overview

This page contains lecture notes, problem sets and videos of the three short courses:

"Ergodic Theory of Hyperbolic Groups" by Danny Calegari

"Invariants of Hyperbolic 3-Manifolds" by Walter Neumann

"Function Theory on Symplectic Manifolds" by Leonid Polterovich

Ergodic Theory of Hyperbolic Groups

Lecturer: Danny Calegari (CalTech)

Assistant: Alden Walker (CalTech)

An introduction to the use of dynamical and probabilistic methods in geometric group theory, especially as applied to hyperbolic groups. I hope to discuss (central) limit theorems for random geodesics and random walks, behavior of characteristic functions (e.g. (stable) commutator length, w-length, etc.) under random homomorphisms, and a few other topics if time permits.

Lecture notes by Danny Calegari:

The ergodic theory of hyperbolic groups (PDF, draft 6 July 2011)

Problem sets by Alden Walker:

Exercises and examples for the workshop (without solutions) (PDF, draft 11 July 2011)

Exercises and examples for the workshop (with solutions) (PDF, draft 11 July 2011)

Invariants of Hyperbolic 3-Manifolds

Lecturer: Walter Neumann (Barnard College, Columbia University)

Assistant: Christian Zickert (Berkeley)

This short course will concentrate on number-theoretic invariants and number theoretic methods in the study of 3-manifolds. The first lecture will be a brief introduction to algebraic number theory, followed by four lectures on 3-manifolds, concentrating mostly on hyperbolic 3-manifolds.

Function Theory on Symplectic Manifolds

Lecturer: Leonid Polterovich (Chicago/Tel Aviv)

Assistant: Daniel Rosen (Chicago)

Function spaces associated to a symplectic manifold exhibit unexpected properties and interesting structures, giving rise to an alternative intuition and new tools in symplectic topology. These phenomena are detected by modern symplectic methods such as Floer theory and are closely related to algebraic and geometric properties of groups of Hamiltonian diffeomorphisms. I shall discuss these developments, their applications as well as links to other areas such as group quasi-morphisms and quantum-classical correspondence. All necessary symplectic preliminaries will be explained.

Lecture notes by Leonid Polterovich:

Part 1: Three wonders of symplectic geometry (PDF, draft 11 July 2011)

Part 2: C⁰–rigidity of the Poisson bracket (PDF, draft 12 July 2011)

Part 3: Calabi (PDF, draft 13 July 2011)

Part 4: Symplectic quasi-states (PDF, draft 13 July 2011)

Problem sets by Daniel Rosen:

Problem session 1 (PDF, draft 13 July 2011)

Problem session 2 (PDF, draft 13 July 2011)

Problem session 3 (PDF, draft 13 July 2011)

Problem session 4 (PDF, draft 13 July 2011)

Problem session 5 (PDF, draft 15 July 2011)

References:

Dusa McDuff and Dietmar Salamon: Introduction to symplectic topology (Oxford University Press, 1998)

Leonid Polterovich: The geometry of the group of symplectic diffeomorphisms (Birkhäuser Verlag, 2001)

videos

Recordings of the lectures can be downloaded below. The size of each file is between 420 and 504 MB.

	Monday, 11.7.	Tuesday, 12.7.	Wednesday, 13.7.	Thursday, 14.7.	Friday, 15.7.
Calegari	Lecture 1	Lecture 2	Lecture 3	Lecture 4	Lecture 5
Neumann	Lecture 1	Lecture 2	Lecture 3	Lecture 4	Lecture 5
Polterovich	Lecture 1	Lecture 2	Lecture 3	Lecture 4	Lecture 5

timetable

	Monday, 11.7.	Tuesday, 12.7.	Wednesday, 13.7.	Thursday, 14.7.	Friday, 15.7.
09:00-10:15	Danny Calegari	Danny Calegari	Danny Calegari	Danny Calegari	Danny Calegari
10:15-10:45	Morning tea	Morning tea	Morning tea	Morning tea	Morning tea
10:45-12:00	Leonid Polterovich	Leonid Polterovich	Leonid Polterovich	Leonid Polterovich	Leonid Polterovich
12:00-13:30	Lunch	Lunch	Lunch	Lunch	Lunch
13:30-14:45	Walter Neumann	Walter Neumann	Walter Neumann	Walter Neumann	Walter Neumann
14:45-15:00	Break	Break	Geoff Prince (AMSI)	Break	Break
15:00-15:45	Problem sessions	Problem sessions	Problem sessions	Problem sessions	Problem sessions
15:45-16:15	Afternoon tea	Afternoon tea	Afternoon tea	Afternoon tea	Afternoon tea
16:15-17:00	Problem sessions	Problem sessions	Problem sessions	Problem sessions	Problem sessions