A seminar on Riemann surfaces by Prof Kapil Hari is organized at IISER Mohali during the summer of 2025. Details follow.
(Possible) pre-requisites:
- Open mapping theorem in complex analysis
- Definition of Riemann surfaces and understanding that one smooth 2-manifold may have multiple Riemann surface structures (such as is not biholomorphic to , different complex tori, etc)
- History of Jacobi and Abelian integrals
notes
The notes for the topics discussed will be kept here: https://dub.sh/riemann
plan
- June to July: Essentially some topics from chapters 4 to 15 from Narasimhan - Compact Riemann Surfaces
- sheaves
- (Etale space) definition of Riemann surface of a global holomorphic function (in Ahlfors)
- Riemann surfaces of algebraic functions
- Riemann-Roch theorem
- The Jacobian of a Riemann surface and Abel’s theorem
- Torelli theorem - Wikipedia
- August to November: The previous plan was to follow Ramanan’s global analysis but we shall consider the interests of the participants.
references
Possible references for topics in Riemann surfaces:
- https://www.mat.univie.ac.at/~armin/lect/advanced_complex_analysis.pdf
- Books on Riemann surfaces by (Narasimhan may be too daunting)
- Forster
- Miranda
- Wilhelm Schlag
- https://mathweb.tifr.res.in/~srinivas/rsfull.pdf (uses ringed spaces definition of Riemann surfaces where the rings are rings of functions on it)
- https://math.berkeley.edu/~teleman/math/Riemann.pdf
- Donaldson
- Jost
- Renzo Cavalieri, Eric Miles (Hurwitz theory)
- …a hundred more books and notes available online
- These lectures assume covering space theory (algebraic topology) and uniformization theory and does (pre-moduli space) classification of Riemann surfaces: https://www.youtube.com/playlist?list=PLbMVogVj5nJSm4256vuITlsovUT1xVkUL
- Elliptic Functions and Integrals, Uniformization Theorem etc: https://www.bilibili.com/video/BV1fW41197nr/?spm_id_from=333.337.search-card.all.click
more events to attend in summer
epilogue
- A message from Kapil Hari, the Dean academics: https://iisermmag.wordpress.com/2012/08/23/message-from-the-dean-academics/