Harmonic maps of Painlevé-type: the loop group point of view


Martin Guest

10:00:00 - 10:50:00

101 , Mathematics Research Center Building (ori. New Math. Bldg.)

Harmonic maps from a Riemann surface to a Lie group or symmetric space have appeared (explicitly or implicitly) in classical differential geometry for a hundred years. From the 1980\'s it was recognised that the harmonic map equation is “integrable” and that loop groups are the symmetry groups responsible for this special structure. Special cases related to mathematical physics (sigma models, Hitchin/Higgs equations, tt* geometry,...) have provided crucial motivation and interesting examples. We shall sketch the loop group framework, together with some of the history.

Then we shall discuss some examples of “Painlevé type”. Here the Riemann surface is just C or C* and a scaling reduction of the harmonic map equation is an ordinary differential equation.

Our main example (a joint project with Alexander Its and Chang-Shou Lin) is the radial Toda equation. This is also an example of the tt* equations.

Our main results concern the asymptotics of solutions near 0 and infinity. Loop groups provide a convenient language, and partial proofs. To complete the proofs we need p.d.e. theory and the Riemann-Hilbert method. In fact, it is only by combining all 3 methods that we obtain the full story.