Error message

Monday, 01 March 2021

Miklos Abert
Title: Graph convergence, spectral theory and Lück approximation (Lecture 1)
Abstract:

TBA

Tom Hutchcroft
Title: Percolation on nonamenable groups, old and new (Lecture 1)
Abstract:
I will give an overview of both the classical theory of percolation on nonamenable Cayley graphs and of more modern developments. Some topics I intend to cover include:
  • Basics of uniqueness and non-uniqueness: The Burton-Keane theorem and uniqueness monotonicity.
  • Criteria for mean-field critical behaviour; the operator-theoretic approach; perturbative criteria for non-uniqueness.
  • The Aizenman-Kesten-Newman method and its applications.
  • The Magic Lemma and percolation on hyperbolic graphs.

Tuesday, 02 March 2021

Miklos Abert
Title: Invariant Random Subgroups (Lecture-2)
Abstract:

TBA

Tom Hutchcroft
Title: Percolation on Nonamenable Groups, Old and New (Lecture-2)
Abstract:
I will give an overview of both the classical theory of percolation on nonamenable Cayley graphs and of more modern developments. Some topics I intend to cover include:
 
  • Basics of uniqueness and non-uniqueness: The Burton-Keane theorem and uniqueness monotonicity.
  • Criteria for mean-field critical behaviour; the operator-theoretic approach; perturbative criteria for non-uniqueness.
  • The Aizenman-Kesten-Newman method and its applications.
  • The Magic Lemma and percolation on hyperbolic graphs.

Wednesday, 03 March 2021

Subhajit Goswami
Title: GFF Level-Set Percolation (Lecture-1)
Abstract:

In this minicourse we will review some recent developments in the study of percolation of Gaussian free field level-sets on the hypercubic lattice in three and more dimensions. As a canonical percolation model with slow algebraic decay of correlations, the new methodologies that emerged in the course of these developments form a robust toolkit for answering some classical questions in percolation theory in non-classical contexts. One of the main focuses of the course is to work towards a proof of equality of several natural critical parameters associated with this model. This result yields, among other important properties of the model, the sharpness of the associated phase transition, which corresponds to classical results in the context of Bernoulli percolation due to Menshikov and Aizenman-Barsky (in the subcritical phase) and Grimmett-Marstrand (in the supercritical phase).  To the best of our knowledge, the only instances in which an analogous result is known to hold, in all dimensions greater or equal to three, are the random cluster representation of the Ising model and the Bernoulli percolation.  In the last of the four lectures, we will discuss a very recent result where --- in dimension 3 --- we pin down the precise large deviations behavior of the probability of connecting two points through a finite cluster (the so-called ``truncated two-point function'') at all but the critical parameter. It is possible to derive such precise rates ---- which are often impossible to obtain in the non-planar case even for the more classical percolation models --- partially thanks to the rich interplay the model enjoys with potential theory of random walks which is its another appealing feature. The proofs involve several instances of the application of rigorous renormalization methods and we will try to go through the details of at least one of them. No prior knowledge of percolation theory is necessary. The contents are mostly based on the joint works of the speaker with H. Duminil-Copin, Pierre-F Rodriguez and Franco Severo.

Thursday, 04 March 2021

Gourab Ray
Title: Universality of Dimers Via Imaginary Geometry (Lecture-1)
Abstract:

The dimer model is a model of uniform perfect matching and is one of the fundamental models of statistical physics. It has many deep and intricate connections with various other models in this field, namely the Ising model and  the six-vertex model. I will focus on dimers on general (locally) planar graphs in this course.

This model has received a lot of attention in the mathematics community in the past two decades. The primary reason behind such popularity is that this model is integrable, in particular, the correlation functions can be represented exactly in a determinental form. This gives rise to a rich interplay between algebra, geometry, probability and theoretical physics. However, using the integrability seem only possible on graphs with special local structures, while the dimer model is supposed to have GFF (a canonical conformally invariant Gaussian field) type fluctuations in a much more general setting.

I will explain a new approach to understanding the dimer problem which uses the connection of this model to uniform spanning trees. In particular, it turns out, that the “winding” of uniform spanning trees (UST) is related to the height function. Using well-known SLE technology pioneered by Schramm, the scaling limit of UST is well understood. I will explain how to use this technology to also understand its limiting winding field, and then relate it to the Gaussian free field via the theory of imaginary geometry. This methods seems to be robust enough to extend to the somewhat uncharted territory of dimers on more general surfaces, e.g., the pair of pants.

I will give a general overview of the method, and ongoing and upcoming results on this approach in the first lecture.

In the second lecture, I will give the details of the proof in the simply connected case, in particular the relation to imaginary geometry, and then explain what is still left to be understood in general Riemann surfaces.

This talk is based on the following papers and some works in progress with N. Berestycki (U. Vienna) and B. Laslier (U. Paris--Diderot.).

-https://projecteuclid.org/euclid.aop/1585123322

-https://arxiv.org/abs/1610.07994

-https://arxiv.org/abs/1908.00832

Tom Hutchcroft
Title: Percolation on Nonamenable Groups, Old and New (Lecture-3)
Abstract:
I will give an overview of both the classical theory of percolation on nonamenable Cayley graphs and of more modern developments. Some topics I intend to cover include:
 
  • Basics of uniqueness and non-uniqueness: The Burton-Keane theorem and uniqueness monotonicity.
  • Criteria for mean-field critical behaviour; the operator-theoretic approach; perturbative criteria for non-uniqueness.
  • The Aizenman-Kesten-Newman method and its applications.
  • The Magic Lemma and percolation on hyperbolic graphs.
Ian Biringer
Title: The Chabauty Topology 2 (Lecture-1)
Abstract:

Let G be a topological group and Sub(G) be the set of all closed subgroups of G. In this talk, we will define the Chabauty topology on Sub(G) and discuss its basic properties, including its relationship with geometric convergence of manifolds. We'll also spend some time discussing the topology of Sub(G) for specific G, e.g. G=R^n and G=PSL(2,R).

Friday, 05 March 2021

Gourab Ray
Title: Universality of Dimers Via Imaginary Geometry (Lecture-2)
Abstract:

The dimer model is a model of uniform perfect matching and is one of the fundamental models of statistical physics. It has many deep and intricate connections with various other models in this field, namely the Ising model and  the six-vertex model. I will focus on dimers on general (locally) planar graphs in this course.

 This model has received a lot of attention in the mathematics community in the past two decades. The primary reason behind such popularity is that this model is integrable, in particular, the correlation functions can be represented exactly in a determinental form. This gives rise to a rich interplay between algebra, geometry, probability and theoretical physics. However, using the integrability seem only possible on graphs with special local structures, while the dimer model is supposed to have GFF (a canonical conformally invariant Gaussian field) type fluctuations in a much more general setting.

I will explain a new approach to understanding the dimer problem which uses the connection of this model to uniform spanning trees. In particular, it turns out, that the “winding” of uniform spanning trees (UST) is related to the height function. Using well-known SLE technology pioneered by Schramm, the scaling limit of UST is well understood. I will explain how to use this technology to also understand its limiting winding field, and then relate it to the Gaussian free field via the theory of imaginary geometry. This methods seems to be robust enough to extend to the somewhat uncharted territory of dimers on more general surfaces, e.g., the pair of pants.

I will give a general overview of the method, and ongoing and upcoming results on this approach in the first lecture.

In the second lecture, I will give the details of the proof in the simply connected case, in particular the relation to imaginary geometry, and then explain what is still left to be understood in general Riemann surfaces.

This talk is based on the following papers and some works in progress with N. Berestycki (U. Vienna) and B. Laslier (U. Paris--Diderot.).

-https://projecteuclid.org/euclid.aop/1585123322

-https://arxiv.org/abs/1610.07994

-https://arxiv.org/abs/1908.00832

 
Tom Hutchcroft
Title: Percolation on Nonamenable Groups, Old And New (Lecture-4)
Abstract:
 I will give an overview of both the classical theory of percolation on nonamenable Cayley graphs and of more modern developments. Some topics I intend to cover include:
 
  • Basics of uniqueness and non-uniqueness: The Burton-Keane theorem and uniqueness monotonicity.
  • Criteria for mean-field critical behaviour; the operator-theoretic approach; perturbative criteria for non-uniqueness.
  • The Aizenman-Kesten-Newman method and its applications.
  • The Magic Lemma and percolation on hyperbolic graphs.
Itai Benjamini
Title: Bounded Harmonic Functions on Graphs
Abstract:

TBA

Saturday, 06 March 2021

Miklos Abert
Title: Invariant random subgroups and Kesten's theorem
Abstract:

TBA

Ian Biringer
Title: Invariant Random Subgroups of Lie Groups (Lecture-2)
Abstract:

Let G be a Lie group. An invariant random subgroup (IRS) of G is a Borel probability measure on Sub(G) that is invariant under the conjugation action of G on Sub(G). We'll show how IRSs of Lie groups are a common generalization of both normal subgroups and lattices and discuss their basic properties. We'll then present the Stuck-Zimmer theorem, which classifies IRSs in higher rank Lie groups, and give some examples of exotic IRSs in rank 1 Lie groups.

