Mathematics

Based on joint work with B. Dubrovin and D. Guzzetti. In the first part of this seminar I will recall basic notions of the general theory of meromorphic linear ODEs in the complex plane and of their Isomonodromic deformations. After this, I will show how under minimal and sharp conditions one of the main assumptions of the theory of M. Jimbo, T. Miwa and K. Ueno can  be relaxed. This allows to treat the case of systems with “coalescing” irregular singularities, namely irregular singularities at which the leading term of the Laurent expansion of the coefficients is diagonal but not with simple spectrum. As an application, I will show the computations of the monodromy data at points of the Maxwell stratum of the Frobenius structure defined on the base space of universal unfolding of the $A_3$-singularity, and I will reinterpret them in terms of the associated algebraic solutions of Painlevé VI equation.