Office: 4-112 Mathematics Building
Phone: (631)-632-8274
Dept. Phone: (631)-632-8290
FAX: (631)-632-7631
Homepage
Tuesdays and Thursdays, 12:30 to 1:50. Physics P-116. Although we meet in person, I will try to put the meetings on Zoom and make the recordings available within a few days. If you would like to join the class remotely, contact me for the Zoom information.
Tue 2-3:30, Thur 11-12:30. You can always email me a question, or contact me to set up an appointment for an in-person or Zoom meeting. If you can find me in my office, I am usually happy to speak with you.
I will try to produce PDF slides for each class. These will be posted here, usually grouped by topic and each collection covering several lectures.
First day: Brief introduction to quasiconformal mappings and some of the topics I hope to cover.
Rapid review of complex analysis: concise review of material from the first year complex analysis course that will be useful in this class: conformal maps, Riemann mapping theorem, the uniformization theorem.
Extremal length and conformal modulus: definition and basic properties of conformal modulus, an extremal length proof of Koebe 1/4-theorem.
Logarithmic capacity: definition and basic properties of log-capacity. Pfluger's theorem, harmonic measure estimates.
Geometric definition of QC maps: definition of QC maps in terms of modulus, compactness of K-QC maps, Gehring-Hayman theorem, localness of QC maps, 1-QC maps are conformal.
Consequences of compactness: Properties following from compactness: characterization of quasicircles, quasisymmetric maps, removable sets, conformal welding.
Analytic definition of QC maps: QC maps are differentiable a.e., derivatives in L^p, measurable Riemann mapping theorem, holomorphic dependence
Cauchy and Beurling transforms : formula for a QC map in terms of dilatation, analytic dependence of map on dilatation
Distortion of area and dimension: Astala's theorems on distortion of area and Hausdorff dimension, Smirnov's k^2 theorem for quasilines.
Brief introduction to holomorphic motions: the definition a holomorphic motion and proof of the λ-lemma. State but not prove the extended λ-lemma.
Degenerate Beltrami equations: Lehto's theorem for solving certain Beltrami equations where dilation has sup norm 1.
All the slides in one big file (721 pages)
Class recordings are posted HERE
I will update this as the semester proceeds.
Tue, Jan 28: first day, introduction to course
Thur, Jan 30: review of complex analysis, Mobius transformations, normal families, Riemann mapping thm
Tue, Feb 4:
Defn of conformal modulus and extremal length, basic properties
Thur, Feb 6:
Symmetry of modulus, proof of Koebe 1/4-theorem using extremal length, intro to logarithmic capacity
Tue, Feb 11: More on logarithmic capacity,
Thur, Feb 13: Pfluger's theorem
Tue, Feb 18: Gehring-Hayman theorem, radial limits of conformal maps
Thur, Feb 20:
harmonic measure, algebra related to derivatives and dilatations
Tue, Feb 25: Geometric definition of K-QC maps. Holder continuity
Thur, Feb 27: Compactness of K-QC maps, localness of quasiconformality
Tue, Mar 4: quasisymmetric maps, extension to QC maps
Thur, Mar 6: the 3-point condition, conformal removability
Tue, Mar 11: Proof of Jones-Smirnov theorem
Thur, Mar 13: consequences of Jones-Smirnov theorem, biLipschitz reflections
Tue, Mar 18: No class, Spring Break
Thur, Mar 20: No class, Spring Break
Tue, Mar 25: Conformal welding
Thur, Mar 27: Overview of MRMT, review of real analysis, absolute continuity on lines
Tue, Apr 1: QC maps are differentiable a.e.. partial derivatives are square integrable
Thur, Apr 3: reverse Holder inequality, Bojarski's inequality via Gehring's proof, Pompeiu formula
Tue, Apr 8: weak convergence of dilatations, finish MRMT
Thur, Apr 10: Cauchy and Beurling transforms
Tue, Apr 15: Analytic dependence on parameters
Thur, Apr 17: Class canceled
Tue, Apr 22:
Astala's dimension distortion theorem,
Thur, Apr 24: Optimal p(K), Astala's dimension distortion theorem, Conformal dimension
Tue, Apr 29: Smirnov's theorem on dimensions of quasicircles,
Thur, May 1: Holomorphic motions, Degenerate Beltrami equations, Lehto's approach I
Tue, May 6:
Degenerate Beltrami equations, Lehto's approach II
Thur, May 8: last day,
We will not follow any particular book, but in the first part of the course I plan to discuss basic material found in several sources, such as Ahlfors' "Lectures on Quasiconformal Mappings". This is short, dense a book that covers a lot of material quickly, and I will try to cover some of the material somewhat more verbosely than Ahlfors does. Below is a list of relevant books on QC mappings (the links are mostly to publisher webpages). The later part of the course will touch on several topics discussed in greater depth in the book of Astala, Iwaniec and Martin (an excellent reference with lots of details included; highly recommended).
