Jul 11, 2022
Monday

09:30 AM  10:30 AM


Lecture & MiniCourse 1: "Geometric Inequalities: Homotopies, Fillings and Geodesics"
Regina Rotman (University of Toronto)

 Location
 
 Video

 Abstract
We will discuss various geometric inequalities motivated by famous existence theorems of various minimal objects in differential geometry proven by topological methods. Let M be a closed Riemannian manifold. Quantitative versions of such theorems as the existence of a periodic geodesic on M due to A. Fet and L. Lusternik, the existence of infinitely many geodesics between an arbitrary pair of points on M (J. P. Serve) and the existence of three simple closed geodesics ona Riemannian 2sphere (L. Lusternik and L. Schnirelmann) will be presented.
We will begin with a discussion of surfaces, next explore how the results for surfaces can be generalized to curvaturefree estimates on higher dimensional manifolds. We will next discuss geometric inequalities that involve curvature bounds. If time permits, we will also talk about the case of noncompact complete manifolds with some geometric constraints, like finite volume.
In terms of the prerequisites, in addition to Do Carmo's Riemannian Geometry, I would expect students to know some fundamentals of Algebraic Topology, such as Homology and Homotopy Groups, which can be found in Hatcher's textbook.
 Supplements



10:30 AM  11:00 AM


Coffee Break

 Location
 
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Lecture & Mini Course 2: Isoperimetric Filling Inequalities in CAT(0) Spaces
Urs Lang (ETH Zurich)

 Location
 
 Video

 Abstract
The minicourse will start with a quick introduction, essentially from scratch, to currents in metric spaces in the sense of AmbrosioKirchheim. This will be followed by a proof of the isoperimetric filling inequality of Euclidean type for cycles in CAT(0) spaces. This important inequality is due to FedererFleming for Euclidean space and to Gromov and Wenger in the general case. Some applications will be discussed. If time permits, an improvement of the isoperimetric inequality for cycles of dimension greater than or equal to the asymptotic rank of the underlying CAT(0) space, also due to Wenger, will be sketched. This pertains to notions of higherrank hyperbolicity studied recently in work of Kleiner, the lecturer, and others.
 Supplements



12:00 PM  01:00 PM


Lunch

 Location
 
 Video


 Abstract
 
 Supplements



01:00 PM  02:00 PM


Research Talk: "Smooth and NonSmooth Aspects of Ricci Curvature Lower Bounds: an Optimal Transport Point of View"
Andrea Mondino (University of Warwick)

 Location
 
 Video

 Abstract
After recalling the basic notions coming from differential geometry, the talk will be focused on spaces satisfying Ricci curvature lower bounds. The idea of compactifying the space of Riemannian manifolds satisfying Ricci curvature lower bounds goes back to Gromov in the ‘80s and was pushed by Cheeger and Colding in the ‘90s who investigated the fine structure of possibly nonsmooth limit spaces.
A completely new approach via optimal transportation was proposed by LottVillani and Sturm around 15 years ago. Via such an approach one can give a precise notion of Ricci curvature lower bounds for a nonsmooth space, without appealing to smooth approximations. Such an approach has been refined in the last years giving new insights to the theory and yielding applications which seem to be new even for smooth Riemannian manifolds. The goal of the talk is to give an introduction to the topic meant to nonspecialists, arriving up to the most recent applications across differential geometry, metric geometry and physics.
 Supplements



02:00 PM  02:30 PM


Coffee Break

 Location
 
 Video


 Abstract
 
 Supplements



02:30 PM  03:30 PM


Research Talk: "Random Walks on GromovHyperbolic Spaces: a Survey and Results in Large Deviations"
Cagri Sert (University of Zurich)

 Location
 
 Video

 Abstract
We will start by defining random walks on Gromovhyperbolic spaces and surveying basic results. In particular, we will talk about such notions and results as the drift (average escape rate), stationary measures, central limit theorem etc. In a second part, we will focus on the theory of large deviations in this context and give an overview of results obtained in collaboration with several coauthors: R. Aoun, A. Boulanger, P. Mathieu and A. Sisto.
 Supplements



