**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.