Quasiconformal Mappings (incomplete) lecture notes by Christopher Bishop. I will convert these notes into lecture slides for the course, and (I hope) eventually update the lecture notes based on the lectures.
Complex Analysis by Don Marshall. This was the textbook for MAT 536 last Spring (a prerequisite for this class).
Geometric Function Theory by Tom Carroll. This book covers in detail much about conformal maps that I will review rapidly in the first few lectures. Much of the same material was covered in MAT 536 last Spring, when we used Don Marshall's textbook.
Conformal Maps and Geometry by Dmitry Beliaev. World Scientific, 2020. Link is to first 60 pages from author's website.
Fractals in Probability and Analysis by Christopher Bishop and Yuval Peres. Cambridge University Press, 2017.
Real Analysis, 2nd edition by B.S. Bruckner, J.B. Bruckner and A.M. Thomson, ClassicalRealAnalysis.com, 2008.
Lectures on Quasiconformal mappings by Lars Ahlfors. Very concise introduction to most of the main topics we will discuss.
Quasiconformal mappings in the plane by Olli Lehto and Kalle Virtanen
Conformal Welding by David Hamilton, Chapter 4 of Handbook of Complex analysis (see next entry), 2002.
Handbook of Complex Analysis edited by Reiner Kuhnau, North-Holland, 2002.
Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane by Kari Astala, Tadeusz Iwaniec, and Gaven Martin
Teichmüller Theory and Applications to Geometry, Topology, and Dynamics, Vol. 1: Teichmuller Theory by John Hubbard
Quasiconformal Maps and Teichmüller Theory by Alastair Fletcher and Vladimir Markovic
The Ubiquitous Quasidisk by Fred Gehring and Kari Hag, AMS Mathematical Surveys and Monographs, vol 184, 2012.
There are many, many papers on quasiconformal maps, but below are a few of my personal favorites. If time permits after covering the basic properties of QC mappings, I will try to discuss a few of these. Let me know if there are other papers you would like me to insert here.
What is a quasiconformal map? by Juha Heinonen. Notices of the AMS, vol 53, no 11, pages 1334-1335.
A historical survey of quasiconformal mappings by Olli Lehto the AMS, vol 53, no 11, 1984, pages 1334-1335.
Quasiconformal Mappings, with applications to differential equations, function theory and topology by Lipman Bers, Bull, AMS vol 83 No 6, Nov 1977, 1083--1100.
Quasiconformal Mapping by Seppo Rickman, Annales Academia Scientiarum Fennica, Series A. I. Mathematica Volumen L3, 1988, 371-385.
Area distortion of quasiconformal mappings by Kari Astala, Acta Math 173(1994) 37-60.
On the area distortion by quasiconformal mappings by A. Eremenko and D.H. Hamilton. PAMS 123(1995) 2793-2797.
On a holder constant in the theory of quasiconformal mappings by Istvan Prause, Computational Methods and Function Theory (CMFT), 2014, vol 14, page 483-486.
Area distortion under quasiconformal mappings, by F.W. Gehring and E. Reich, Annals Acad. Sci. Fenn. vol 388, 1966..