04:00 PM  05:00 PM


Meet Your Mentor Session

 Location
 
 Video


 Abstract
 
 Supplements



05:00 PM  05:30 PM


Further Explanation of Course Material by a Mentor or a Lecturer

 Location
 
 Video


 Abstract
 
 Supplements




Jul 12, 2022
Tuesday

09:30 AM  10:30 AM


Lecture & MiniCourse 1: "Geometric Inequalities: Homotopies, Fillings and Geodesics"
Regina Rotman (University of Toronto)

 Location
 
 Video

 Abstract
We will discuss various geometric inequalities motivated by famous existence theorems of various minimal objects in differential geometry proven by topological methods. Let M be a closed Riemannian manifold. Quantitative versions of such theorems as the existence of a periodic geodesic on M due to A. Fet and L. Lusternik, the existence of infinitely many geodesics between an arbitrary pair of points on M (J. P. Serve) and the existence of three simple closed geodesics ona Riemannian 2sphere (L. Lusternik and L. Schnirelmann) will be presented.
We will begin with a discussion of surfaces, next explore how the results for surfaces can be generalized to curvaturefree estimates on higher dimensional manifolds. We will next discuss geometric inequalities that involve curvature bounds. If time permits, we will also talk about the case of noncompact complete manifolds with some geometric constraints, like finite volume.
In terms of the prerequisites, in addition to Do Carmo's Riemannian Geometry, I would expect students to know some fundamentals of Algebraic Topology, such as Homology and Homotopy Groups, which can be found in Hatcher's textbook.
 Supplements



10:30 AM  11:00 AM


Coffee Break

 Location
 
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Lecture & Mini Course 2: Isoperimetric Filling Inequalities in CAT(0) Spaces
Urs Lang (ETH Zurich)

 Location
 
 Video

 Abstract
The minicourse will start with a quick introduction, essentially from scratch, to currents in metric spaces in the sense of AmbrosioKirchheim. This will be followed by a proof of the isoperimetric filling inequality of Euclidean type for cycles in CAT(0) spaces. This important inequality is due to FedererFleming for Euclidean space and to Gromov and Wenger in the general case. Some applications will be discussed. If time permits, an improvement of the isoperimetric inequality for cycles of dimension greater than or equal to the asymptotic rank of the underlying CAT(0) space, also due to Wenger, will be sketched. This pertains to notions of higherrank hyperbolicity studied recently in work of Kleiner, the lecturer, and others.
 Supplements



12:00 PM  01:00 PM


Lunch

 Location
 
 Video


 Abstract
 
 Supplements



01:00 PM  02:00 PM


Research Talk: Hyperbolic Actions and Relative Free Factor Complexes
Richard Wade (University of British Columbia)

 Location
 
 Video

 Abstract
We shall look at Gromov’s classification of group actions on hyperbolic spaces and I’ll give a brief tour of how it is being used to study outer automorphism groups of free groups using relative free factor complexes.
 Supplements



02:00 PM  02:30 PM


Coffee Break

 Location
 
 Video


 Abstract
 
 Supplements



02:30 PM  04:30 PM


TA Session

 Location
 
 Video


 Abstract
 
 Supplements



05:00 PM  05:30 PM


Further Explanations of Course Material by a Mentor or a Lecturer

 Location
 
 Video


 Abstract
 
 Supplements




Jul 13, 2022
Wednesday

09:30 AM  10:30 AM


Lecture & MiniCourse 1: "Geometric Inequalities: Homotopies, Fillings and Geodesics"
Regina Rotman (University of Toronto)

