Hermitian, symmetric and symplectic random ensembles: PDEs for the distribution of the spectrum.
Irrationality of the roots of the Yablonskii-Vorob'ev polynomials and relations between them.
Justification of the existence of a group of asymptotics of the general fifth Painlevé transcendent.
Kovalevska vs. Kovacic-two different notions of integrability and their connections
Ordinary differential equations all share the same common root-real physical problems. But, although the physical motivation remains the most important one, the way the subject develops does depend highly on the methods available. In the exposition I would like to show some connections between two methods of checking the ODE for integrability (whatever it should mean), with distant motivations and techniques. These are the so-called Painlevé tests and the methods originating in Ziglin's theory and...
Middle convolution and Heun's equation.
Moduli spaces for linear differential equations and the Painlevé equations
A systematic construction of isomonodromic families of connections of rank two on the Riemann sphere is obtained by considering the analytic Riemann–Hilbert map , where is a moduli space of connections and , the monodromy space, is a moduli space for analytic data (i.e., ordinary monodromy, Stokes matrices and links). The assumption that the fibres of (i.e., the isomonodromic families) have dimension one, leads to ten moduli spaces . The induced Painlevé equations are computed explicitly....
Non-existence criteria for Laurent polynomial first integrals.
Note on the uniqueness of a global positive solution to the second Painlevé equation.
On commuting matrix differential operators.
On deficiencies of small functions for Painlevé transcendents of the fourth kind.
On the asymptotic representation of the solutions to the fourth general Painlevé equation.
On the asymptotics of the real solutions to the general sixth Painlevé equation.
On the identity of two -discrete Painlevé equations and their geometrical derivation.
Painlevé equations and complex reflections
We will explain how some new algebraic solutions of the sixth Painlevé equation arise from complex reflection groups, thereby extending some results of Hitchin and Dubrovin-- Mazzocco for real reflection groups. The problem of finding explicit formulae for these solutions will be addressed elsewhere.
Painlevé's Theorem on automorphic functions.
Periodic integrals and tautological systems
We study period integrals of CY hypersurfaces in a partial flag variety. We construct a regular holonomic system of differential equations which govern the period integrals. By means of representation theory, a set of generators of the system can be described explicitly. The results are also generalized to CY complete intersections. The construction of these new systems of differential equations has lead us to the notion of a tautological system.
Polynomial solutions of a generalization of the first Painlevé differential equation.
Polynomial solutions to the Hele--Shaw problem.
Pure imaginary solutions of the second Painlevé equation.
Quadratic transformations for the third and fifth Painlevé equations.