Isomonodromy and integrability
Seminar Room 1, Newton Institute
The property of having no movable critical points for an ordinary differential equation was linked with integrable systems via theta functions in the 19th century, and more recently, since the 1970s, with integrable partial differential equations via similarity reduction. A geometric integration of these features will be explored in the first part of the talk, after work by H. Flaschka (1980), which suggests a deformation of the spectral curve. This provides the segue to the second part of the talk, concerning a joint project with F.W. Nijhoff. The isomonodromy equations for spectral data (e.g., the Baker function) are studied as systems of ODEs, following R. Garnier (1912). Special functions, specifically the Kleinian sigma function, are implemented in the equations, to seek the Gauss-Manin-connection counterpart of the Legendre equation, by which R. Fuchs (1906) had connected the isomonodromy property and the absence of movable critical points. Work by Nijhoff et al. on discrete and Schwarzian equations would be related to this higher-genus Legendre version of the isomonodromy condition.