lehigh University, Bethlehem, Pennsylvania.
Title 1 (2 lectures): GFF, GMC, SLE, LQG and other Conformally Invariant TLAs.
Abstract 1: I will introduce some key concepts used in conformally invariant systems. The starting point will be the Gaussian Free Field and its relationship to Markov chains and processes. There will be an emphasis on the idea of "exploration'". I will start with discrete and then go the continuous. Exploration leads to considering particular curves, Schramm-Loewner evolutions (SLE) and (local) martingales for SLE can be written as exponentials of a GFF. These lead to random metrics and measures on domains. I will relate them to another measure, Minkowski content.
Note: GFF is Gaussian free field, GMC is Gaussian multiplicative chaos, SLE is Schramm-Loewner evolution, LQG is Liouville quantum gravity, and TLA is Temporal Logic of Actions.
Title 3: Minkowski content, quantum length, and "zippers".
Abstract 3: This can be considered a continuation of the two tutorial lectures (GFF, GMC, SLE, LQG and other Conformally Invariant TLAs) focusing on a particular idea - the quantum length of an SLE curve and how this is related to the reverse Loewner flow and Minkowski content. (I will not assume that people have attended the Wednesday tutorial lectures!)
2023 Probability Seminar, Institute for Advanced Study by Gregory F Lawler.
Title 1 (2 lectures): GFF, GMC, SLE, LQG and other Conformally Invariant TLAs.
Abstract 1: I will introduce some key concepts used in conformally invariant systems. The starting point will be the Gaussian Free Field and its relationship to Markov chains and processes. There will be an emphasis on the idea of "exploration'". I will start with discrete and then go the continuous. Exploration leads to considering particular curves, Schramm-Loewner evolutions (SLE) and (local) martingales for SLE can be written as exponentials of a GFF. These lead to random metrics and measures on domains. I will relate them to another measure, Minkowski content.
Note: GFF is Gaussian free field, GMC is Gaussian multiplicative chaos, SLE is Schramm-Loewner evolution, LQG is Liouville quantum gravity, and TLA is Temporal Logic of Actions.
Title 3: Minkowski content, quantum length, and "zippers".
Abstract 3: This can be considered a continuation of the two tutorial lectures (GFF, GMC, SLE, LQG and other Conformally Invariant TLAs) focusing on a particular idea - the quantum length of an SLE curve and how this is related to the reverse Loewner flow and Minkowski content. (I will not assume that people have attended the Wednesday tutorial lectures!)
Venue: Institute for Advanced Study, Princeton.
Title: SLE/GFF Coupling, Zipping Up, and Quantum Length.
Abstract: I will present a somewhat novel approach to known relationships (in works of Sheffield, Miller, and others) between SLE and GFF, the exponential of the GFF (quantum length/area), and Minkowski content of paths. The Neumann GFF is defined as the real part of a stochastic integral with respect to a complex Brownian motion. This viewpoint helps illuminate the relationship between boundary length and the stationary object invariant under "zipping up".
Title: SLE/GFF Coupling, Zipping Up, and Quantum Length.
Abstract: I will present a somewhat novel approach to known relationships (in works of Sheffield, Miller, and others) between SLE and GFF, the exponential of the GFF (quantum length/area), and Minkowski content of paths. The Neumann GFF is defined as the real part of a stochastic integral with respect to a complex Brownian motion. This viewpoint helps illuminate the relationship between boundary length and the stationary object invariant under "zipping up".
Last Updated March 2024