Conformal and Quasiconformal Maps

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Topics for the participants' lectures

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 4-6 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.

  1. The convergence of circle packings to the Riemann map (B. Rodin, D. Sullivan, J. Diff. Geom. 26, 349-360, 1987) (Appendix 1 needs not be done in detail, because it will be superceded by topic 2.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 103-115 (1990)) [[Klein]]

  2. An estimate for hexagonal circle packings (Z. He, J. Diff. Geom. 33 no 2 pp 395-412 (1991) (Up to (including) paragraph 3). Remark: Beautiful pictures of circle packings can be obtained from Gareth Mc Caughan's page [[Mohanty]]

  3. Quasiconformal homeomorphisms and dynamics I Solution of the Fatou-Julia problem on wandering domains. (D Sullivan, Ann. of Math. 122 pp 401-418 (1985)) (We care about non-existence of wandering domains. The result of this paper has already made it into text books. A maybe easier-to-read source for this theorem is in the book "Complex Dynamics" by Carleson and Gamelin) [[Lucas]]

  4. Area distortion of quasiconformal mappings (K. Astala, Acta Math. 173 pp 37-60 (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]]

  5. Holomorphic motions and polynomial hulls, (Z. Slodkowski, Proc. Amer. Math. Soc. 111 (1991) 347-355) and Holomorphic motions (K. Astala and G. Martin, can be retrieved from , 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]]

  6. A proof of the Bieberbach conjecture (L. de Branges. Acta Math 154 (1985) no 1-2 pp 137-152) (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 203-214) The addition theorem for Legrendre polynomials can be found in the Bateman project "Higher transcendental functions" by Erdelyi, Magnus, Oberhettinger, Tricomi.). [[Grinshpan]]

  7. The Loewner differential equation and slit mappings (D. Marshall, S. Rohde, available through 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]]

  8. Basic properties of SLE (S. Rohde, O. Schramm, available through 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]]

  9. Basic properties of SLE (S. Rohde, O. Schramm, available through 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 self-intersecting 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]]

  10. Conformal invariance of planar loop-erased random walks and uniform spanning trees (G. Lawler, O. Schramm, W. Werner, can be retrieved from ) 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 1-33 that is not geared towards Theorem 1.1. [[Meyer]]

  11. Critical Percolation in the plane. I conformal invariance and cardy's formula. II continuum scaling limit. (S. Smirnov. available through (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]]