 Location
 
 Video

 Abstract
We will discuss various geometric inequalities motivated by famous existence theorems of various minimal objects in differential geometry proven by topological methods. Let M be a closed Riemannian manifold. Quantitative versions of such theorems as the existence of a periodic geodesic on M due to A. Fet and L. Lusternik, the existence of infinitely many geodesics between an arbitrary pair of points on M (J. P. Serve) and the existence of three simple closed geodesics ona Riemannian 2sphere (L. Lusternik and L. Schnirelmann) will be presented.
We will begin with a discussion of surfaces, next explore how the results for surfaces can be generalized to curvaturefree estimates on higher dimensional manifolds. We will next discuss geometric inequalities that involve curvature bounds. If time permits, we will also talk about the case of noncompact complete manifolds with some geometric constraints, like finite volume.
In terms of the prerequisites, in addition to Do Carmo's Riemannian Geometry, I would expect students to know some fundamentals of Algebraic Topology, such as Homology and Homotopy Groups, which can be found in Hatcher's textbook.
 Supplements



10:30 AM  11:00 AM


Coffee Break

 Location
 
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Lecture & Mini Course 2: Isoperimetric Filling Inequalities in CAT(0) Spaces
Urs Lang (ETH Zurich)

 Location
 
 Video

 Abstract
The minicourse will start with a quick introduction, essentially from scratch, to currents in metric spaces in the sense of AmbrosioKirchheim. This will be followed by a proof of the isoperimetric filling inequality of Euclidean type for cycles in CAT(0) spaces. This important inequality is due to FedererFleming for Euclidean space and to Gromov and Wenger in the general case. Some applications will be discussed. If time permits, an improvement of the isoperimetric inequality for cycles of dimension greater than or equal to the asymptotic rank of the underlying CAT(0) space, also due to Wenger, will be sketched. This pertains to notions of higherrank hyperbolicity studied recently in work of Kleiner, the lecturer, and others.
 Supplements



12:00 PM  01:00 PM


Lunch

 Location
 
 Video


 Abstract
 
 Supplements



01:00 PM  02:00 PM


Research Talk: Rigidity for Automorphism Groups of Free Groups
Martin Bridson (University of Oxford; Clay Mathematics Institute )

 Location
 
 Video

 Abstract
I will discuss various results describing how aspects of rigidity familiar from the setting of lattices in semisimple Lie groups can be transported to the setting of automorphism groups of free groups.
 Supplements



02:00 PM  02:30 PM


Coffee Break

 Location
 
 Video


 Abstract
 
 Supplements



02:30 PM  04:30 PM


TA Session

 Location
 
 Video


 Abstract
 
 Supplements



05:00 PM  05:30 PM


Further Explanations of Course Material by a Mentor or a Lecturer

 Location
 
 Video


 Abstract
 
 Supplements




Jul 14, 2022
Thursday

09:30 AM  10:30 AM


Lecture & MiniCourse 1: "Geometric Inequalities: Homotopies, Fillings and Geodesics"
Regina Rotman (University of Toronto)

 Location
 
 Video

 Abstract
We will discuss various geometric inequalities motivated by famous existence theorems of various minimal objects in differential geometry proven by topological methods. Let M be a closed Riemannian manifold. Quantitative versions of such theorems as the existence of a periodic geodesic on M due to A. Fet and L. Lusternik, the existence of infinitely many geodesics between an arbitrary pair of points on M (J. P. Serve) and the existence of three simple closed geodesics ona Riemannian 2sphere (L. Lusternik and L. Schnirelmann) will be presented.
We will begin with a discussion of surfaces, next explore how the results for surfaces can be generalized to curvaturefree estimates on higher dimensional manifolds. We will next discuss geometric inequalities that involve curvature bounds. If time permits, we will also talk about the case of noncompact complete manifolds with some geometric constraints, like finite volume.
In terms of the prerequisites, in addition to Do Carmo's Riemannian Geometry, I would expect students to know some fundamentals of Algebraic Topology, such as Homology and Homotopy Groups, which can be found in Hatcher's textbook.
 Supplements



10:30 AM  11:00 AM


Coffee Break

 Location
 
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Lecture & Mini Course 2: Isoperimetric Filling Inequalities in CAT(0) Spaces
Urs Lang (ETH Zurich)