Monday, 08 March 2021

Subhajit Goswami
Title: Percolation of Level-Sets of the Gaussian Free Field (Lecture-2)
Abstract:

In this minicourse we will review some recent developments in the study of percolation of Gaussian free field level-sets on the hypercubic lattice in three and more dimensions. As a canonical percolation model with slow algebraic decay of correlations, the new methodologies that emerged in the course of these developments form a robust toolkit for answering some classical questions in percolation theory in non-classical contexts. One of the main focuses of the course is to work towards a proof of equality of several natural critical parameters associated with this model. This result yields, among other important properties of the model, the sharpness of the associated phase transition, which corresponds to classical results in the context of Bernoulli percolation due to Menshikov and Aizenman-Barsky (in the subcritical phase) and Grimmett-Marstrand (in the supercritical phase).  To the best of our knowledge, the only instances in which an analogous result is known to hold, in all dimensions greater or equal to three, are the random cluster representation of the Ising model and the Bernoulli percolation.  In the last of the four lectures, we will discuss a very recent result where --- in dimension 3 --- we pin down the precise large deviations behavior of the probability of connecting two points through a finite cluster (the so-called ``truncated two-point function'') at all but the critical parameter. It is possible to derive such precise rates ---- which are often impossible to obtain in the non-planar case even for the more classical percolation models --- partially thanks to the rich interplay the model enjoys with potential theory of random walks which is its another appealing feature. The proofs involve several instances of the application of rigorous renormalization methods and we will try to go through the details of at least one of them. No prior knowledge of percolation theory is necessary. The contents are mostly based on the joint works of the speaker with H. Duminil-Copin, Pierre-F Rodriguez and Franco Severo.

Remi Coulon
Title: Twisted Patterson-Sullivan Measure and Applications to Growth Problems (Lecture-1)
Abstract:

Given a group G acting properly by isometries on a metric space X, the exponential growth rate of G with respect to X measures "how big" the orbits of G are. If H is a subgroup of G, its exponential growth rate is bounded above by the one of G. We are interested in the following question: when do H and G have the same exponential growth rate ?

This problem has both a combinatorial and a geometric origin. For the combinatorial part, Grigorchuk and Cohen proved in the 80's that a group Q = F/N (written as a quotient of the free group) is amenable if and only if N and F have the same exponential growth rate (with respect to the word length in F). About the same time Brooks gave a geometric interpretation of Kesten's amenability criterion in terms of the bottom of the spectrum of the Laplace operator. He obtained in this way a statement analogue to the one of Grigorchuk and Cohen for the deck automorphism group of the cover of certain compact hyperbolic manifolds. These works initiated many fruitful developments in geometry, dynamics and group theory.

In this series of lectures, we will revisit this problem in the general context of a group G acting on a CAT(-1) space, or more generally Gromov hyperbolic space X. We will first introduce measure-theoretic tools to study the dynamics of the geodesic flow on X. Then, given a subgroup H of G, we will explain the construction of a Hilbert-valued measure on the boundary at infinity of X. The « dimension » of this measure will allow us to characterize the equality of the exponential growth rates of H and G, in terms of the algebraic properties of the quotient G/H.

Barbara Dembin
Title: Maximal flows and minimal cutsets in first passage percolation in $\mathbb Z^d$
Abstract:
We consider the standard first passage percolation model in the lattice $\mathbb Z^d$ for $d\geq 2$: with each edge $e$ we associate a random capacity $t(e)\geq 0$ such that the family $(t(e))_e$ is independent and identically distributed with a common law $G$. We interpret this capacity as a rate of flow, \textit{i.e.}, it corresponds to the maximal amount of water that can cross the edge per unit of time.
For a given unitary vector $v$, a law of large numbers is known thanks to subadditive ergodic theorems: we can define the flow constant $\nu_G(v)$ in the direction $v$ as the asymptotic maximal amount of water that can flow per unit of time and unit of surface in the direction $v$. The problem of maximal flow is dual to the problem of finding minimal cutsets. A minimal cutset is a set of edges separating the sinks from the sources that limits the flow propagation by acting as a bottleneck: all its edges are saturated. In this talk, we will first present a law of large numbers and large deviations results for maximal flows in cylinders and we will present a generalization for any bounded connected domain. 

