SUMMER SCHOOL
Conformal and Quasiconformal Maps
Home page
Every participant will lecture on one of the following papers.
One paper is longer and will be discussed by two of the
participants. Some of the papers will only be read partially,
more detailed instructions are below.
The papers are not listed in the order of difficulty
nor in the order they will be presented, but somewhat
ordered according to the subject.
Every participant will lecture for 2 hours, split into two
lectures at different times of the schedule.
Every participant submits a 46 page summary of his or her
topic prior to the summer school. Preferrably in some
form of tex or latex. A
sample file
can be retrieved here.

The convergence of circle packings to the
Riemann map
(B. Rodin, D. Sullivan, J. Diff. Geom. 26, 349360, 1987)
(Appendix 1 needs not be done in detail, because it will
be superceded by topic 2. Appendix 2 should be done in detail,
a more detailed description of the construction can be found in
"On Thurston's Formulation and Proof of Andreev's
Theorem" by A. Marden and B. Rodin in Proceedings of Valparaiso,
Springer Lecture Notes 1435, pp 103115 (1990))
[[Klein]]

An estimate for hexagonal circle packings (Z. He, J. Diff. Geom.
33 no 2 pp 395412 (1991) (Up to (including) paragraph 3).
Remark: Beautiful pictures of circle packings can be
obtained from
Gareth Mc Caughan's page
[[Mohanty]]

Quasiconformal homeomorphisms and dynamics
I Solution of the FatouJulia problem on wandering
domains. (D Sullivan, Ann. of Math. 122 pp 401418 (1985))
(We care about nonexistence of wandering domains. The
result of this paper has already made it into text books.
A maybe easiertoread source for this theorem is
in the book "Complex Dynamics" by Carleson and Gamelin)
[[Lucas]]

Area distortion of quasiconformal mappings
(K. Astala, Acta Math. 173 pp 3760 (1994))
and On the area distortion by quasiconformal mappings
(A. Eremenko and D. Hamilton, Proc. AMS Vol 123
number 9, (1995))
(The article by Astala was a major breakthrough.
Eremenko and Hamilton wrote a shorter and sharper proof
of the main result in Astala, mostly following Astala.
The goal for the workshop is to discuss the topic
to the extend as it appears in the Eremenko/Hamilton
paper: the Astala paper is only listed as source of
better understanding and good side remarks, and also
to give due credit. This topic requires a bit of reading
on the side, such as the Gehring/Reich paper and the
Lehto/Virtanen book or other sources e.g. Ahlfors/Bers
(see references in the articles).)
[[Savic]]

Holomorphic motions and polynomial hulls,
(Z. Slodkowski, Proc.
Amer. Math. Soc. 111 (1991) 347355)
and
Holomorphic motions (K. Astala and G. Martin,
can be retrieved from
www.math.jyu.fi/research/report83.html , papers
on analysis dedicated to Olli Martio)
The presenter really should present the second paper,
which is a mostly expository paper explaining the
result of the first paper.
We care about a proof of Theorem 3.3.
The paper is basically self cvontained, only the proof
of theorem 4.1. may require some reading on the side.
[[Cai]]

A proof of the Bieberbach conjecture (L. de Branges. Acta Math 154
(1985) no 12 pp 137152)
(We really care about the simplified proof as presented
in "The Bieberbach conjecture" (L. Weinstein. Intern. Math. Res. Notices
(Appendix to Duke Math Journal) no. 5 1991 pp. 61.64.).
The proof of the Milin result and further details can be traced down
via the survey article "The Bieberbach conjecture and Milin
functionals" (A. Grinshpan. Amer. Math. Monthly 106 (1999) no. 3.
pp 203214) The addition theorem for Legrendre polynomials can be
found in the Bateman project "Higher transcendental functions"
by Erdelyi, Magnus, Oberhettinger, Tricomi.).
[[Grinshpan]]

The Loewner differential equation and slit
mappings (D. Marshall, S. Rohde, available through
http://www.math.washington.edu/~rohde)
We are mainly interested in the fact that Hoelder 1/2
continuous driving force generates slits, but if time
allows the converse can also be discussed.
Section 2 needs some side reading on quasiconformal
maps etc., some of that can be presented in the lecture
but probably not all.
[[Muscalu]]

Basic properties of SLE (S. Rohde, O. Schramm, available
through http://www.math.washington.edu/~rohde) Part I
The second paragraph of the abstract of the paper starts
"The present paper attempts a first systematic study of SLE.
It is proved that for all kappa not equal 8 the SLE trace is
a path" The purpose of the first part of this series is to
prove this statement, which is basically Theorem 5.1.
This requires most of the article prior to Theorme 5.1, but
some parts such ass Lemma 3.3 can be skipped.
[[Tsai]]

Basic properties of SLE (S. Rohde, O. Schramm, available
through http://www.math.washington.edu/~rohde) Part II
The second paragraph of the abstract continues
"for kappe in [0,4] it is a simple path; for
kappa in (4,8) it is a selfintersecting path; and
for kappa>8 it is space filling"
The second part of the series is to discuss this sentence.
This requires mostly presentation of sections 6 and 7 of the
article.
[[Li]]

Conformal invariance of planar looperased
random walks and uniform spanning trees
(G. Lawler, O. Schramm, W. Werner, can be retrieved from
front.math.ucdavis.edu/math.PR/0112234 )
We only care about the first part, which is LERW
(Theorem 1.1). That is, up to page 33 only (still a lot),
save for everything on pages 133 that is not geared
towards Theorem 1.1.
[[Meyer]]

Critical Percolation in the plane. I conformal
invariance and cardy's formula. II continuum scaling limit.
(S. Smirnov. available through http://www.math.kth.se/~stas
(long version))
Sections 1 and 2. Lemma 4.1 is needed and can be looked up
in Grimmetts book, (11.70 in second edition, 9.70 in first edition.)
[[Molnar]]