Computational Aspects of Riemann Surfaces and Integrable Systems
On a spectral curve for a symmetric monopole.
In the study of a symmetric SU(2) monopole of charge 3 via finite gap integration methods, a spectral curve plays a fundamental role. I will present some of its interesting properties, and, exploiting the symmetries of the system, explicitly construct several quantities intrinsic to the curve. This will be relevant for the solution of the monopole equation, and moreover provides a concrete realization of some abstract results in the theory of symmetric Riemann surfaces.
Abelian Functions: New results for functions associated with a cyclic tetragonal curve of genus six.
This presentation will be on the topic of Abelian (multiply periodic) functions. These functions are defined in analogy to the Weierstrass $\wp$-function, and we use them to generalise the elliptic theory to higher genus functions. After a review of the lower genus cases I will discuss the similar functions associated with a cyclic tetragonal curve of genus six. The computations involved in working with such functions are significantly greater than in the lower genus cases, with some latter computations being run in parallel using the Distributed Maple package. I will present some new results for these functions including the associated partial differential equations satisfied by the functions, a series expansion for the sigma-function and a new addition formula. I will also demonstrate that such functions can give a solution to the KP-equation and outline how a general set of solutions could be generated using a wider class of curves.
The tau and sigma-functions of an algebraic curve.
Since the classic articles of M.Sato and Y.Sato in 1980, 1981 it has been known that KP-flows are described by the Plucker embedding of the Grassmanian Gr(n,infty) into projective space. I will describe this embedding using multidimensional sigma-functions and their logarithmic derivatives (that give Kleinian zeta and P-functions as coordinates). I will also compare the derivation of differential and algebraic relations associated to the Abelian functions of the curve that come from the Plucker relations and the Bilinear Identity Method (that is equivalent to finding residues). The evaluation is exemplified by the lower genera curves - genus two hyperelliptic and genus three trigonal.
Riemann Surfaces with Symmetry: New results for Klein's curve.
Given a Riemann surface with symmetry one often wishes to choose cycles and construct period matrices reflecting this. Although an algorithm exists for choosing a homology basis for an arbitrary plane curve (and this is implemented in Maple's 'Algcurves') this does not reflect such symmetry considerations. I will describe some tools I have developed to do this using Klein's curve as an example. Some new results are obtained in the process.