We consider an n{\texttimes}n linear system of ODEs with an irregular singularity of Poincar\'e rank 1 at z=$\infty$, holomorphically depending on parameter t within a polydisc in Cn centred at t=0. The eigenvalues of the leading matrix at z=$\infty$ coalesce along a locus Δ contained in the polydisc, passing through t=0. Namely, z=$\infty$ is a resonant irregular singularity for t∈Δ. We analyse the case when the leading matrix remains diagonalisable at Δ. 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 Δ. When the deformation is isomonodromic away from Δ, it is well known that a fundamental matrix solution has singularities at Δ. 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 Δ, in such a way that the fundamental matrix solutions and the constant monodromy data are well defined in the whole polydisc. These data can be computed just by considering the system at fixed t=0. Conversely, if the t-dependent system is isomonodromic in a small domain contained in the polydisc not intersecting Δ, if the entries of the Stokes matrices with indices corresponding to coalescing eigenvalues vanish, then we show that Δ is not a branching locus for the fundamental matrix solutions. The importance of these results for the analytic theory of Frobenius Manifolds is explained. An application to Painlev\'e equations is discussed.

}, doi = {10.1215/00127094-2018-0059}, url = {https://doi.org/10.1215/00127094-2018-0059}, author = {Giordano Cotti and Boris Dubrovin and Davide Guzzetti} } @article {2018, title = {Local moduli of semisimple Frobenius coalescent structures}, number = {arXiv;1712.08575}, year = {2018}, institution = {SISSA}, abstract = {There is a conjectural relation, formulated by the second author, between the enumerative geometry of a wide class of smooth projective varieties and their derived category of coherent sheaves. In particular, there is an increasing interest for an explicit description of certain local invariants, called monodromy data, of semisimple quantum cohomologies in terms of characteristic classes of exceptional collections in the derived categories. Being intentioned to address this problem, which, to our opinion, is still not well understood, we have realized that some issues in the theory of Frobenius manifolds need to be preliminarily clarified, and that an extension of the theory itself is necessary, in view of the fact that quantum cohomologies of certain classes of homogeneous spaces may show a coalescence phenomenon.

}, url = {http://preprints.sissa.it/handle/1963/35304}, author = {Giordano Cotti and Boris Dubrovin and Davide Guzzetti} } @article {doi:10.1142/S2010326317400044, title = {Analytic geometry of semisimple coalescent Frobenius structures}, journal = {Random Matrices: Theory and Applications}, volume = {06}, number = {04}, year = {2017}, pages = {1740004}, abstract = {We present some results of a joint paper with Dubrovin (see references), as exposed at the Workshop {\textquotedblleft}Asymptotic and Computational Aspects of Complex Differential Equations{\textquotedblright} at the CRM in Pisa, in February 2017. The analytical description of semisimple Frobenius manifolds is extended at semisimple coalescence points, namely points with some coalescing canonical coordinates although the corresponding Frobenius algebra is semisimple. After summarizing and revisiting the theory of the monodromy local invariants of semisimple Frobenius manifolds, as introduced by Dubrovin, it is shown how the definition of monodromy data can be extended also at semisimple coalescence points. Furthermore, a local Isomonodromy theorem at semisimple coalescence points is presented. Some examples of computation are taken from the quantum cohomologies of complex Grassmannians.

