Advanced Courses and Seminars


Semester A:


Course number 80894 (Unipotent Flows - a graduate course in the framework of Kazhdan's
Sunday seminar)

Lecturer: Elon Lindenstrauss

Raghunathan conjectured that If G is a Lie group, Gamma a lattice, p in G/Gamma, and U an (ad-)unipotent group then the closure of U.p is homogeneous (a periodic orbit of a subgroup of G). This conjecture was proved by Ratner in the early 90's via the classification of invariant measures; significant special cases were proved earlier by Dani and
Margulis using a different, topological dynamics approach.

The proof of the Raghunathan conjecture given by Ratner as well as the proofs given by Dani and Margulis are not effective, nor do they provide rates --- e.g. if p is generic in the sense that it does not lie on a periodic orbit of any proper subgroup L<G with U<=L, these proofs do
not give an estimate (possibly depending on diophantine-type properties of the pair (p,U))) how large a piece of an orbit is needed so that it comes within distance epsilon of any point in a given
compact subset of G/Gamma.

The main purpose of my course will be to preset work in progress with Margulis, Mohammadi and Shah giving an effective and quantitative density theorem for orbits of unipotent groups (though the actual quantitative estimates are very modest). I will try to keep it essentially self-contained.



Topology   80674

Lecturer: Tomer Schlank

Time: Monday 9:00-11:45

This is a seminar (the students will give the lectures) on Morse Theory. Morse Theory is a beautiful theory which can be used to connect geometry, analysis and algebraic topology. The theory has many applications especially in the study of manifolds. The theory can be used to obtain many results on the homotopy, diffeomorphism and cobordism type of manifolds. We will largely follow the book: morse Theory by Milnor but might use occasionally other resources  



Group Theory  80549

Lecturer: Alex Lubotzky

Time: Tuesday 11-13, Ross A70


We will study together a few advanced topics relates to discrete groups, profinite groups, sofic groups etc.



Geometric Group Theory  80614


Lecturer: Chloe Perin

Time: Mondays 9-11 and Wednesdays 15-16

Place: Shprinzak 102


A gentle introduction to geometric group theory: presentations of groups, Cayley graphs, decision problems, quasiisometry and quasiisometry invariants, hyperbolic groups, limit groups.




Percolation Theory - 80988 

Lecturer: Ori Gurel-Gurevich

Time: Thursdays 12-2

Place: Shprinzak 102


This course is an introduction to Percolation theory. The material coverd: Definition of percolation. Existence of critical probability. Galton-Watson branching process. Invariant percolation on trees. Bernoulli percolation on Z^d. Russo's formula, FKG, BK inequalities. Uniqueness of infinite cluster. The critical probability of bond percolation on the square lattice is 1/2. Smirnov's proof of conformal invariance and Cardy's formula. Overview of Schramm-Loewner Evolution.




Semester B:


Seminar in Analysis 80742

Lecturer: Raz Kuperman

Time: Monday 10:00-11:45
The seminar will be an introduction to variational calculus. We will read together an introductory text on the subject. The pre-requisites are background in functional analysis, preferably with some knowledge in the theory of Sobolev spaces.



High dimensional expanders  80923

Lecturer: Alex Lubotzky

Time: Sunday 10-12, Math 209    


The course will deal with simplicial complexes and their homology/cohomology theory. We will be mainly interested in their expansion/mixing properties and the connection to the spectral theory of the higher dimensional Laplacians. The most important examples, which will be studied in detailed, will be affine buildings and their finite quotients.




Ergodic Theory  80615

Lecturer: Mike Hochman

The course is an introduction to ergodic theory. The goal is to introduce the student to the main ingredients of the classical theory, including results on recurrence, space- vs. time-averages and ergodic decomposition, spectral invariants, mixing, and entropy. In detail, the topics we will cover are: Poincare recurrence, existence of invariant measures in topological systems, the ergodic theorem (mean and pointwise), ergodic decomposition, unique ergodicity, weak mixing (characterizations via spectral theory, multiplier property and isometric factors), strong mixing, Shannon and Kolmogorov-Sinai entropy, Shannon-McMillan-Breiman theorem, systems with completely positive entropy and Pinsker factor. If time permit we will discuss Furstenberg disjointness and the Rohlin lemma and its applications.