Office: 4-112 Mathematics Building
Phone: (631)-632-8274
Dept. Phone: (631)-632-8290
FAX: (631)-632-7631
Homepage
Mondays and Wednesdays, 11:00am to 12:20pm. Math Tower 5-127. (time and place may change; check the math department website for latest details). 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.
To be announced. You can always email me a question, or contact me to set up an appointment for an in-person or Zoom meeting. If I am in my office, I am usually happy to speak with you anytime.
I will assume the first year core courses in real and complex analysis. There is a large overlap between the first few weeks of this course, and the first few weeks of my Spring 2025 topics course on harmonic measure. I won't assume this material, but you can get a head start on this course by looking at the slides and recordings from MAT 627, Spring 2025 . The main overlap occurs in the first eight meetings that concerned extremal length, logarithmic capacity and conformal mappings. In this course, We will reprove some of the same facts, but in a slightly different way.
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.
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Introduction to harmonic measure .
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Table of contents and preface of 'Harmonic Measure' by Garnett and Marshall.
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Chapter 1 .
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Chapter 2 .
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Chapter 3 .
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Chapter 4 .
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Chapter 5, Sections 1 and 3 .
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Chapter 6, .
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Chapter 7, .
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Chapter 8, .
Class recordings will eventually be posted HERE (usually within 24 hours of class).
I will update this as the semester proceeds.
Mon, Jan 26; University closed (snowday).
Wed, Jan 28:
First class, introduction to course (due to technical problems and a fire alarm, class only lasted 50 minutes)
Mon, Feb 2: Continue introduction to harmonic measure, Start textbook, Section I.1
harmonic measure on unit disk and half-plane.
Wed, Feb 4: Sections I.1 and I.2, Dirichlet problem on half-plane, Fatou's theorem on non-tangential limits.
Mon, Feb 9: Class canceled
Wed, Feb 11: Sections I.3 and I.4: Caratheodory's theorem, the hyperbolic metric, Koebe's 1/4 theorem.
Mon, Feb 16: Section I.5: The Hayman-Wu theorem
Wed, Feb 18: Sections II.1 and II.2: Schwarz alternating method and Green's function
Mon, Feb 23: University closed (snowday).
Wed, Feb 25: Sections III.1 and III.2: Capacity and the logarithmic potential
Mon, Mar 2: Sections III.3 and III.4: The energy integral and
equilibrium distribution.
Wed, Mar 4: No class.
Mon, Mar 9: Sections III.5, III.6 and III.7: Wiener's solution of Dirichlet problem, regular points, Wiener's series.
Wed, Mar 11: Sections III.8 and III.9:
Polar sets, sets of harmonic measure zero, and estimates for harmonic measure.
March 16-20: No classes, Spring Break:
Mon, Mar 23: (pre-recorded) Sections IV.1, III.2 and III.3:
Extremal length.
Wed, Mar 25:
Sections IV.5, IV.6 and V.1:
Extremal distance, harmonic measure estimates,
and Denjoy conjecture.
Mon, Mar 30:
Wed, Apr 1:
Mon, Apr 6:
Wed, Apr 8:
Mon, Apr 13:
Wed, Apr 15:
Mon, Apr 20:
Wed, Apr 22:
Mon, Apr 27:
Wed, Apr 29:
Mon, May 4:
Wed, May 6: Last class
Notes on Harmonic Measure by Marti Prats and Xavier Tolsa. Covers the higher dimensional version of much of what we will discuss in the plane. Much is less is known in higher dimensions, but much of that is due to Tolsa.
Complex Analysis by Don Marshall. This was the textbook for MAT 536 in Srping 2024.
Geometric Function Theory by Tom Carroll. This undergraduate text covers in more length and detail much about conformal maps that I will review rapidly. Much of this book lies in between a standard undergraduate course in complex analysis and the more advanced book by Garnett and Marshall.
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.
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.
The Ubiquitous Quasidisk by Fred Gehring and Kari Hag, AMS Mathematical Surveys and Monographs, vol 184, 2012.
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
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.
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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