Spring 2014 Goals

• Finish Abel Map

  It’s been a long time coming but the necessary infrastructure is now in place for implementing the Abel map: given a genus $g$ compact Riemann surface $X$, a basis for its first homology group $\\{a_1, \ldots, a_g, b_1, \ldots, b_g\\}$, a basis for its space of holomorphic differentials $\\{\omega_1, \ldots, \omega_g\\}$, and the Jacobian $J(X)$ the Abel map $A : X \to J(X)$ is defined $A(P) = \left( \int_{P_0}^P \omega_1, \ldots, \int_{P_0}^P \omega_g \right)$ where $P_0 \in X$ is some fixed place.

  Some recent refactorization of my code not only made path construction easier to implement and read but also improved the performance of my Riemann surface path construction and evaluation approximately ten-fold.

• Read Bernard’s Thesis

  This is another goal that has been a long time coming and I’m ashamed to admit I haven’t done yet. His thesis (the paper version is found here) contains much of the mathematics needed to go from some initial data of the KP equation to a finite genus solution of the form I want to compute. Some discussions with Bernard lead us to believe that the initial value problem of the KP equation can lead to a possible approach to the Constructive Schottky problem. This is a lofty guess so I would like to see if it’s remotely possible.

• Read (some) of Griffiths and Harris

  I’d eventually like to read Chapters 1 and 2 of Griffiths’ and Harris’ Principles of Algebraic Geometry. This is the kind of algebraic geometry that seems most useful to what I’m studying and am interested in. In particular, Chapter 2 discusses the key players in the Constructive Schottky Problem. At the very least, having an understanding of these two chapters will make me feel like I can be conversant in the field.

• Write High-Performance Object-Oriented Analogues of Puiseux, Places / Divisors, etc.

  Ondrzej Certik is working on a C-implementation of sympy which should drastically improve baseline performance of abelfunctions symbolics. In the meantime, there are some algorithmic improvements to Puiseux series and other objects that need working on.

  Generally fleshing out the remaining OOP infrastructure of abelfunctions is medium to high priority.

• Finish Abelfunctions documentation.