|XXIX Workshop on Geometric Methods in Physics||27.06-03.07.2010|
Sigma function and dispersionless hierarchies
Riemann surfaces (smooth compact curves) traditionally provide
explicit solutions to equations of mathematical physics via
the Riemann theta function. Klein and H.F. Baker in the 19th century
introduced a version of it, the sigma function, which is
more directly related to algebraic data of the curve.
The sigma function, its order of vanishing on the Wirtinger
stratification of the Jacobian, and its (characteristic)
differential equations have been the focus of much recent work.
In this talk we illustrate the current knowledge of the sigma
function and present an untraditional way of giving explicit
solutions to dispersionless hierarchies (e.g., dKP, Burgers-Hopf).