NOTE: The notes appear in separate sections, which can be accessed
through the underlined titles in the table of contents.
<BR><BR><BSR>
Riemann Surfaces and Second-Order Theta Functions 
<BR>
Robert C. Gunning
<BR> 
<a href="theta2/A"> A: Theta functions </a>
<ol>
<li> Complex tori and their period matrices </li> 
<li> Theta Functions </li>
<li> The functional equation for theta functions </li>
<li> Even and odd theta functions </li>
<li> The addition theorem </li>
<li> The theta locus </li>
<li> Singularities of the theta locus </li>
<li> Wirtinger varieties </li>
</ol> 
<a href="theta2/B"> B: Riemann Surfaces </a>
<ol>
<li> The topology of surfaces </li>
<li> Abelian differentials </li>
<li> The cross-ratio function </li>
<li> Factors of automorphy </li>
<li> Theta factors of automorphy </li>
<li> The prime function </li>
<li> Canonical meromorphic differentials </li>
<li> Quadratic period functions </li>
<li> Varieties of special positive divisors </li>
<li> The canonical curve </li>
</ol>
<a href="theta2/C"> C: First-order Riemannian theta functions
</a>
<ol>
<li> Riemann's theorem  </li>
<li> Prime function expansion </li>
<li> The Gauss mapping </li>
<li> Semicanonical functions </li>
<li> Prime function expansion for theta derivatives </li>
<li> The second-order Gauss mapping </li>
<li> Semicanonical functions of second ordeer </li>
</ol>
<a href="theta2/D"> D: Second-order Riemannian theta functions
</a>
<ol>
<li> The analogues of Riemann's theorem </li>
<li> Translation by half-periods </li>
<li> Fay's trisecant identity </li>
<li> Limiting forms of the trisecant identity </li>
<li> A special limiting form: the KP equation </li>
<li> Multisecant identities  </li>
</ol>
<a href="theta2/E"> E: Special Riemann surfaces </a>
<ol>
<li> The hyperelliptic multisecant identities </li>
<li> Limiting forms of the hyperelliptic multisecant identities </li>
<li> A special limiting form: the KDV equation </li>
<li> The trigonal multisecant identities </li>
</ol>
F:  Differential expansions of Riemannian theta functions 
<ol>
<li><a href="theta2/F:1-3"> The general expansion formula </a> </li>
<li> Dominant terms in the expansion </li>
<li> The primary and binary expansion: dominant terms </li>
<li> <a href="theta2/F:4-6"> The binary expansion: subdominant terms </a> </li>
<li> The ternary expansion </li>
<li> The quaternary expansion </li>
</ol> 
 G:  Invariants sof the differential expansion
<ol>
<li><a href="theta2/F:1-3"> The general expansion formula </a> </li>
<li> Differential invariants and the addition theorem </li>
<li> Fourth order derivatives at the origin </li>
<li> Hyperelliptic Riemann surfaces 
<li> <a href="theta2/G:4"> Examples (preliminary version) </a> </h3>  </li>
</ol>