Geometry and Topology Seminar: D7 Integrable systems, Painlevé VI and explicit solutions to the anti-self dual Einstein equation via radicals

Abstract: Though Einstein's equation is well studied, relatively few Einstein metrics have been written in terms of explicit formulae via radicals. In this talk we discuss many such examples that occur as anti-self dual Einstein metrics and describe their singularities. The construction heavily relies upon the theory of isomonodromic deformation and related algebraic geometry developed by N.J. Hitchin in the 1990s and the equivalence of the anti-self dual Einstein equation to a certain Painlevé VI equation under some symmetry assumptions discovered by K.P. Tod. The solution to Painlevé VI is achieved through a relation of its solution to pairs of conics obeying the Poncelet's porism by exploiting Cayley's criterion. In this talk we discuss some important cases that are not well fleshed out in the literature, such as the solution of Painlevé VI associated with the Poncelet porism where the inscribing-circumscribing polygons have an even number of sides. Moreover, we provide some explicit metrics with unusual cone angle singularities along a singular real projective plane that were speculated about by Atiyah and LeBrun and discuss their sectional curvature.

Host: Quo-Shin Chi

Note: the speaker will be presenting via Zoom