We consider an n × n linear system of ODEs with an irregular singularity of Poincaré rank 1 at z = ∞, holomorphically depending on parameter t within a polydisc in C n centred at t = 0. The eigenvalues of the leading matrix at z = ∞ coalesce along a locus ∆ contained in the polydisc, passing through t = 0. Namely, z = ∞ is a resonant irregular singularity for t ∈ ∆. We analyse the case when the leading matrix remains diagonalisable at ∆. We discuss the existence of fundamental matrix solutions, their asymptotics, Stokes phenomenon and monodromy data as t varies in the polydisc, and their limits for t tending to points of ∆. When the deformation is isomonodromic away from ∆, it is well known that a fundamental matrix solution has singularities at ∆. When the system also has a Fuchsian singularity at z = 0, we show under minimal vanishing conditions on the residue matrix at z = 0 that isomonodromic deformations can be extended to the whole polydisc, including ∆, in such a way that the fundamental 1

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In this paper we compute Stokes matrices and monodromy for the quantum cohomology of projective spaces. This problem can be formulated in a "classical" framework, as the problem of computation of Stokes matrices and monodromy of (systems of) differential equations with regular and irregular singularities. We prove that the Stokes' matrix of the quantum cohomology coincides with the Gram matrix in the theory of derived categories of coherent sheaves. We also study the monodromy group of the quantum cohomology and we show that it is related to hyperbolic triangular groups.The inverse to this matrix has entries a ijThis matrix is equivalent to the one above with respect to the action of the braid group. We will also call it "canonical". The mentioned conjecture claims that the Stokes matrix of the quantum cohomology of CP d is equal to the above Gram matrix (modulo the action of the braid group: remarkably, this action on the Stokes matrix for the Frobenius manifold coincides with the natural action of the braid group on the collections of exceptional objects [25] [23]).This conjecture has its origin in the paper by Cecotti and Vafa [6], where another Stokes matrix introduced in [8] for the tt * equations was found in the case of the CP 2 topological σ model. It was suggested, on physical arguments, that the entries of the Stokes' matrix S =They must satisfy a Diophantine equation x 2 + y 2 + z 2 − xyz = 0 whose integer solutions (x, y, z) are all equivalent to (3,3,3) modulo the action of the braid group. The authors of [6] also suggested that their matrix must coincide with the Stokes matrix defined in the theory of WDVV equations of associativity, that is, in the geometrical theory of Frobenius manifolds for 2D topological field theories [7], [9].Later, in [25], the links between N = 2 supersymmetric field theories and the theory of derived categories were further investigated and the coincidence of χ(E i , E j ) with the Stokes matrix of tt * for CP d was conjectured.The conjecture may probably be derived from more general conjectures by Kontsevich in the framework of categorical mirror symmetry. To my knowledge, the subject was discussed in [18] (I thank B. Dubrovin for this reference).The main result of this paper is the proof (Theorem 2, 2 ′ ) that the conjecture about coincidence of the Stokes matrix for quantum cohomology of CP d and the Gram matrix χ(E i , E j ) of a full exceptional collection in Der b (Coh(CP d )) is true. In this way, we generalize to any d the result obtained in [10] for d = 2.We remark that it has not yet been proved that the Stokes' matrix for tt * equations and the Stokes' matrix for the corresponding Frobenius manifold coincide. This point deserves further investigation.We also study the structure of the monodromy group of the quantum cohomology of CP d . The notion of monodromy group of a Frobenius manifold was introduced in [9]. We prove (Theorem 3) that for d = 3 the group is isomorphic to the subgroup of orientation preserving transformations in the hyperbolic triangular group ...

The paper provides the tables of the critical behaviours at x = 0, 1, ∞ for the Painlevé 6 functions. The connection formulae for the basic solutions are also provided, in parametric form.MSC: 34M55 (Painlevé and other special functions)

We extend the analytic theory of Frobenius manifolds to semisimple points with coalescing eigenvalues of the operator of multiplication by the Euler vector field. We clarify which freedoms, ambiguities and mutual constraints are allowed in the definition of monodromy data, in view of their importance for conjectural relationships between Frobenius manifolds and derived categories. Detailed examples and applications are taken from singularity and quantum cohomology theories. We explicitly compute the monodromy data at points of the Maxwell Stratum of the A 3 -Frobenius manifold, as well as at the small quantum cohomology of the Grassmannian G 2 C 4 . In the latter case, we analyse in details the action of the braid group on the monodromy data. This proves that these data can be expressed in terms of characteristic classes of mutations of Kapranov's exceptional 5-block collection, as conjectured by one of the authors.