Bohan Fang

BICMR, Peking University

Course on Moduli of Curves, Ribbon Graphs, and Witten’s Conjecture

Detailed Syllabus (tentative)

I will try to keep the course accessible without assuming too much background knowledge. Some basic understanding of manifolds and Riemann surfaces should be enough. I will list more details about the Topological Recursion later in the course.

Part I: The Geometry of Moduli Spaces

This first part of the course builds the geometric foundations. The primary goal is to construct the moduli space $\mathcal M_{g,n}$ and its compactification​ as a geometric object, using the language of hyperbolic geometry and complex analysis.

Key Definitions & Theorems:

  • Hyperbolic metric on a surface  

  • Teichmüller Space

  • Pair of pants decomposition  

  • Fenchel-Nielsen coordinates (length and twist parameters)

  • Mapping Class Group, Nielsen-Thurston classification of mapping classes

  • Nodal curve

  • Stability condition

  • Deligne-Mumford compactification  

  • Boundary strata and their recursive structure  

Part II: The Combinatorics of Moduli Spaces

This part of the course introduces the second pillar: the combinatorial description of $\mathcal M_{g,n}$​ via ribbon graphs. This will provide the essential tool for Kontsevich’s proof.

Key Definitions & Theorems:

  • Ribbon graph / Fat graph  

  • Geometric realization of a ribbon graph into a surface  

  • Cell decomposition of the moduli space indexed by ribbon graphs

Part III: Intersection Theory and Matrix Models

This part introduces the intersection-theoretic problem that motivated Witten, and solves it using the combinatorial machinery and a surprising connection to physics. We will start with tautological classe, and then define intersection numbers on $\overline{\mathcal M}_{g,n}$ and their generating functions.

Witten’s Conjeture states that this generating function satisfies a series of partial differential equations, i.e. KdV hierarchy. Kontsevich’s formula express this generating function in terms of ribbon graphs, which are also expressed by a matrix model. Kontsevich’s proof eventually boils down to showing the integrable hierarchy for this matrix model.

Key Definitions & Theorems:

  • Universal curve and tautological classes

  • Intersection numbers, generating function

  • KdV Hierarchy

  • Witten’s Conjecture

  • Hermitian matrix model

  • Feynman diagrams, propagators, vertices

  • Wick’s Theorem for Gaussian integrals

  • Kontsevich’s proof on Witten’s Conjecture

References:

Textbooks and survey articles:

  • Harris, J., & Morrison, I. (1998). Moduli of Curves. Springer.

  • Farb, B., & Margalit, D. (2012). A Primer on Mapping Class Groups. Princeton University Press.

  • Zvonkine, D. (2012). “An introduction to moduli spaces of curves and their intersection theory”. In Handbook of Teichmüller Theory, Vol. III.

  • Arbarello, E., Cornalba M., & Griffiths, P. A. (2011). Geometry of algebraic curves, vol 2. Springer.

Original research articles (The primary sources):

  • Witten, E. (1991). “Two-dimensional gravity and intersection theory on moduli space”. Surveys in Differential Geometry.

  • Kontsevich, M. (1992). “Intersection theory on the moduli space of curves and the matrix Airy function”. Communications in Mathematical Physics.

  • Deligne, P., & Mumford, D. (1969). “The irreducibility of the space of curves of given genus”. Publications Mathématiques de l’IHÉS.

  • Penner, R. C. (1987). “The decorated Teichmüller space of punctured surfaces”. Communications in Mathematical Physics.