Jan 24, 2022
Monday
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08:00 AM - 08:10 AM
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Welcome
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
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- Supplements
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08:10 AM - 09:00 AM
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An Elementary Introduction to Multiplicative Chaos
Eero Saksman (University of Helsinki)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
We introduce Gaussian multiplicative chaos (GMC) and discuss its basic properties. The talks are intended especially to people having more an analysis background. No a priori knowledge of GMC is required.
- Supplements
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09:10 AM - 10:00 AM
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Stochastic Analysis on Diff(S^1) Revisited
Maria Gordina (University of Connecticut)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
The talk will review several approaches to constructing a Brownian motion on such spaces. Not only these spaces are equipped with a Fréchet topology, they might be viewed as both Riemannian and sub-Riemannian infinite-dimensional manifolds.
- Supplements
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10:20 AM - 11:10 AM
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Loewner Energy, SLE and Weil-Petersson Quasicircles
Wang Yilin (Massachusetts Institute of Technology)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
This is an introductory lecture on the Loewner energy and its links to Schramm-Loewner evolutions and Weil-Petersson quasicircles, a class of Jordan curves arising from the Kahler geometry on the universal Teichmuller space. This class of curves is also of interest in geometric function theory and has many drastically different but equivalent definitions. These links show that the Loewner energy sits at the crossroad of random conformal geometry and analysis. After explaining the basic definitions and the main connections, I will focus on the related open questions.
- Supplements
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11:40 AM - 12:30 PM
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Geodesics in the Brownian Map: Strong Confluence and Geometric Structure
Wei Qian (Université Paris-Saclay)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
I will talk about a joint work with Jason Miller where we establish results on all geodesics in the Brownian map, including those between exceptional points. First, we prove a strong and quantitative form of the confluence of geodesics phenomenon which states that any pair of geodesics which are sufficiently close in the Hausdorff distance must coincide with each other except near their endpoints. Then, we show that the intersection of any two geodesics minus their endpoints is connected, the number of geodesics which emanate from a single point and are disjoint except at their starting point is at most 5, and the maximal number of geodesics which connect any pair of points is 9. For each k=1,…,9, we obtain the Hausdorff dimension of the pairs of points connected by exactly k geodesics. For k=7,8,9, such pairs have dimension zero and are countably infinite. Further, we classify the (finite number of) possible configurations of geodesics between any pair of points, up to homeomorphism, and give a dimension upper bound for the set of endpoints in each case. Finally, we show that every geodesic can be approximated arbitrarily well and in a strong sense by a geodesic connecting typical points. In particular, this gives an affirmative answer to a conjecture of Angel, Kolesnik, and Miermont that the geodesic frame, the union of all of the geodesics in the Brownian map minus their endpoints, has dimension one, the dimension of a single geodesic.
- Supplements
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Jan 25, 2022
Tuesday
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08:00 AM - 08:50 AM
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Loewner Dynamics for Real Rational Functions and the Multiple SLE(0) Process
Thomas Alberts (University of Utah)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Recently Peltola and Wang (see Eveliina's talk of last week!) introduced the multiple SLE(0) process as the deterministic limit of the random multiple SLE(κ) curves as κ goes to zero. They also showed that the limiting curves have important geometric characterizations that are independent of their relation to SLE(κ): they are the real locus of real rational functions, and they can be generated by a deterministic Loewner evolution driven by multiple points. We prove that the Loewner evolution is a very special family of commuting SLE(0, rho) processes (with commutation holding in a very strong sense), and use this to directly show that the curves satisfy a geodesic multichord property. We also show that our SLE(0, rho) processes lead to relatively simple solutions for the degenerate versions of the BPZ equations in terms of the poles and critical points of the rational function, and that the dynamics of these poles and critical points come from the Calogero-Moser integrable system. Although our results are purely deterministic they are again motivated by taking limits of probabilistic constructions in conformal field theory.
- Supplements
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09:00 AM - 09:50 AM
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Conformal Welding in Liouville Quantum Gravity
Nina Holden (ETH Zurich)
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- Location
- SLMath: Online/Virtual
- Video
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- Abstract
Liouville quantum gravity (LQG) is a model for a random surface with origin in the physics literature. A very powerful tool in the study of LQG is conformal welding. We will give an overview of recent and more classical conformal welding results, and will also mention a few applications.
- Supplements
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10:20 AM - 11:10 AM
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Gaussian Free Field: An Introduction
Nathanael Berestycki (University of Vienna)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
I will give an elementary introduction to the Gaussian free field and some of its main properties (including conformal invariance in two dimensions, domain Markov property, and the behaviour of its circle average process). No advanced knowledge of probability will be assumed.
- Supplements
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11:40 AM - 12:30 PM
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An Elementary Introduction to Multiplicative Chaos Pt II
Eero Saksman (University of Helsinki)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
We introduce Gaussian multiplicative chaos (GMC) and discuss its basic properties. The talks are intended especially to people having more an analysis background. No a priori knowledge of GMC is required.
