Robert C. Gunning

A: Theta functions

- Complex tori and their period matrices
- Theta Functions
- The functional equation for theta functions
- Even and odd theta functions
- The addition theorem
- The theta locus
- Singularities of the theta locus
- Wirtinger varieties

- The topology of surfaces
- Abelian differentials
- The cross-ratio function
- Factors of automorphy
- Theta factors of automorphy
- The prime function
- Canonical meromorphic differentials
- Quadratic period functions
- Varieties of special positive divisors
- The canonical curve

- Riemann's theorem
- Prime function expansion
- The Gauss mapping
- Semicanonical functions
- Prime function expansion for theta derivatives
- The second-order Gauss mapping
- Semicanonical functions of second ordeer

- The analogues of Riemann's theorem
- Translation by half-periods
- Fay's trisecant identity
- Limiting forms of the trisecant identity
- A special limiting form: the KP equation
- Multisecant identities

- The hyperelliptic multisecant identities
- Limiting forms of the hyperelliptic multisecant identities
- A special limiting form: the KDV equation
- The trigonal multisecant identities

- The general expansion formula
- Dominant terms in the expansion
- The primary and binary expansion: dominant terms
- The binary expansion: subdominant terms
- The ternary expansion
- The quaternary expansion

- The general expansion formula
- Differential invariants and the addition theorem
- Fourth order derivatives at the origin
- Hyperelliptic Riemann surfaces
- Examples (preliminary version)