THEMATIC PROGRAMS

November 26, 2014

Fall 2005 Program on Renormalization and Universality in Mathematics and Mathematical Physics
&
Winter/Spring2006 Program on Holomorphic Dynamics, Laminations, and Hyperbolic Geometry

Graduate Courses


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

As part of the Affiliation agreement with some Canadian Universities, 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:

  • Students outside the greater Toronto area may apply for travel support. Please submit a proposed budget outlining expected costs if public transit is involved, otherwise a mileage rate is used to reimburse travel costs. We recommend that groups coming from one university travel together, or arrange for car pooling (or car rental if applicable).

  • Students outside the commuting distance of Toronto may submit an application for a term fellowship. Support is offered up to $1000 per month.

    For more details on the thematic year, see Program Page or contact holodynamics@fields.utoronto.ca

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