- Supplements
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Jan 26, 2022
Wednesday
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08:30 AM - 09:20 AM
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Fractal Measures
Gregory Lawler (University of Chicago)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
This will be a tutorial talk on the construction of fractal measures with an emphasis on Minkowski content measures.
- Supplements
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09:50 AM - 10:40 AM
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Gaussian Free Field: An Introduction Pt II
Nathanael Berestycki (University of Vienna)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
I will give an elementary introduction to the Gaussian free field and some of its main properties (including conformal invariance in two dimensions, domain Markov property, and the behaviour of its circle average process). No advanced knowledge of probability will be assumed.
- Supplements
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11:10 AM - 12:00 PM
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A Pedagogical Introduction to Critical Percolation via the Hierarchical Lattice
Tom Hutchcroft (California Institute of Technology)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
I will give an introduction to critical exponents in Bernoulli percolation by analyzing the model on the hierarchical lattice, a toy model that has much more symmetry than the Euclidean lattice and permits exact computation of several interesting quantities. Along the way I will review some classical topics such as the sharpness of the phase transition.
- Supplements
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Jan 27, 2022
Thursday
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08:00 AM - 08:50 AM
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Random Surfaces
Scott Sheffield (Massachusetts Institute of Technology)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
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- Supplements
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09:00 AM - 09:50 AM
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A Pedagogical Introduction to Critical Percolation via the Hierarchical Lattice Pt II
Tom Hutchcroft (California Institute of Technology)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
I will give an introduction to critical exponents in Bernoulli percolation by analyzing the model on the hierarchical lattice, a toy model that has much more symmetry than the Euclidean lattice and permits exact computation of several interesting quantities. Along the way I will review some classical topics such as the sharpness of the phase transition.
- Supplements
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10:20 AM - 11:10 AM
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Removability of Planar Sets
Malik Younsi (University of Hawaii at Manoa)
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- Location
- SLMath: Online/Virtual
- Video
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- Abstract
Ever since the seminal work of Ahlfors and Beurling in the 1950's, the study of removable planar sets with respect to various classes of holomorphic functions has proven over the years to be of fundamental importance for a wide variety of problems in complex analysis and related areas. Questions revolving around necessary and sufficient geometric conditions for removability have held a prominent role in the development of valuable techniques and applications. In recent years, attention has been drawn to the more modern notion of (quasi)conformal removability, in view of applications to an ever-growing variety of central problems in complex analysis, probability and dynamics. In this talk, I will discuss various results related to conformal removability, focusing on applications to conformal welding and to Koebe's uniformization conjecture.
- Supplements
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11:40 AM - 12:30 PM
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Introduction to Liouville Conformal Field Theory
Rémi Rhodes (Université d'Aix-Marseille (AMU))
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- Location
- SLMath: Online/Virtual
- Video
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- Abstract
In this talk I will review the probabilistic construction of the Liouville theory (by F. David, C. Guillarmou, A. Kupiainen, V. Vargas and myself) on Riemann surfaces. In the second part of my talk I will review some recent progress related to the conformal bootstrap for the Liouville theory.
- Supplements
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Jan 28, 2022
Friday
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08:30 AM - 09:20 AM
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Introduction to Liouville Conformal Field Theory Pt II
Rémi Rhodes (Université d'Aix-Marseille (AMU))
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- Location
- SLMath: Online/Virtual
- Video
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- Abstract
In this talk I will review the probabilistic construction of the Liouville theory (by F. David, C. Guillarmou, A. Kupiainen, V. Vargas and myself) on Riemann surfaces. In the second part of my talk I will review some recent progress related to the conformal bootstrap for the Liouville theory.
- Supplements
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09:50 AM - 10:40 AM
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Removability of Planar Sets Pt II
Malik Younsi (University of Hawaii at Manoa)
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- Location
- SLMath: Online/Virtual
- Video
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- Abstract
Ever since the seminal work of Ahlfors and Beurling in the 1950's, the study of removable planar sets with respect to various classes of holomorphic functions has proven over the years to be of fundamental importance for a wide variety of problems in complex analysis and related areas. Questions revolving around necessary and sufficient geometric conditions for removability have held a prominent role in the development of valuable techniques and applications.
In recent years, attention has been drawn to the more modern notion of (quasi)conformal removability, in view of applications to an ever-growing variety of central problems in complex analysis, probability and dynamics. In this talk, I will discuss various results related to conformal removability, focusing on applications to conformal welding and to Koebe's uniformization conjecture.
- Supplements
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11:10 AM - 12:00 PM
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Random Conductances and the Chromatic Polynomial
Richard Kenyon (Brown University)
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- Location
- SLMath: Online/Virtual
- Video
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- Abstract
For a graph G, the chromatic polynomial at negative integer values is the expected value of a certain quantity, related to Dirichlet energy, over the simplex of random positive conductances (of sum 1). Talk based on joint work with Wei Yeung Lam.
- Supplements
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