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

Riemann surfaces §

introductory textbooks §

• 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 https://mathoverflow.net/questions/313254/references-for-riemann-surfaces

introductory lecture videos §

• A full course on Riemann Surfaces by M Khalkhali with videos on YouTube
• 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 for MSRI summer school 2014: Complex geometry and geometric analysis on complex manifolds ^yg7hmg
  • 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.
    • Syllabus
  • 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

moduli of Riemann surfaces §

🤔

Genus zero Riemann surfaces, flat tori, lattices and elliptic curves §

Higher genus Riemann surfaces, hyperbolic surfaces, Fuschian groups and Teichmuller theory §

• Fuschian groups, Teichmuller theory and MCGs
  • Farb Margalit - Mapping class groups
  • GTM091.The.Geometry.of.Discrete.Groups Beardon.A.F..(Springer.1995)
• geodesic flow
• spectra
  • Geometry and Spectra of Compact Riemann Surfaces - Google Books
  • arithmetic quantum unique ergodicity

modular surfaces §

• https://web.math.princeton.edu/~sarnak/Preprints/baltimore.pdf
• Lectures on Diophantine approximation and Dynamics

Automorphism groups §

• Riemann surfaces and their automorphism groups - GAC2010 workshop, HRI, India
• Isometry groups of hyperbolic surfaces – SPP 2026
• Hyperbolic Surfaces with Prescribed Infinite Symmetry Groups on JSTOR
• http://www.kurims.kyoto-u.ac.jp/EMIS/journals/DMJDMV/vol-06/16.pdf
• https://math.uchicago.edu/~may/REU2013/REUPapers/Benson-Tilsen.pdf

morphisms §

https://en.wikipedia.org/wiki/De_Franchis_theorem says there are only finitely many non-constant holomorphic mappings between two fixed compact Riemann surfaces of genus greater than 1.

bounds on that number: https://www.cambridge.org/core/services/aop-cambridge-core/content/view/4F55ECA45BC877D46534577A07CA2C56/S0013091507000223a.pdf

|

_{g}X\to \! _{h}X

| $g=0$ | 1 | 2 | 3 | 4 | 5 | 6 | | ----------------------- | --------------------- | -------------------------------- | --------- | ------------ | --- | --- | --- | | $h=0$ | rational maps $\text{C}(X)$ | | | | | | | | 1 | | translations and a few isogenies | | | | | | | 2 | | | $\le 84$ | 0 | 0 | 0 | 0 | | 3 | | | | $\le 83(2)$ | 0 | 0 | 0 | | 4 | | | | | | 0 | 0 | | 5 | | | | | | | 0 | | 6 | | | | | | | |

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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