2022 Schedule: Metric Geometry and Geometric Analysis Summer School
Monday 11 July
9.30-10.30 LECTURE mini-course 1: Regina Rotman
Coffee Break
11.00 - 12 LECTURE mini course 2: Urs Lang
Lunch Break
13.00-14.00 Research Talk: Andrea Mondino.
Title. Smooth and non-smooth aspects of Ricci curvature lower bounds: an optimal transport point of view.
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 non-smooth limit spaces.
A completely new approach via optimal transportation was proposed by Lott-Villani and Sturm around 15 years ago. Via such an approach one can give a precise notion of Ricci curvature lower bounds for a non-smooth 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 non-specialists, arriving up to the most recent applications across differential geometry, metric geometry and physics.
Coffee Break
14.30 - 15.30 Research Talk: Cagri Cert.
Title: Random walks on Gromov-hyperbolic spaces: a survey and results in large deviations.
Abstract: We will start by defining random walks on Gromov-hyperbolic 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 co-authors: R. Aoun, A. Boulanger, P. Mathieu and A. Sisto.
16.00 - 17-00 Meet your mentor session
17.00 - 17.30 Further explanation of course material by a mentor or a lecturer
Tuesday 12 July
9.30-10.30 LECTURE mini-course 1: Regina Rotman
Coffee Break
11.00 - 12 LECTURE mini course 2: Urs Lang
Lunch Break
13.00 - 14.00 Research Talk: Ric Wade
Title: Hyperbolic actions and relative free factor complexes.
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.
Coffee Break
14.30 - 15.30 TA Session
15.30 - 16.30 TA Session
17.00 - 17.30 Further explanations of course material by a mentor or a lecturer
Wednesday 13 July
9.30-10.30 LECTURE mini-course 1: Regina Rotman
Coffee Break
11.00 - 12 LECTURE mini course 2: Urs Lang
Lunch Break
13.00 - 14.00 Research Talk: Martin Bridson
Title: Rigidity for automorphism groups of free groups
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.
Coffee Break
14.30 - 15.30 TA Session
15.30 - 16.30 TA Session
17.00 - 17.30 Further explanations of course material by a mentor or a lecturer
Thursday 14 July
9.30-10.30 LECTURE mini-course 1: Regina Rotman
Coffee Break
11.00 - 12 LECTURE mini course 2: Urs Lang
Lunch Break
13.00 - 14.00 Research Talk: Alex Nabutovsky
Title: Old and new trends in systolic geometry
Abstract: Let $M$ be a non-simply connected Riemannian manifold. The least length of a closed curve on $M$ that is not contractible to a point is called the systole of $M$. Yu. Burago and V. Zalgaller and, independently, J. Hebda proved that the systole of any closed Riemannian surface $M$ does not exceed $2\sqrt{Area(M)}$. In 1983 M. Gromov discovered that for a special class of essential manifolds the systole does not exceed $c(n)volume(M)^{\frac {1}{n}}$, where $n$ 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.
Coffee Break
14.30 - 15.30 TA Session
15.30 - 16.30 TA Session
17.00 - 17.30 Further explanations of course material by a mentor or a lecturer
Friday 15 July
9.30-10.30 LECTURE mini-course 1: Regina Rotman
Coffee Break
11.00 - 12 LECTURE mini course 2: Urs Lang
Lunch Break
13.00 - 14.00 Research Talk: Roman Sauer
Title: Balls in essential manifolds and actions on Cantor spaces
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.
Coffee Break
14.30 - 15.30 TA Session
15.30 - 16.30 TA Session
17.00 - 17.30 Further explanations of course material by a mentor or a lecturer
Monday 18 July
9.30-10.30 LECTURE mini-course 1: Bruce Kleiner
Coffee Break
11.00 - 12 LECTURE mini course 2: Mladen Bestvina
Lunch Break
13.00-14.00 Research Talk: Alessandro Sisto
Title: An introduction to hierarchical hyperbolicity
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 3-manifolds, 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.
Coffee Break
14.30 - 15.30 Research Talk: Marc Lackenby
Title: Knot theory and machine learning
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 4-manifolds, 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 real-valued invariant called the natural slope of a hyperbolic knot in the 3-sphere, 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 4-ball 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.
16.00 - 17-00 Meet your mentor session
17.00 - 17.30 Further explanation of course material by a mentor or a lecturer
Tuesday 19 July
9.30-10.30 LECTURE mini-course 1: Bruce Kleiner
Coffee Break
11.00 - 12 LECTURE mini course 2: Mladen Bestvina
Lunch Break
13.00 - 14.00 Research Talk: Alice Kerr
Title: Product set growth in mapping class groups
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.
Coffee Break
14.30 - 15.30 TA Session
15.30 - 16.30 TA Session
17.00 - 17.30 Further explanations of course material by a mentor or a lecturer
Wednesday 20 July
9.30-10.30 LECTURE mini-course 1: Bruce Kleiner
Coffee Break
11.00 - 12 LECTURE mini course 2: Mladen Bestvina
Lunch Break
13.00 - 14.00 Research Talk: Bin Sun
Title: Every countable group is an outer automorphism group of an acylindrically hyperbolic group with
Kazhdan's property (T)
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.
14.30 - 15.30 TA Session
15.30 - 16.30 TA Session
17.00 - 17.30 Further explanations of course material by a mentor or a lecturer
Thursday 21 July
9.30-10.30 LECTURE mini-course 1: Bruce Kleiner
Coffee Break
11.00 - 12 LECTURE mini course 2: Mladen Bestvina
Lunch Break 12.00 - 13.00
13.00 - 14.00 Research Talk: Clement dell'Aiera
Title : Large scale geometry of Hecke pairs
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 S-arithmetic 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.
Coffee Break
14.30 - 15.30 TA Session
15.30 - 16.30 TA Session
17.00 - 17.30 Further explanations of course material by a mentor or a lecturer
Friday 22 July
9.30-10.30 LECTURE mini-course 1: Bruce Kleiner
CONFERENCE GROUP PHOTOGRAPH - PENROSE PAVEMENT
Coffee Break
11.00 - 12 LECTURE mini course 2: Mladen Bestvina
Lunch Break
13.00 - 14.00
Research Talk: John Mackay
Title: Boundaries of Random Groups
Abstract: 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.
Coffee Break
14.30 - 15.30 TA Session
15.30 - 16.00. Further explanations of course material by a mentor or a lecturer