Summer 2025 seminar course on Riemann surfaces (and beyond)

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 $\mathbb{C}$ is not biholomorphic to $D$, 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

(Narasimhan may be too daunting)

Inculcation: Riemann surfaces, that is, complex 1-manifolds, complex algebraic curves, round/flat/hyperbolic surfaces and so on

books §

• 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

lecture videos §

• 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

Riemann Surfaces by Jacob Bernstein ( bernstein@math.jhu.edu)

https://www.bilibili.com/video/BV1fW41197nr/?spm_id_from=333.337.search-card.all.click For MSRI summer school 2014: Complex geometry and geometric analysis on complex manifolds

• Prerequisites:
  • Knowledge of basic complex analysis—at the level of Ahlfors, Complex Analysis, Chapters 1-5—will be assumed. Some basic familiarity with (abstract) surface theory and differential forms will be helpful. However, I will review this material as needed.
• Reading:
  • The main text will be Donaldson, Riemann Surfaces; get at http://www2.imperial.ac.uk/~skdona/RSPREF.PDF.
  • Syllabus: https://www.slmath.org/ckeditor_assets/attachments/106/bernstein_hein_naber_syllabus.pdf
• Other useful references:
  • Farkas and Kra, Riemann Surfaces; a classical text on the subject.
  • Miranda, Algebraic Curves and Riemann Surfaces; a more algebraic perspective.
• Week 1: Introduction to Riemann Surfaces
  • Surfaces and Topology
  • Riemann Surfaces and Holomorphic Maps
  • Maps between Riemann Surfaces
  • Calculus on Riemann Surfaces
  • De Rham Cohomology
• Week 2: Geometric Analysis on Riemann Surfaces
  • Elliptic Functions and Integrals
  • Meromorphic Functions
  • Inverting the Laplacian
  • The Uniformization Theorem
  • Riemann Surfaces and Minimal Surfaces

• A full course on Riemann Surfaces by M Khalkhali with videos on YouTube

Link to original

more events to attend in summer §

• Summer School on Rigidity of Discrete Groups, June 30 – July 4, 2025 at IISER Mohali

epilogue §

• A message from Kapil Hari, the Dean academics: https://iisermmag.wordpress.com/2012/08/23/message-from-the-dean-academics/

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