Chris Eilbeck (Heriot-Watt University)

10 Dec 2015 - W01 0-012 (Wechloy), 4 pm c.t.

Generalisations of the Weierstrass sigma function, discriminants, and heat equations

Abstract: The solutions of many interesting nonlinear wave equations can be written down in terms of Weierstrass $\wp$ functions and their generalizations to higher genus. In turn the $\wp$ functions are just the 2nd logarithmic derivatives of an entire function, the Weierstrass $\sigma$ function generalised to genus $g$. We discuss the properties of the $\sigma$ function associated with a plane curve of genus $g$. These functions satisfy sets of interesting linear parabolic PDEs. These PDEs can be used to form linear but complicated recurrence relations for the coefficients of the sigma expansion. A connection with the discriminant of the associated curve is also highlighted.