Removability theorems for Sobolev functions and quasiconformal mappings, by P.W. Jones and S.K. Smirnov, Ark. Mat. 38(2000), 263-279.
Non-removability of the Sierpinski gasket by D. Ntalampekos, Invent. Math., 216(2019), 519-595.
On removable sets for holomorphic functions by M. Younsi, EMS 2(2015), 219-254 .
Conformal removability is hard by C.J. Bishop, preprint.
Constructing entire functions by quasiconformal folding by C.J. Bishop, Acta Math 214(2015) 1-60..
Conformal welding and Koebe's theorem , by C.J. Bishop, Annals Math, 166(2007), 613-656 .
Models for the Eremenko-Lyubich class, by C.J. Bishop, JLMS 92(2015), 202-221 .
Dimensions of quasicircles by S.K. Smirnov, Acta Math. 205(2010) 189-197.
Hausdorff measure of quasicircles by I.Prause, X. Tolsa and I. Uriarte-Tuero, Adv. in Math., 229(2012), 1313-1228.
Quasisymmetric dimension distortion of Ahlfors regular subsets of a metric space . C.J. Bishop, H.Hakobyan and M. Williams, GAFA 26(2016), 379-421.
Metric definition of quasiconformality and exceptional sets by D. Ntalampekos, Math. Ann., 389(2024), 3231-3253.
Definitions of Quasiconformality by J. Heinonen and P. Koskela, Invent. Math., 120(1995) 61-79.
Solutions de L'equation de Beltrami avec |mu| =1 by Guy David, AASF Series A.I. Mathematica Vol 13, 1988, 25-70.
The Theory of Quasiconformal Mappings in Higher Dimensions, I by Gavin Martin, Handbook of Teichmüller theory. Vol. IV, 619–677, IRMA Lect. Math. Theor. Phys., 19, Eur. Math. Soc., Zürich, 2014.
Characterizations of Quasidisks by Fred Gehring, Banach Center Publications, vol 48, Polish Academy of Sciences, Warszawa, 1999.
Conformal dimension via $p$-resistance : Sierpinski carpet by Jaroslaw Kwapisz, AASF, vol 45, 2020, 3-51.
On the distortion of boundary sets under conformal mappings by N.G. Makarov, Proc. London Math. Soc. (3) 51 (1985), no. 2, 369–384.
Hausdorff dimension of harmonic measures in the plane by Peter Jones and Tom Wolff, Acta Math. 161 (1988), no. 1-2, 131–144.
On the Hausdorff dimension of harmonic measure in higher dimension. by Jean Bourgain, Invent. Math. 87 (1987), no. 3, 477–483.
Bloch functions, asymptotic variance, and geometric zero packing. by Haaken Hedenmalm, American Journal of Mathematics, Volume 142, Number 1, February 2020, pp. 267-321
Quasicircles of dimension $1 + k^2$ do not exist by Oleg Ivrii, preprint 2017.
Definitions of quasiconformality by Juha Heinonen and Pekka Koskela, Invent. math. 120, 61-79 (1995).
Quasiconformal maps in metric spaces with controlled geometry by Juha Heinonen and Pekka Koskela, Acta Math., 181 (1998), 1–61.
Julia and John by Lennart Carleson, Peter Jones and Jean-Chistophe Yoccoz, Bulletin/Brazilian Mathematical Society, Volume 25, pages 1–30, (1994)
Collet, Eckmann and Holder by Jacek Graczyk and Stas Smirnov, Inventiones vol 133, 69-96, 1998.
Extending homeomorphisms of the circle to quasiconformal homeomorphisms of the disk by David Epstein and Vladimir Markovic, Geometry & Topology, 11 (2007), pages 517–595.
Quasiconformal homeomorphisms and the convex hull boundary by David Epstein, Al Marden and Vladimir Markovic, Annals of Math., 159(2004), pages 305-336.
Conformal welding and Koebe's theorem by Christopher Bishop, 166(2007), pages 613-656.
The not too short introduction to LaTex
Mathematical biographies at the St. Andrews MacTutor website.
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