 Location
 
 Video

 Abstract
The minicourse will start with a quick introduction, essentially from scratch, to currents in metric spaces in the sense of AmbrosioKirchheim. This will be followed by a proof of the isoperimetric filling inequality of Euclidean type for cycles in CAT(0) spaces. This important inequality is due to FedererFleming for Euclidean space and to Gromov and Wenger in the general case. Some applications will be discussed. If time permits, an improvement of the isoperimetric inequality for cycles of dimension greater than or equal to the asymptotic rank of the underlying CAT(0) space, also due to Wenger, will be sketched. This pertains to notions of higherrank hyperbolicity studied recently in work of Kleiner, the lecturer, and others.
 Supplements



12:00 PM  01:00 PM


Lunch

 Location
 
 Video


 Abstract
 
 Supplements



01:00 PM  02:00 PM


Research Talk: Old and New Trends in Systolic Geometry
Alexander Nabutovsky

 Location
 
 Video

 Abstract
Let MM be a nonsimply connected Riemannian manifold. The least length of a closed curve on MM that is not contractible to a point is called the systole of MM. Yu. Burago and V. Zalgaller and, independently, J. Hebda proved that the systole of any closed Riemannian surface MM does not exceed 2Area(M)−−−−−−−−√2Area(M). In 1983 M. Gromov discovered that for a special class of essential manifolds the systole does not exceed c(n)volume(M)1nc(n)volume(M)1n, where nn denotes the dimension. His paper established connections between the systolic inequality and higher codimension isoperimetric inequalities in Banach spaces, introduced new natural metric invariants of Riemannian manifolds and became a starting point for development of the area of systolic geometry. We are going to sketch essential elements of Gromov's proof and then review some newer developments in the study of systoles including a much simpler proof of Gromov's result recently found by P. Papazoglou. Other topics include isoperimetric inequalities for Hausdorff contents and upper bounds for the systole in terms of Hausdorff contents discovered by Y. Liokumovich, B. Lishak, R. Rotman and the speaker.
 Supplements



02:00 PM  02:30 PM


Coffee Break

 Location
 
 Video


 Abstract
 
 Supplements



02:30 PM  04:30 PM


TA Session

 Location
 
 Video


 Abstract
 
 Supplements



05:00 PM  05:30 PM


Further Explanations of Course Material by a Mentor or a Lecturer

 Location
 
 Video


 Abstract
 
 Supplements




Jul 15, 2022
Friday

09:30 AM  10:30 AM


Lecture & MiniCourse 1: "Geometric Inequalities: Homotopies, Fillings and Geodesics"
Regina Rotman (University of Toronto)

 Location
 
 Video

 Abstract
We will discuss various geometric inequalities motivated by famous existence theorems of various minimal objects in differential geometry proven by topological methods. Let M be a closed Riemannian manifold. Quantitative versions of such theorems as the existence of a periodic geodesic on M due to A. Fet and L. Lusternik, the existence of infinitely many geodesics between an arbitrary pair of points on M (J. P. Serve) and the existence of three simple closed geodesics ona Riemannian 2sphere (L. Lusternik and L. Schnirelmann) will be presented.
We will begin with a discussion of surfaces, next explore how the results for surfaces can be generalized to curvaturefree estimates on higher dimensional manifolds. We will next discuss geometric inequalities that involve curvature bounds. If time permits, we will also talk about the case of noncompact complete manifolds with some geometric constraints, like finite volume.
In terms of the prerequisites, in addition to Do Carmo's Riemannian Geometry, I would expect students to know some fundamentals of Algebraic Topology, such as Homology and Homotopy Groups, which can be found in Hatcher's textbook.
 Supplements



10:30 AM  11:00 AM


Coffee Break

 Location
 
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Lecture & Mini Course 2: Isoperimetric Filling Inequalities in CAT(0) Spaces
Urs Lang (ETH Zurich)

 Location
 
 Video

 Abstract
The minicourse will start with a quick introduction, essentially from scratch, to currents in metric spaces in the sense of AmbrosioKirchheim. This will be followed by a proof of the isoperimetric filling inequality of Euclidean type for cycles in CAT(0) spaces. This important inequality is due to FedererFleming for Euclidean space and to Gromov and Wenger in the general case. Some applications will be discussed. If time permits, an improvement of the isoperimetric inequality for cycles of dimension greater than or equal to the asymptotic rank of the underlying CAT(0) space, also due to Wenger, will be sketched. This pertains to notions of higherrank hyperbolicity studied recently in work of Kleiner, the lecturer, and others.
 Supplements



