Winter /Spring Term 2006

MAT 1847HS

**HOLOMORPHIC DYNAMICS**

*Instructor*: **M. Lyubich**

MAT 1846HS

**SEVERAL GEMS OF COMPLEX DYNAMICS**

*Instructor*: **M. Yampolsky**

Fall Term 2005:

MAT 1502HF

**STOCHASTIC LOEWNER EVOLUTION**

*Instructor*: **I. Binder**

MAT 1739HF

**RENORMALIZATIONS: FROM CIRCLE DIFFEOMORPHISMS
TO KAM THEORY**

*Instuctor*: **K. Khanin**

MAT 1844HF

**RENORMALIZATION IN ONE-DIMENSIONAL DYNAMICS**

*Instructor*: **M. Yampolsky**

MAT 1847HS

**HOLOMORPHIC DYNAMICS**

*Instructor*: **M. Lyubich**

**Start date: Tuesday, January 17, 2006**

**Tuesdays & Thursdays, 11:10 - 12:30**

The central theme of this course will be the Rigidity Conjecture in
Holomorphic Dynamics that asserts that any two rational maps (except
one special class of maps covered by torus automorphisms) which are
topologically conjugate must be conjugate by a Moebius transformation.
This Conjecture is intimately related to the Mostow Rigidity phenomenon
in hyperbolic geometry. In the quadratic case, it is related to the
MLC Conjecture asserting that the Mandelbrot set is locally connected.
After covering necessary background in basic holomorphic dynamics and
renormalization theory, recent advances in the problem will be discussed.

MAT 1846HS

**SEVERAL GEMS OF COMPLEX DYNAMICS**

*Instructor*: **M. Yampolsky**

**Start date: Tuesday, January 17, 2006**

**Tuesdays & Thursdays, 3:30 - 5:00**

In this course we plan to present a few beautiful and fundamental results
of modern complex dynamics. The course will be structured as a series
of mini-courses, which will be loosely related to each other. We plan
to make presentation self-contained, and accessible to a graduate student
with knowledge of basic complex analysis and differential geometry,
and interest in dynamics.

Some of the theorems we plan to cover are:

- Discontinuities in the dependence of Julia sets on parameters; Lavaurs
theorem.

- Theorem of Shishikura on Hausdorff dimension of the boundary of the
Mandelbrot set.

- A. Epstein's proof of Fatou-Shishikura bound on the number of non-repelling
cycles.

A common theme in the above results is the study of perturbations of
parabolic orbits. Further topics may include computability of Julia
sets, properties of Siegel disks, or other themes suggested by the audience.

MAT 1502HF

**STOCHASTIC LOEWNER EVOLUTION**

*Instructor*:** I. Binder**

**Start date: Tuesday, September 27th**

**Tuesdays & Thursdays, 3:00 - 4:30 p.m.**

Stochastic (or Schramm) Loewner Evolution (SLE) is a family of conformally
invariant random processes conjectured to describe the scaling limit
of various combinatorial models arising in Statistical Mechanics and
Conformal Field Theory, such as Loop Erased random walk, Self-avoiding
random walk, percolation, and the Ising model. SLE proved to be an important
link between Complex Analysis, Probability, and Theoretical Physics.
SLE has also been used by Lawler, Schramm, and Werner to verify the
Mandelbrot's conjecture about the dimension of the Brownian Frontier.
We start with a careful discussion of the necessary background from
Stochastic Analysis and the Geometric Function Theory. Than we move
to the proof of Mandelbrot's conjecture. Other topics that might be
covered are the dimension properties of the SLE and the proof of the
Smirnov's theorem about the critical limit of percolation.

References:

1. "Random Planar Curves and Schramm-Loewner Evolution" by
Wendelin

Werner. (http://arxiv.org/abs/math.PR/0303354)

2. "Conformal Restriction and Related Questions" by Wendelin
Werner.

(http://arxiv.org/abs/math.PR/0307353)

3. Conformally Invariant Processes in the Plane by Greg Lawler

MAT 1739HF

**RENORMALIZATIONS: FROM CIRCLE DIFFEOMORPHISMS TO KAM THEORY**

*Instructor*: **K. Khanin**

Start date: Week of September 26th

**Mondays 1:00 - 3:00 p.m. and/or Fridays 1:00 - 2:00 p.m.**

In the first part of the course we shall discuss the renormalization
approach to Herman theory and prove rigidity theorem for circle diffeomorphisms
with Diophantine rotation numbers. We then extend the whole construction
to circle diffeomorphisms with break points. Finally, in the third part
of the course we use renormalizations to prove a KAM-type theorem for
area-preserving twist maps.

MAT 1844HF

**RENORMALIZATION IN ONE-DIMENSIONAL DYNAMICS**

*Instructor*: **M. Yampolsky**

*Start date: Week of September 26th*

**Mondays 1:00 - 3:00 p.m. and/or Fridays 1:00 - 2:00 p.m.**

Renormalization ideas entered one-dimensional dynamics in the late 1970's
with the discovery of Feigenbaum universality. After seminal works of
Sullivan, and Douady and Hubbard it has revolutionized the field. The
course will serve as a self-contained introduction to this beautiful
subject, only familiarity with complex analysis will be assumed.

We will describe two main examples of renormalization in dynamics -
unimodal maps and critical circle maps. For the latter we will outline
the construction of the renormalization theory, culminating with Lanford
universality. For the former we will explain the connection of renormalization
to self-similarity of the Mandelbrot set.

### Taking the Institute's Courses for Credit

As graduate students at any of the Institute's University Partners,
you may discuss the possibility of obtaining a credit for one or more
courses in this lecture series with your home university graduate
officer and the course instructor. Assigned reading and related projects
may be arranged for the benefit of students requiring these courses
for credit.

### Financial Assistance

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graduate students are eligible to apply for financial assistance to
attend graduate courses. To apply for funding, apply
here

Two types of support are available: