Exact WKB analysis of higher order differential equations that is centered around the notion of a virtual turning point

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Along the flow line described in "Toulouse Project', which is designed to clarify the structure of higher order Painleve equations, we have obtained the following results.1. A O-parameter solution of (P_J)m (J= I, II, IV) is locally and formally transformed to a 0-parameter solution of (P_I)_I near a turning point of the first kind.(T. Kawai and Y. Takei: Advances in Math., 203 (2006))2. Construction of a (2m)-parameter solution of instanton-type of (P_J)m (J= I, II, IV) (Y. Takei (2008), T. Koike: Kokyuroku Bessatsu, B2 (2007), B5 (2008))3. For each turning point of (P_J)m (J= I, U, IV) of the first kind, an instanton-type solution associated with the turning point can be locally and formally transformed to a 2-parameter solution of (P_I)_I. (T. Kawai and Y. Takei: Kokyuroku Bessatsu, B5 (2008))4. Each (P_J)m (J= I, II, IV) is isomorphic to an appropriate degenerate Gamier system restricted to some complex line. (T. Koike: Kokyuroku Bessatsu, B2 (2007), B5 (2008))In addition to the progress in the study of (P_J)m, we have made a substantial progress in the study of virtual turning points through the investigation of the Stokes geometry of another higher order Painleve equation called the Noumi-Yamada system. (P. Kawai et al. (2008)) Furthermore a novel treatment of several infinite series in exact WKB analysis with the aid of infinite order differential operator has been found by T. Aoki, T. Kawai and Y. Takei (RIMS Preprint 1616 (2007)).A conference centered around these results was held from January 28 through February 1, 2008 at CIRM (Centre International de Rencontres Mathematiques) in Marseilles (France). It turned out to be an intensive and fruitful meeting with many foreign (neither French nor Japanese) participants.