12:00 PM  01:00 PM


Lunch

 Location
 
 Video


 Abstract
 
 Supplements



01:00 PM  02:00 PM


Research Talk: Balls in Essential Manifolds and Actions on Cantor Spaces
Roman Sauer (Karlsruhe Institute of Technology)

 Location
 
 Video

 Abstract
We discuss a universal lower bound on the volume of balls in essential manifolds. One ingredient is geometric, the other ingredient involves group actions on Cantor spaces that simulate the residual finiteness of fundamental groups.
 Supplements



02:00 PM  02:30 PM


Coffee Break

 Location
 
 Video


 Abstract
 
 Supplements



02:30 PM  04:30 PM


TA Session

 Location
 
 Video


 Abstract
 
 Supplements



05:00 PM  05:30 PM


Further Explanations of Course Material by a Mentor or a Lecturer

 Location
 
 Video


 Abstract
 
 Supplements




Jul 18, 2022
Monday

09:30 AM  10:30 AM


Lecture & Mini Course 1: Metric Geometry and Analysis on Boundaries of Gromov Hyperbolic Spaces, and Applications
Bruce Kleiner (New York University, Courant Institute)

 Location
 
 Video


 Abstract
The minicourse will cover some aspects of metric and analytical structure on boundaries of Gromov hyperbolic spaces, applications to rigidity, and open problem.
Recommended preparatory reading:
(1) Quasiisometries and the MilnorSvarc lemma. BridsonHaefliger I.8; DrutuKapovich
8.18.3.
(2) Gromov hyperbolic spaces: definitions, examples, Morse lemma on stability of
quasigeodesics, definition of the boundary. BridsonHaefliger. III.H.1, III.H.3; Drutu
Kapovich 11.1, 11.10, 11.11, 11.13.
(3) The theorems of Rademacher and Stepanov, Section 3 in Lectures on Lipschitz analysis,
Heinonen, available here:
http://www.math.jyu.fi/research/reports/rep100.pdf#page=18
 Supplements



10:30 AM  11:00 AM


Coffee Break

 Location
 
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Lecture & Mini Course 2: Projection Complexes and Applications to Mapping Class Groups
Mladen Bestvina (University of Utah)

 Location
 
 Video


 Abstract
The main goal will be to present a proof that mapping class groups have finite asymptotic dimension. This will give me a good excuse to talk about projection complexes, asymptotic dimension, curve complexes and subsurface projections. Most of this will be selfcontained, with few "black boxes".
Reading list:
Hyperbolic groups and spaces, from the standard books like
BridsonHaefliger or KapovichDrutu
Some familiarity with mapping class groups, e.g. the first 3 sections
of FarbMargalit
 Supplements



12:00 PM  01:00 PM


Lunch

 Location
 
 Video


 Abstract
 
 Supplements



01:00 PM  02:00 PM


Research Talk: An Introduction to Hierarchical Hyperbolicity
Alessandro Sisto (ETH Zürich)

 Location
 
 Video


 Abstract
Hierarchical hyperbolicity provides a common framework to work with various classes of spaces and groups such as mapping class groups, Teichmueller space, and cubical groups, as well as many fundamental groups of 3manifolds, Artin groups, etc. I will explain what a hierarchically hyperbolic structure is and try to convey a picture of what a hierarchically hyperbolic space looks like.
 Supplements



