Topics in Complex Analysis:

From Conformal Invariants to Percolation

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Math 219 - Tu Th 12-1:15 - Virtual Reality
Harvard University - Spring 2021

Topics may include:

• The hyperbolic metric on the unit disk
• Conformal mapping
• The Poincaré metric on a plane region
• Extremal length
• Quasiconformal mappings
• Transfinite diameter
• Capacity
• Harmonic measures
• Discrete harmonic functions
• Random walks
• Brownian motion
• Random walks in the hyperbolic plane
• Martingales and Furstenberg's theorem
• Percolation
• Cardy's formula

• Löwner's method
• Stochastic Löwner evolution (SLE)
• The Gaussian free field

• C. McMullen, From conformal invariants to percolation
  All assigned homework will appear in these notes. These rough notes will evolve during the course; be sure to use the current version, linked above.
• C. McMullen, Advanced Complex Analysis
  The background on complex analysis needed in this course is largely covered in these notes.
• C. McMullen, Probability Theory
  Basic background on probability theory needed in this course is covered in these notes. See especially the chapter on random walks.

• L. V. Ahflors, Conformal Invariants: Topics in Geometric Function Theory. AMS, 1973.
• L. V. Ahflors, Lectures on Quasiconformal Mappings. AMS, 2006.
• L. V. Ahflors, Complex Analysis. McGraw-Hill, 1979.
• N. Berestycki and J. R. Norris, Lectures on SLE.
• B. Bollobas and O. Riordan, Percolation. Cambridge University Press, 2006.
• Dynkin and Yushkevich, Markov Processes. Plenum Press, 1969.
• Garnett and Marshall, Harmonic Measure. Cambridge University Press, 2005.
• S. Janson, Gaussian Hilbert Spaces. Cambridge University Press, 2009.
• G. F. Lawler, Conformally Invariant Processes in the Plane. AMS, 2005.
• G. F. Lawler and V. Limic, Random Walk: A Modern Introduction. Cambridge University Press, 2012.
• J-F. Le Gall, Brownian Motion, Martingles, and Stochastic Calculus. Springer, 2013.
• J. Milnor, Dynamics in One Complex Variable. Princeton University Press, 2006.
• P. Moerters and Y. Peres, Brownian Motion. Cambridge University Press, 2010.
• Nehari, Conformal Mapping. Dover 1952.

• B. Bollobas and O. Riordan, A short proof of the Harris-Kesten theorem. Bull. London Math. Soc. 2006.
• R. Courant, H. Firedrichs and H. Lewy, On the partial difference equations of mathematical physics. Math. Ann. 1928 (translation).
• B. Derrida and S. Stauffer, Corrections to scaling and phenomenological renormalization for 2-dimensional percolation and lattice animal problems. J. Physique 46 (1985), 1632-1630.
• Duminil-Copin and Smirnov, Conformal invariance of lattice models. Clay Math. Proc., 2012.
• H. Furstenberg Noncommuting random products. Trans. AMS 1963.
• Galton, One-dimensional random walks.
• E. P. Hsu, A brief introduction to Brownian motion on a Riemannian manifold. PSS03, Kyushu, 2003.
• R. Kenyon, Long-range properties of spanning trees. J. Math. Physics, 2000.
• C. McMullen Barycentric subdivision, martingales and hyperbolic geometry Preprint, 2011.
• O. Schramm, Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 2000.
• O. Schramm, Conformally invariant scaling limits. ICM Address, 2006.
• S. Sheffield, Gaussian free fields for mathematicians. Arxiv, 2006.
• S. Smirnov, Critical percolation in the plane. CRAS, 2001.
• S. Smirnov, Critical percolation in the plane (long version). Preprint, 2001.

Hardware requirements. Students are required to have iPads or similar devices to facilitate online collaboration and to participate in online discussions. Click here to request an iPad from Harvard.

Prerequisites. Intended for graduate students and advanced undergraduates. Basic complex analysis is essential (Math 113 and preferrably 213a). Familiarity with real analysis, probability theory, topology and differential geometry on surfaces will be useful, but relevant material will be reviewed in class. Background material on some of these topics will be covered in section (depending on enrollment), and can also be found in the suggested texts and the course notes.

Readings and Lectures. Assigned material should be read before class. Lectures may go beyond the reading, and not every topic in the reading will be covered in class.

Homework. Homework will be assigned every 1 or 2 weeks. Collaboration is encouraged on homework. Late homework is not accepted, but the lowest homework grade will be dropped. You are welcome to come to section and office hours for hints on the homework and to meet up with others for joint work.

Slack. This class has a Slack channel. Please participate!

Expectations. Enrolled students should attend lectures regularly and work on the homework assignments, which are an important part of the course. Students should share their video Zoom lectures whenever possible.

Grades. Letter grades, for those requiring them, will be based on homework. If you are taking the course for a grade, please come to the instructor's office hours early in the term to introduce yourself and share your background.

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Image by Scott Sheffield