| 2010-9-7 Tuesday | │首页│简介│学术活动│新闻│图书馆│下载│ENGLISH│ |
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FROM 2010-6-1 TO 2010-12-31 Organizer: Guizhen Cui, Yunping Jiang and Yuefei Wang |
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●INTRODUCTION
l 沈维孝 (NationalUniversity of Singapore) TAN Lei (Universite de Angers) Walter Bergweiler (Kiel University, Germany) Laurent Bartholdi (Universität zu Göttingen) Michel Zinsmeister (Universite de Orleans) Arnaud Cheritat (Institut de Mathematiques deToulouse) Sebastien Godillon (Universite de Cergy-Pontoise 伍胜健（北京大学） 乔建永（中国矿业大学） 漆 张广远（清华大学） 方丽萍（北京理工大学） 尹永成（复旦大学） 邱维元（复旦大学） 沈玉良（苏州大学） 张高飞（南京大学） 刘劲松 ( 程 范金华（南京理工大学） 彭文娟（数学院） 翟 李保奎（北京理工大学） 高君杨（中国矿业大学） 周 廖良文（南京大学）
l 1. Title: Dynamics of transcendental entire functions 1. Title: One-dimensional Dynamics (I)
2. Title: One-dimensional Dynamics (II)
3. Title: Teichmuller space and geodesic currents
4. Title: The best bound of the area-length ratio in Ahlfors' covering surface theory
5. Title: No invariant line fields on escaping set of a sine family
6. Title: On the convergence of Thurston's algorithm for hyperbolic exponential maps
7. Title: Combinatorial rigidity for unicritical maps
8. Title: Holomorphic motions, homotopy, and homologous
9. Title: Density of hyperbolic rational maps
10. Title: Complex manifold structure on a bundle of universal Teichmuller space
l 1. Title: Dynamics of transcendental entire functions Speaker: Walter Bergweiler
Complex dynamics is concerned with the behavior ofanalytic functions under iteration. The iteration theory of rational functionsbegan with long memoirs by Fatou and Julia in 1918-1920. In the 1980s the areabecame again an area of very active research. This is partially due to thebeautiful computer graphics related to it, but - more importantly - there werealso new and powerful mathematical methods introduced to the subject bySullivan, Douady and Hubbard, and others.
The iteration of transcendental entire functions wasfirst studied by Fatou in 1926, but in recent years there has been majorinterest in the subject again. In part the theory runs parallel to that ofrational functions, but there are also significant differences. For example,while a theorem of Sullivan says that rational functions do not have wanderingdomains, such domains may exist for transcendental entire functions. Besides theJulia set, a major role in transcendental dynamics is played by the escapingset consisting of those points which tend to infinity under iteration. By atheorem of Eremenko, the Julia set is the boundary of the escaping set.
The purpose of this lectures is to give anintroduction to complex dynamics. While the focus will be on the dynamics oftranscendental entire functions, we do not assume familiarity with the dynamicsof rational functions. Some mathematical tools that are particularly helpful inthe iteration theory of transcendental entire functions (e.g. the Ahlforstheory of covering surfaces or Wiman-Valiron theory) will also be reviewed.
2. Title: Algebraic invariants of holomorphicdynamics Speaker: Laurent Bartholdi
Self-similar (or fractal) objects abound ingeometry, but their impact on algebra is relatively new. Groups, associativealgebras, and Lie algebras are called self-similar if they are equippedwith a biset (respectively bimodule), namely a set (resp. module) withcommuting left and right actions, which is free qua right set (resp.module). Important examples include the infinite torsion groups, and groups ofexponential growth, by Grigorchuk inter alia.
A dynamical system may be conveniently encoded as aself-similar group; this yields an extremely potent algebraic invariant of thatdynamical system, and a link between dynamics and algebra.
The purpose of this program is to study the linksbetween group theory and holomorphic dynamics, with particular focus onalgebraization of Thurston's algorithm.
3. Title: Random and deterministic growth processes Speaker: Michel Zinsmeister
In 1923, in order to prove Bieberbach conjecture forunivalent functions for n=3, Lowner has shown how one can associate to acontinuous function (called the driving function) from the positive realhalf-line to the circle an increasing family of compact sets in the plane. In1999, Oded Schramm revived this theory by showing that if we choose as drivingfunction a Brownian motion then one obtains sets that describe the scalinglimit of many interfaces coming from statistical mechanics such as the onesoccurring in percolation theory. This fundamental discovery revived the studyof deterministic Lowner processes as well as growth processes in general. Thereis a random growth process initiated by physicists called DLA which is still acomplete mystery from the mathematical point of view and it is hoped thatLowner theory could yield some light on it.
The purpose of this program is to give a detailedaccount of general Lowner theory followed by its applications. Of course theSLE processes of Schramm will be discussed in details. Hele-Shaw flows and theHastings-Levitov model of DLA will also be discussed, the emphasis being put onthe connection with Lowner theory.
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