02:00 PM  02:30 PM


Coffee Break

 Location
 
 Video


 Abstract
 
 Supplements



02:30 PM  03:30 PM


Research Talk: Knot Theory and Machine Learning
Marc Lackenby

 Location
 
 Video


 Abstract
Knot theory is divided into several subfields. One of these is hyperbolic knot theory, which is focused on the hyperbolic structure that exists on many knot complements. Another branch of knot theory is concerned with invariants that have connections to 4manifolds, for example the knot signature and Heegaard Floer homology. In my talk, I will describe a new relationship between these two fields that was discovered with the aid of machine learning. Specifically, we show that the knot signature can be estimated surprisingly accurately in terms of hyperbolic invariants. We introduce a new realvalued invariant called the natural slope of a hyperbolic knot in the 3sphere, which is defined in terms of its cusp geometry. Our main result is that twice the knot signature and the natural slope differ by at most a constant times the hyperbolic volume divided by the cube of the injectivity radius. This theorem has applications to Dehn surgery and to 4ball genus. We will also present a refined version of the inequality where the upper bound is a linear function of the volume, and the slope is corrected by terms corresponding to short geodesics that have odd linking number with the knot. My talk will outline the proofs of these results, as well as describing the role that machine learning played in their discovery.
 Supplements



04:00 PM  05:00 PM


Meet Your Mentor Session

 Location
 
 Video


 Abstract
 
 Supplements



05:00 PM  05:30 PM


Further Explanations of Course Material by a Mentor or a Lecturer

 Location
 
 Video


 Abstract
 
 Supplements




Jul 19, 2022
Tuesday

09:30 AM  10:30 AM


Lecture & Mini Course 1: Metric Geometry and Analysis on Boundaries of Gromov Hyperbolic Spaces, and Applications
Bruce Kleiner (New York University, Courant Institute)

 Location
 
 Video


 Abstract
The minicourse will cover some aspects of metric and analytical structure on boundaries of Gromov hyperbolic spaces, applications to rigidity, and open problem.
Recommended preparatory reading:
(1) Quasiisometries and the MilnorSvarc lemma. BridsonHaefliger I.8; DrutuKapovich
8.18.3.
(2) Gromov hyperbolic spaces: definitions, examples, Morse lemma on stability of
quasigeodesics, definition of the boundary. BridsonHaefliger. III.H.1, III.H.3; Drutu
Kapovich 11.1, 11.10, 11.11, 11.13.
(3) The theorems of Rademacher and Stepanov, Section 3 in Lectures on Lipschitz analysis,
Heinonen, available here:
http://www.math.jyu.fi/research/reports/rep100.pdf#page=18
 Supplements



10:30 AM  11:00 AM


Coffee Break

 Location
 
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Lecture & Mini Course 2: Projection Complexes and Applications to Mapping Class Groups
Mladen Bestvina (University of Utah)

 Location
 
 Video


 Abstract
The main goal will be to present a proof that mapping class groups have finite asymptotic dimension. This will give me a good excuse to talk about projection complexes, asymptotic dimension, curve complexes and subsurface projections. Most of this will be selfcontained, with few "black boxes".
Reading list:
Hyperbolic groups and spaces, from the standard books like
BridsonHaefliger or KapovichDrutu
Some familiarity with mapping class groups, e.g. the first 3 sections
of FarbMargalit
 Supplements



12:00 PM  01:00 PM


Lunch

 Location
 
 Video


 Abstract
 
 Supplements



01:00 PM  02:00 PM


Research Talk: Product Set Growth in Mapping Class Groups
Alice Kerr (University of Oxford)

 Location
 
 Video


 Abstract
A standard question in group theory is to ask if we can categorise the subgroups of a group in terms of their growth. In this talk we will be asking this question for uniform product set growth, a property that is stronger than the more widely understood notion of uniform exponential growth. We will see how considering acylindrical actions on hyperbolic spaces can help us, and give a particular application to mapping class groups.
 Supplements



02:00 PM  02:30 PM


Coffee Break

 Location
 
 Video


 Abstract
 
 Supplements



02:30 PM  04:30 PM


TA Session

 Location
 
 Video


 Abstract
 
 Supplements



05:00 PM  05:30 PM


Further Explanations of Course Material by a Mentor or a Lecturer

 Location
 
 Video


 Abstract
 
 Supplements




Jul 20, 2022
Wednesday

09:30 AM  10:30 AM


Lecture & Mini Course 1: Metric Geometry and Analysis on Boundaries of Gromov Hyperbolic Spaces, and Applications
Bruce Kleiner (New York University, Courant Institute)

 Location
 
 Video

 Abstract
The minicourse will cover some aspects of metric and analytical structure on boundaries of Gromov hyperbolic spaces, applications to rigidity, and open problem.
Recommended preparatory reading:
(1) Quasiisometries and the MilnorSvarc lemma. BridsonHaefliger I.8; DrutuKapovich
8.18.3.
(2) Gromov hyperbolic spaces: definitions, examples, Morse lemma on stability of
quasigeodesics, definition of the boundary. BridsonHaefliger. III.H.1, III.H.3; Drutu
Kapovich 11.1, 11.10, 11.11, 11.13.
(3) The theorems of Rademacher and Stepanov, Section 3 in Lectures on Lipschitz analysis,
Heinonen, available here:
http://www.math.jyu.fi/research/reports/rep100.pdf#page=18
 Supplements



10:30 AM  11:00 AM


Coffee Break

 Location
 
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Lecture & Mini Course 2: Projection Complexes and Applications to Mapping Class Groups
Mladen Bestvina (University of Utah)

 Location
 
 Video

 Abstract
The main goal will be to present a proof that mapping class groups have finite asymptotic dimension. This will give me a good excuse to talk about projection complexes, asymptotic dimension, curve complexes and subsurface projections. Most of this will be selfcontained, with few "black boxes".
Reading list:
Hyperbolic groups and spaces, from the standard books like
BridsonHaefliger or KapovichDrutu
Some familiarity with mapping class groups, e.g. the first 3 sections
of FarbMargalit
 Supplements



12:00 PM  01:00 PM


Lunch

 Location
 
 Video


 Abstract
 
 Supplements



01:00 PM  02:00 PM


Research Talk: Every Countable Group is an Outer Automorphism Group of an Acylindrically Hyperbolic Group with Kazhdan's Property (T)
Bin Sun (University of Oxford)

 Location
 
 Video

 Abstract
The combination of Kazhdan’s property (T) and negative curvature typically limits the amount of outer automorphisms. Indeed, it is a result of Paulin that every property (T) hyperbolic group has a finite outer automorphism group. Belegradek and Szczepan ́ski extends Paulin’s result to property (T) relatively hyperbolic groups. We prove that for every countable group Q there is an acylindrically hyperbolic group G such that Out(G) = Q. Therefore the combination of property (T) and acylindrical hyperbolicity is much more flexible in terms of outer automorphisms.
 Supplements



02:00 PM  02:30 PM


Coffee Break

 Location
 
 Video


 Abstract
 
 Supplements



02:30 PM  04:30 PM


TA Session

 Location
 
 Video


 Abstract
 
 Supplements



05:00 PM  05:30 PM


Further Explanations of Course Material by a Mentor or a Lecturer

 Location
 
 Video


 Abstract
 
 Supplements




Jul 21, 2022
Thursday

09:30 AM  10:30 AM


Lecture & Mini Course 1: Metric Geometry and Analysis on Boundaries of Gromov Hyperbolic Spaces, and Applications
Bruce Kleiner (New York University, Courant Institute)

 Location
 
 Video

 Abstract
The minicourse will cover some aspects of metric and analytical structure on boundaries of Gromov hyperbolic spaces, applications to rigidity, and open problem.
Recommended preparatory reading:
(1) Quasiisometries and the MilnorSvarc lemma. BridsonHaefliger I.8; DrutuKapovich
8.18.3.
(2) Gromov hyperbolic spaces: definitions, examples, Morse lemma on stability of
quasigeodesics, definition of the boundary. BridsonHaefliger. III.H.1, III.H.3; Drutu
Kapovich 11.1, 11.10, 11.11, 11.13.
(3) The theorems of Rademacher and Stepanov, Section 3 in Lectures on Lipschitz analysis,
Heinonen, available here:
http://www.math.jyu.fi/research/reports/rep100.pdf#page=18
 Supplements



