### Takahiro Kawai and Yoshitsugu Takei

RIMS, Kyoto University

### On the Toulouse Project - how much we have understood the structure of
solutions of higher order Painleve equations

**Wednesday 10th September 14:05-14:55pm,
Eastern Avenue Lecture Theatre. **
The Toulouse Project is a collection of problems/proposals whose solutions
should help us to understand the structure of solutions of higher order
Painlev\'e equations $(P_J)_m$ ($J=I,II,IV$) introduced by P.R. Gordoa,
N. Joshi and A. Pickering (Publ. RIMS, 37(2001), 327-347) and S. Shimomura
(Ann. Scuola Norm. Sup. Pisa, 29(2000), 1-17). Its original form consists
of seven steps, and we are in the middle stage of the fifth step, that is,
we have clarified the structure of instanton-type solutions of $(P_J)_m$
near a turning point of the first kind, whose definition will be given
in the talk.

In this talk we discuss WKB theoretic aspects of our theory, emphasizing
the importance of WKB analysis of the underlying Lax pair, the simultaneous
equations whose compatibility condition coincides with $(P_J)_m$. We also
plan to discuss what points are peculiar to higher order Painlev\'e equations,
i.e., not observed in the second order ones.