Introduction to Riemann Surfaces

Introduction to Riemann Surfaces
מבוא למשטחי רימן

Spring semester 2017
March 26, 2017 - July 14, 2017

 

  • Lecturer
    Bo'az Klartag 
    Room 264, Ziskind building
    Phone: 08-9342844 
    e-mail:  

     
  • Classes
    Monday, 11:15 - 13:00, Ziskind 155.
     
  • TA sessions
    Wednesday, 16:15 - 17:00, Ziskind 155.
     
  • Syllabus
    Riemann surfaces are complex manifolds of one complex dimension or equivalently two real dimensions. They were discovered by Riemann in the 1850s in his studies of multivalued holomorphic functions such as the square root or the logarithm.

    The subject has evolved to a rather pretty branch of mathematics, with many connections to other areas. We offer an introductory course on the subject from the point of view complex analysis, geometry and topology.

    Here is a rather detailed plan of the course, each topic will probably be discussed  for 1-2 weeks:

  1. Holomorphic functions and monodromy
  2. Riemann surfaces, examples
  3. Degree of a map, algebraic Riemann surfaces
  4. Covering maps, Riemann existence theorem
  5. Normalization of an algebraic curve
  6. Riemann-Hurwitz, holomorphic 1-forms
  7. Elliptic functions and integrals
  8. Harmonic and subharmonic functions, Harnack's inequality
  9. Perron's method for the Dirichlet problem, Green's function, Riemann mapping.
  10. Elliptic, parabolic and hyperbolic surfaces, proof of the uniformisation theorem.
  • Prerequisites
    undergraduate complex analysis (say, Cauchy's theorem).
    undergraduate topology.
     
  • Final grade
    It will be based on the solution of the homework exercises. The ideas is that the students will be asked to present homework solutions during the TA sessions. 
     
  • Related literature
    There are tens of books on this classical topic, with great variation in emphasis and style. Among them we recommend:
  1. Ahlors, Sario, Riemann Surfaces, 1960.
  2. Beardon, A primer on Riemann Surfaces, 1984 (easy to read).
  3. Donaldson, Riemann Surfaces, 2011 (another book we will follow).
  4. Farkas, Kra, Riemann Surfaces, 1980.
  5. Forster, Lectures on Riemann Surfaces, 1981.
  6. Schlag, A course in complex analysis and Riemann Surfaces, 2014.
  7. For uniformization: Ahlors, Conformal Invariants: Topics in Geometric Function Theory, 1973. 
     

 

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