10:30 AM  11:00 AM


Coffee Break

 Location
 
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Lecture & Mini Course 2: Projection Complexes and Applications to Mapping Class Groups
Mladen Bestvina (University of Utah)

 Location
 
 Video

 Abstract
The main goal will be to present a proof that mapping class groups have finite asymptotic dimension. This will give me a good excuse to talk about projection complexes, asymptotic dimension, curve complexes and subsurface projections. Most of this will be selfcontained, with few "black boxes".
Reading list:
Hyperbolic groups and spaces, from the standard books like
BridsonHaefliger or KapovichDrutu
Some familiarity with mapping class groups, e.g. the first 3 sections
of FarbMargalit
 Supplements



12:00 PM  01:00 PM


Lunch

 Location
 
 Video


 Abstract
 
 Supplements



01:00 PM  02:00 PM


Research Talk: Large Scale Geometry of Hecke Pairs
Clément Dell'Aiera (Université de Metz)

 Location
 
 Video

 Abstract
In this talk, we study almost normal subgroups from a geometric point of view. When a group G is equipped with a proper left invariant length, we characterize the subgroups H whose coset space G/H, with the induced metric, is a locally finite space coarsely embeddable into a Hilbert space. We will give examples that i find interesting : these are Sarithmetic groups with quite exotic properties. If time allows, we will present the main application : if H and G/H admit a coarse embeding into a Hilbert space, then G satisfies the Novikov conjecture.
 Supplements



02:00 PM  02:30 PM


Coffee Break

 Location
 
 Video


 Abstract
 
 Supplements



02:30 PM  04:30 PM


TA Session

 Location
 
 Video


 Abstract
 
 Supplements



05:00 PM  05:30 PM


Further Explanations of Course Material by a Mentor or a Lecturer

 Location
 
 Video


 Abstract
 
 Supplements




Jul 22, 2022
Friday

09:30 AM  10:30 AM


Lecture & Mini Course 1: Metric Geometry and Analysis on Boundaries of Gromov Hyperbolic Spaces, and Applications
Bruce Kleiner (New York University, Courant Institute)

 Location
 
 Video

 Abstract
The minicourse will cover some aspects of metric and analytical structure on boundaries of Gromov hyperbolic spaces, applications to rigidity, and open problem.
Recommended preparatory reading:
(1) Quasiisometries and the MilnorSvarc lemma. BridsonHaefliger I.8; DrutuKapovich
8.18.3.
(2) Gromov hyperbolic spaces: definitions, examples, Morse lemma on stability of
quasigeodesics, definition of the boundary. BridsonHaefliger. III.H.1, III.H.3; Drutu
Kapovich 11.1, 11.10, 11.11, 11.13.
(3) The theorems of Rademacher and Stepanov, Section 3 in Lectures on Lipschitz analysis,
Heinonen, available here:
http://www.math.jyu.fi/research/reports/rep100.pdf#page=18
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10:30 AM  11:00 AM


Coffee Break

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11:00 AM  12:00 PM


Lecture & Mini Course 2: Projection Complexes and Applications to Mapping Class Groups
Mladen Bestvina (University of Utah)

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The main goal will be to present a proof that mapping class groups have finite asymptotic dimension. This will give me a good excuse to talk about projection complexes, asymptotic dimension, curve complexes and subsurface projections. Most of this will be selfcontained, with few "black boxes".
Reading list:
Hyperbolic groups and spaces, from the standard books like
BridsonHaefliger or KapovichDrutu
Some familiarity with mapping class groups, e.g. the first 3 sections
of FarbMargalit
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12:00 PM  01:00 PM


Lunch

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01:00 PM  02:00 PM


Research Talk: Boundaries of Random Groups
John Mackay (University of Bristol)

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In many models of random groups, the groups are typically hyperbolic. I'll survey some of what's known about these groups, with a particular focus on the properties of their boundaries at infinity.
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02:00 PM  02:30 PM


Coffee Break

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02:30 PM  04:30 PM


TA Session

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05:00 PM  05:30 PM


Further Explanations of Course Material by a Mentor or a Lecturer

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