Math 570: Teichmüller Theory and Geometric Structures

Emily Dumas

University of Illinois at Chicago
Spring 2013

General information

Instructor	Emily Dumas (ddumas@math.uic.edu)
Office hours	Monday 11-12 and Wednesday 12:30-1:30
CRN	34260
Lectures	MWF 2:00 - 2:50pm in Taft Hall 313
Texts	Complex projective structures, a chapter from the Handbook of Teichmüller Theory. (EMS Publishing House, 2009)
	C. McMullen, Riemann Surfaces, Dynamics, and Geometry (Graduate lecture notes)
	R. Benedetti and C. Petronio, Lectures on Hyperbolic Geometry. (Springer, 1992)
	W. Goldman, Locally homogeneous geometric manifolds, from the proceedings of the 2010 International Congress of Mathematicians. (World Scientific, 2011)
	J. H. Hubbard, Teichmüller Theory, vol. 1. (Matrix Editions, 2006)
	F. Labourie, Lectures on Representations of Surface Groups. (Graduate lecture notes)
	O. Lehto, Univalent Functions and Teichmüller Spaces. (Springer, 1987)
	W. P. Thurston, Geometry and Topology of 3-Manifolds (Graduate lecture notes)

Course description

In this course we will discuss several types of geometric structures on surfaces and their deformation theories, emphasizing how complex-analytic and geometric aspects of these theories interact. Complex structures, hyperbolic structures, and complex projective (CP¹) structures will be discussed in detail.

The course is intended for graduate students who have completed a complex analysis course (e.g. math 535) and a first course in differential geometry (e.g. math 549) at the graduate level.

The final list of topics to be covered may be altered somewhat to accommodate the interests of the students in the course and the pace that suits their common mathematical background. A tentative syllabus follows.

• Prologue: The Schwarzian derivative a geometric function theory
• Compact Riemann surfaces and the complex Teichmüller space
• Compact hyperbolic surfaces and the hyperbolic Teichmüller space
• The uniformization theorem and identification between Teichmüller spaces
• The notion of locally homogeneous geometric structure (or (G,X) structure)
• Complex projective (CP¹) structures on surfaces
• Schwarzian derivative parameterization of complex projective structures
• Hyperbolic geometry of CP¹ structures - grafting and bending
• Comparing projective, hyperbolic, and conformal deformations
• Results and questions about holonomy of CP¹ structures
• Sections of the bundle of CP¹ structures, relation to Weil-Petersson geometry
• Other geometric structures: Flat affine, RP², ambient structures in higher-dimensional models.

Course materials

• Problem list   -   Last updated
• Conformal mapping handout
  • Images created by Donald Marshall using the numerical conformal mapping program Zipper.
• Hyperbolic tilings handout
  • Pictures of hyperbolic tilings created with Curt McMullen's program lim.
• Simple closed curve handout
  • A quasi-Fuchsian limit set, created with David Wright's program kleinian.
• Irreducible holonomy

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