Tuesday, 09 March 2021

Remi Coulon
Title: Twisted Patterson-Sullivan measure and applications to growth problems
Abstract:

Given a group G acting properly by isometries on a metric space X, the exponential growth rate of G with respect to X measures "how big" the orbits of G are. If H is a subgroup of G, its exponential growth rate is bounded above by the one of G. We are interested in the following question: when do H and G have the same exponential growth rate ?

This problem has both a combinatorial and a geometric origin. For the combinatorial part, Grigorchuk and Cohen proved in the 80's that a group Q = F/N (written as a quotient of the free group) is amenable if and only if N and F have the same exponential growth rate (with respect to the word length in F). About the same time Brooks gave a geometric interpretation of Kesten's amenability criterion in terms of the bottom of the spectrum of the Laplace operator. He obtained in this way a statement analogue to the one of Grigorchuk and Cohen for the deck automorphism group of the cover of certain compact hyperbolic manifolds. These works initiated many fruitful developments in geometry, dynamics and group theory.

In this series of lectures, we will revisit this problem in the general context of a group G acting on a CAT(-1) space, or more generally Gromov hyperbolic space X. We will first introduce measure-theoretic tools to study the dynamics of the geodesic flow on X. Then, given a subgroup H of G, we will explain the construction of a Hilbert-valued measure on the boundary at infinity of X. The « dimension » of this measure will allow us to characterize the equality of the exponential growth rates of H and G, in terms of the algebraic properties of the quotient G/H.

Ian Biringer
Title: Unimodular Random Manifolds (Lecture-3)
Abstract:

We'll define a version of Benjamini-Schramm convergence for sequences of finite volume Riemannian manifolds, explain its relationship with weak convergence of IRSs, and outline how to use this notion of convergence to understand the Betti numbers of locally symmetric spaces.

Wednesday, 10 March 2021

Subhajit Goswami
Title: Percolation of Level-Sets of the Gaussian Free Field (Lecture-2)
Abstract:
In this minicourse we will review some recent developments in the study of percolation of Gaussian free field level-sets on the hypercubic lattice in three and more dimensions. As a canonical percolation model with slow algebraic decay of correlations, the new methodologies that emerged in the course of these developments form a robust toolkit for answering some classical questions in percolation theory in non-classical contexts. One of the main focuses of the course is to work towards a proof of equality of several natural critical parameters associated with this model. This result yields, among other important properties of the model, the sharpness of the associated phase transition, which corresponds to classical results in the context of Bernoulli percolation due to Menshikov and Aizenman-Barsky (in the subcritical phase) and Grimmett-Marstrand (in the supercritical phase).  To the best of our knowledge, the only instances in which an analogous result is known to hold, in all dimensions greater or equal to three, are the random cluster representation of the Ising model and the Bernoulli percolation.  In the last of the four lectures, we will discuss a very recent result where --- in dimension 3 --- we pin down the precise large deviations behavior of the probability of connecting two points through a finite cluster (the so-called ``truncated two-point function'') at all but the critical parameter. It is possible to derive such precise rates ---- which are often impossible to obtain in the non-planar case even for the more classical percolation models --- partially thanks to the rich interplay the model enjoys with potential theory of random walks which is its another appealing feature. The proofs involve several instances of the application of rigorous renormalization methods and we will try to go through the details of at least one of them. No prior knowledge of percolation theory is necessary. The contents are mostly based on the joint works of the speaker with H. Duminil-Copin, Pierre-F Rodriguez and Franco Severo.
Remi Coulon
Title: Twisted Patterson-Sullivan Measure and Applications to Growth Problems (Lecture-3)
Abstract:

Given a group G acting properly by isometries on a metric space X, the exponential growth rate of G with respect to X measures "how big" the orbits of G are. If H is a subgroup of G, its exponential growth rate is bounded above by the one of G. We are interested in the following question: when do H and G have the same exponential growth rate ?

This problem has both a combinatorial and a geometric origin. For the combinatorial part, Grigorchuk and Cohen proved in the 80's that a group Q = F/N (written as a quotient of the free group) is amenable if and only if N and F have the same exponential growth rate (with respect to the word length in F). About the same time Brooks gave a geometric interpretation of Kesten's amenability criterion in terms of the bottom of the spectrum of the Laplace operator. He obtained in this way a statement analogue to the one of Grigorchuk and Cohen for the deck automorphism group of the cover of certain compact hyperbolic manifolds. These works initiated many fruitful developments in geometry, dynamics and group theory.

