Alexander V. Kitaev, Steklov Mathematical Institute, St. Petersburg, Russia

Special Functions of the Isomonodromy Type. The Painlev\'e Equations. I, II

Abstract

A huge source of functions describing isomonodromy deformations of linear ODEs with rational coefficients (SFITs) can be viewed as natural generalizations of the classical special functions of the hypergeometric type.  For SFITs one can obtain many analytic results known in the theory  of classical special functions. In this lectures a general overview of the state of art of the theory of SFITs and some of their applications in differential geometry and asymptotic analysis will be given.

Content of the lectures:

1. The monodromy group and the manifold of the monodromy data for linear matrix ODEs with rational coefficients. We mainly deal with $2\times2$ matrix ODEs.

2. Isomonodromy deformations of linear matrix ODEs as special functions of the isomonodromy type (SFITs).

3. Examples of SFITs:

a) The Gamma Function;

b) The Gau{\ss} Hypergeometric functions  and the Sixth Painlev\'e transcendent;

c) Other Painlev\'e Transcendents;

d) "Higher" Painlev\'e transcendents,  some further discussions.

4. Brief historical remarks about the discovery of the Painlev\'e equations.

5. Brief overview of the results known for the Painlev\'e equations: transformations, special solutions, asymptotics.

6. Applications of SFITs in asymptotic analysis: "caustics" and coalescence of singularities in integrable systems.