Riemann surfaces

Bertrand Eynard (IPhT)

2018-01-12 10:00, Salle Itzykson, IPhT
2018-01-19 10:00, Salle Itzykson, IPhT
2018-01-26 10:00, Salle Itzykson, IPhT
2018-02-02 10:00, Salle Itzykson, IPhT
2018-02-09 10:00, Salle Itzykson, IPhT
Blurb:

Algebraic equations are widespread in mathematics and physics, and the geometry of their spaces of solutions can be complicated. In the case of an equation of two complex variables, the space of solutions is a Riemann surface.

Abstract:
We will provide some basic tools (going back to Riemann) for studying algebraic equations and describing the geometry of compact Riemann surfaces.

The solution locus of a polynomial equation $P(x,y)=0$ defines a two-dimensional Riemann surface in $\mathbb C\times \mathbb C$, and the solutions define a multivalued function $x\mapsto Y(x)$ that has several branches $Y_1(x),Y_2(x),\dots$.

We will study the topology and geometry of that surface, the closed contours that can be drawn on it (i.e. its homology), and how to integrate differential forms along contours. For example, we will compute integrals of the type $\int_{\gamma} R(x,Y(x))dx$ where $R$ is a bivariate rational function, and $\gamma$ a contour on the Riemann surface.

Then we will  describe the moduli space of all Riemann surfaces with a given topology: its dimension, topology, etc.

We will introduce some of the many tools that have been invented since the time of Riemann for studying these objects. We will partly follow the Mumford Tata lectures, the Fay lectures, and the Farkas-Kra book.

The plan is:
1. Compact Riemann surfaces, charts, atlas, toplogy. Meromorphic functions and one-forms. Theorems on poles and residues. Newton’s polygon.

2. Integrals, periods, Abel map, Jacobian, divisors. Theta functions, prime form, fundamental form. Basis of cycles, homology and cohomology.

3. Moduli spaces of Riemann surfaces. Deligne-Mumford compactification, Chern classes, tautological ring. Kontsevich integral and KdV hierarchy.

4. If times permits: fiber bundles, Hitchin systems, link to integrable systems.

Series:
IPhT Courses
Short course title:
Riemann surfaces
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