In this series of lectures, we will revisit this problem in the general context of a group G acting on a CAT(-1) space, or more generally Gromov hyperbolic space X. We will first introduce measure-theoretic tools to study the dynamics of the geodesic flow on X. Then, given a subgroup H of G, we will explain the construction of a Hilbert-valued measure on the boundary at infinity of X. The « dimension » of this measure will allow us to characterize the equality of the exponential growth rates of H and G, in terms of the algebraic properties of the quotient G/H.

Thursday, 11 March 2021

Remi Coulon
Title: Twisted Patterson-Sullivan Measure and Applications to Growth Problems (Lecture-4)
Abstract:

Given a group G acting properly by isometries on a metric space X, the exponential growth rate of G with respect to X measures "how big" the orbits of G are. If H is a subgroup of G, its exponential growth rate is bounded above by the one of G. We are interested in the following question: when do H and G have the same exponential growth rate ?

This problem has both a combinatorial and a geometric origin. For the combinatorial part, Grigorchuk and Cohen proved in the 80's that a group Q = F/N (written as a quotient of the free group) is amenable if and only if N and F have the same exponential growth rate (with respect to the word length in F). About the same time Brooks gave a geometric interpretation of Kesten's amenability criterion in terms of the bottom of the spectrum of the Laplace operator. He obtained in this way a statement analogue to the one of Grigorchuk and Cohen for the deck automorphism group of the cover of certain compact hyperbolic manifolds. These works initiated many fruitful developments in geometry, dynamics and group theory.

In this series of lectures, we will revisit this problem in the general context of a group G acting on a CAT(-1) space, or more generally Gromov hyperbolic space X. We will first introduce measure-theoretic tools to study the dynamics of the geodesic flow on X. Then, given a subgroup H of G, we will explain the construction of a Hilbert-valued measure on the boundary at infinity of X. The « dimension » of this measure will allow us to characterize the equality of the exponential growth rates of H and G, in terms of the algebraic properties of the quotient G/H.

Ian Biringer
Title: Benjamini-Schramm Limits of Finite Volume Manifolds (Lecture-4)
Abstract:

Unimodular random manifolds (URMs) are the natural Riemannian analogue of unimodular random graphs, and when we restrict to the setting of manifolds that are quotients of a fixed symmetric space X=G/K, URMs are in 1-1 correspondence with IRSs of G. We'll describe some of the basic theory of URMs, including the No Core Principle and the fact that URMs are all induced by completely invariant measures on a foliated space. As an application, we'll characterize finitely generated IRSs of PSL(2,C). 

Friday, 12 March 2021

Subhajit Goswami
Title: GFF Level-Set Percolation (Lecture-4)
Abstract:

In this minicourse we will review some recent developments in the study of percolation of Gaussian free field level-sets on the hypercubic lattice in three and more dimensions. As a canonical percolation model with slow algebraic decay of correlations, the new methodologies that emerged in the course of these developments form a robust toolkit for answering some classical questions in percolation theory in non-classical contexts. One of the main focuses of the course is to work towards a proof of equality of several natural critical parameters associated with this model. This result yields, among other important properties of the model, the sharpness of the associated phase transition, which corresponds to classical results in the context of Bernoulli percolation due to Menshikov and Aizenman-Barsky (in the subcritical phase) and Grimmett-Marstrand (in the supercritical phase).  To the best of our knowledge, the only instances in which an analogous result is known to hold, in all dimensions greater or equal to three, are the random cluster representation of the Ising model and the Bernoulli percolation.  In the last of the four lectures, we will discuss a very recent result where --- in dimension 3 --- we pin down the precise large deviations behavior of the probability of connecting two points through a finite cluster (the so-called ``truncated two-point function'') at all but the critical parameter. It is possible to derive such precise rates ---- which are often impossible to obtain in the non-planar case even for the more classical percolation models --- partially thanks to the rich interplay the model enjoys with potential theory of random walks which is its another appealing feature. The proofs involve several instances of the application of rigorous renormalization methods and we will try to go through the details of at least one of them. No prior knowledge of percolation theory is necessary. The contents are mostly based on the joint works of the speaker with H. Duminil-Copin, Pierre-F Rodriguez and Franco Severo.

 

Miklos Abert
Title: Quantum Unique Ergodicity (Lecture-4)
Abstract:

TBA