Stat. Field Theory Course
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**Riemann Surfaces and Theta-Functions**

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Tamara Grava

**CONTENTS**

Riemann surfaces, definition and examples.

Topology of a Riemann surface. Compact Riemann surfaces.

Harmonic differentials. Meromorphic functions and Differentials of a
compact Riemann surface. Riemann Bilinear Relations.

Divisors and Riemann Roch Theorem. Divisors and holomorphic line bundles.

Abel Theorem and Jacobi inversion problem.

Complex tori and Jacobi variety.

Theta functions.

The theta function associated with a Riemann surface.

Zeros of theta function and Theta divisor.

Theta functions as sections of line bundles on complex tori.

Applications: quasi periodic solution of the KP equation in terms of theta
functions.

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