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Organizing Committee
Abstract
This workshop focuses on the interplay between dynamics, rigidity, and arithmetic in hyperbolic geometry and related areas. There have been many striking developments in recent years, particularly related to totally geodesic submanifolds in both finite and infinite volume hyperbolic and even complex hyperbolic manifolds.
One aim of this workshop is to expose young researchers to these breakthroughs providing them with the necessary background from dynamics, and geometry to allow them to appreciate some of these recent advances, and prepare them to make new original contributions. For this purpose, we will have minicourses on "Arithmeticity, Superrigidity and totally geodesic manifolds", and "Rigidity and geodesic planes in infinite volume hyperbolic manifolds". These courses will be preceded by an introductory minicourse on Hyperbolic geometry. We will also have a minicourse on "Understanding of geodesic planes in hyperbolic 3-manifolds via computations and visualization". In addition, we wish to bring together experts in these fields to discuss the recent developments and open problems that lie at the crossroads of these different fields and to encourage more interaction among people working in these diverse areas.
This workshop is partially funded by NSF CAREER award DMS-1651563
One aim of this workshop is to expose young researchers to these breakthroughs providing them with the necessary background from dynamics, and geometry to allow them to appreciate some of these recent advances, and prepare them to make new original contributions. For this purpose, we will have minicourses on "Arithmeticity, Superrigidity and totally geodesic manifolds", and "Rigidity and geodesic planes in infinite volume hyperbolic manifolds". These courses will be preceded by an introductory minicourse on Hyperbolic geometry. We will also have a minicourse on "Understanding of geodesic planes in hyperbolic 3-manifolds via computations and visualization". In addition, we wish to bring together experts in these fields to discuss the recent developments and open problems that lie at the crossroads of these different fields and to encourage more interaction among people working in these diverse areas.
This workshop is partially funded by NSF CAREER award DMS-1651563
Image credit: C. McMullen
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Workshop Schedule
Monday, May 15, 2023
9:30 - 9:50 AM EDT
Check In / AM Coffee Break
Check In - 11th Floor Collaborative Space
9:50 - 10:00 AM EDT
Welcome
11th Floor Lecture Hall
10:00 - 10:45 AM EDT
Hyperbolic geometry - An Introduction
11th Floor Lecture Hall
Speaker
Ian Biringer, Boston College
Session Chair
David Fisher, Rice University
Abstract
The aim of this minicourse is to introduce fundamental concepts in hyperbolic geometry, such as limit sets, geometric finiteness, and critical exponent, and Mostow rigidity. We will discuss examples of arithmetic hyperbolic manifolds, and illustrate flexible geometric constructions like Dehn filling, quasi-conformal deformation, and the gluing constructions of non-arithmetic lattices by Gromov and Piatetski-Shapiro.
11:00 - 11:45 AM EDT
Totally geodesic subvarieties via Hodge theory
11th Floor Lecture Hall
Speaker
Gregorio Baldi, Institut des Hautes Études Scientifiques
Session Chair
David Fisher, Rice University
Abstract
In this Mini Course, following a joint work with E.Ullmo, I will explain how (integral) Hodge theory naturally comes up in the study of totally geodesic subvarieties of a complex hyperbolic ball quotient S. From such a point of view, the finiteness of the maximal totally geodesics of S becomes a consequence of a very general conjecture about 'unlikely intersections'. The two lectures will give some motivations and an introduction to such techniques (no prior knowledge in Hodge theory will be assumed).
12:00 - 1:30 PM EDT
Lunch/Free Time
1:30 - 2:15 PM EDT
Geometric, dynamical and arithmetic properties of Anosov representations
11th Floor Lecture Hall
Speaker
Anna Wienhard, Heidelberg University
Session Chair
Nattalie Tamam, University of Michigan
Abstract
Anosov representations provides a rich class of discrete embeddings of hyperbolic groups into semisimple Lie groups, which generalizes the classes of convex cocompact subgroups to the setting of higher rank Lie groups.
In this talk I will discuss some results (old and new) regarding geometric, dynamical and arithmetic properties.
2:30 - 3:15 PM EDT
Hyperbolic geometry - An Introduction
11th Floor Lecture Hall
Speaker
Ian Biringer, Boston College
Session Chair
Nattalie Tamam, University of Michigan
Abstract
The aim of this minicourse is to introduce fundamental concepts in hyperbolic geometry, such as limit sets, geometric finiteness, and critical exponent, and Mostow rigidity. We will discuss examples of arithmetic hyperbolic manifolds, and illustrate flexible geometric constructions like Dehn filling, quasi-conformal deformation, and the gluing constructions of non-arithmetic lattices by Gromov and Piatetski-Shapiro.
3:30 - 4:00 PM EDT
Coffee Break
11th Floor Collaborative Space
4:00 - 4:45 PM EDT
Rigidity and geodesic planes in infinite volume hyperbolic manifolds
11th Floor Lecture Hall
Speaker
Hee Oh, Yale university
Session Chair
Nattalie Tamam, University of Michigan
Abstract
The aim of this mini course is to discuss some basic theory of homogeneous dynamics on the quotient of SO(n,1) by a discrete subgroup and to explain the topological rigidity of geodesic planes in hyperbolic n-manifolds of Fuchsian ends which was proved by McMullen-Mohammadi-O. (for n=3) and by Minju Lee-O. (for
n>3).
5:00 - 6:30 PM EDT
Reception
11th Floor Collaborative Space
Tuesday, May 16, 2023
9:30 - 10:00 AM EDT
Coffee Break
11th Floor Collaborative Space
10:00 - 10:45 AM EDT
Arithmeticity, superrigidity, and totally geodesic manifolds
11th Floor Lecture Hall
Speaker
Nicholas Miller, University of Oklahoma
Session Chair
Alan Reid, Rice University
Abstract
In the 1970s seminal work of Margulis showed that higher rank lattices have superrigid representations, which in particular implies that all such lattices are arithmetic. Since then Gromov–Piatetski-Shapiro and Deligne–Mostow have shown that a similar superrigidity theorem cannot hold for all lattices in the isometry group of real or complex hyperbolic space, i.e., in the rank 1 setting. However in recent work joint with Bader, Fisher, and Stover we show that one can prove certain superrigidity/arithmeticity theorems provided the associated manifold satisfies the geometric condition that it contains infinitely many (maximal) totally geodesic submanifolds. Specifically, we show that this latter criteria forces a hyperbolic manifold to be arithmetic.
The goal of this mini-course will be first to recount the work of Margulis on superrigidity of higher rank lattices and then to go on to discuss the proof of the aforementioned theorem. This will include a discussion of the connections between homogeneous dynamics and geodesic submanifolds, their interaction with superrigidity, and an introduction to techniques introduced by Bader and Furman for studying algebraic representations of ergodic actions. If time permits, we will also discuss the analogous theorem for complex hyperbolic manifolds and the key differences from the real hyperbolic setting.
The goal of this mini-course will be first to recount the work of Margulis on superrigidity of higher rank lattices and then to go on to discuss the proof of the aforementioned theorem. This will include a discussion of the connections between homogeneous dynamics and geodesic submanifolds, their interaction with superrigidity, and an introduction to techniques introduced by Bader and Furman for studying algebraic representations of ergodic actions. If time permits, we will also discuss the analogous theorem for complex hyperbolic manifolds and the key differences from the real hyperbolic setting.