}, doi = {10.1142/S2010326317400044}, url = {https://doi.org/10.1142/S2010326317400044}, author = {Giordano Cotti and Davide Guzzetti} } @article {2014, title = {A Review of the Sixth Painlev{\'e} Equation}, number = {Constructive approximation;volume 41; issue 3; pages 495-527;}, year = {2014}, publisher = {Springer}, abstract = {For the Painlev{\'e} VI transcendents, we provide a unitary description of the critical behaviours, the connection formulae, their complete tabulation, and the asymptotic distribution of poles close to a critical point.}, doi = {10.1007/s00365-014-9250-6}, url = {http://urania.sissa.it/xmlui/handle/1963/34658}, author = {Davide Guzzetti} } @article {2012, title = {Poles Distribution of PVI Transcendents close to a Critical Point (summer 2011)}, journal = {Physica D: Nonlinear Phenomena, Volume 241, Issue 23-24, 1 December 2012, Pages 2188-2203}, number = {arXiv:1104.5066;}, year = {2012}, publisher = {Elsevier}, abstract = {The distribution of the poles of Painlev{\'e} VI transcendents associated to semi-simple Frobenius manifolds is determined close to a critical point. It is shown that the poles accumulate at the critical point,asymptotically along two rays. As an example, the Frobenius manifold given by the quantum cohomology of CP2 is considered. The general PVI is also considered.}, keywords = {Painleve{\textquoteright} equations}, doi = {doi:10.1016/j.physd.2012.02.015}, url = {http://hdl.handle.net/1963/6526}, author = {Davide Guzzetti} } @article {2012, title = {A Review on The Sixth Painlev{\'e} Equation}, number = {arXiv:1210.0311;}, year = {2012}, note = {31 pages, 10 figures}, publisher = {SISSA}, abstract = {For the Painlev\\\'e 6 transcendents, we provide a unitary description of the\r\ncritical behaviours, the connection formulae, their complete tabulation, and\r\nthe asymptotic distribution of the poles close to a critical point.

}, keywords = {Painlev{\'e} equation}, url = {http://hdl.handle.net/1963/6525}, author = {Davide Guzzetti} } @article {2012, title = {Solving the Sixth Painlev{\'e} Equation: Towards the Classification of all the Critical Behaviors and the Connection Formulae}, journal = {Int Math Res Notices (2012) 2012 (6): 1352-1413}, number = {arXiv:1010.1895;}, year = {2012}, note = {53 pages, 2 figures}, publisher = {Oxford University Press}, abstract = {The critical behavior of a three real parameter class of solutions of the\\r\\nsixth Painlev\\\\\\\'e equation is computed, and parametrized in terms of monodromy\\r\\ndata of the associated $2\\\\times 2$ matrix linear Fuchsian system of ODE. The\\r\\nclass may contain solutions with poles accumulating at the critical point. The\\r\\nstudy of this class closes a gap in the description of the transcendents in one\\r\\nto one correspondence with the monodromy data. These transcendents are reviewed in the paper. Some formulas that relate the monodromy data to the critical behaviors of the four real (two complex) parameter class of solutions are\\r\\nmissing in the literature, so they are computed here. A computational procedure to write the full expansion of the four and three real parameter class of solutions is proposed.}, doi = {10.1093/imrn/rnr071}, url = {http://hdl.handle.net/1963/6093}, author = {Davide Guzzetti} } @article {2012, title = {Tabulation of Painlev{\'e} 6 transcendents}, journal = {Nonlinearity, Volume 25, Issue 12, December 2012, Pages 3235-3276}, number = {arXiv:1108.3401;}, year = {2012}, note = {30 pages, 1 figure; this article was published in "Nonlinearity" in 2012}, publisher = {IOP Publishing}, abstract = {The critical and asymptotic behaviors of solutions of the sixth Painlev{\textquoteright}e equation PVI, obtained in the framework of the monodromy preserving deformation method, and their explicit parametrization in terms of monodromy data, are tabulated.}, doi = {10.1088/0951-7715/25/12/3235}, url = {http://hdl.handle.net/1963/6520}, author = {Davide Guzzetti} } @article {2011, title = {An asymptotic reduction of a Painlev{\'e} VI equation to a Painlev{\'e} III}, journal = {J.Phys.A: Math.Theor. 44 (2011) 215203}, number = {arXiv:1101.4705;}, year = {2011}, publisher = {IOP Publishing}, abstract = {When the independent variable is close to a critical point, it is shown that\\r\\nPVI can be asymptotically reduced to PIII. In this way, it is possible to\\r\\ncompute the leading term of the critical behaviors of PVI transcendents\\r\\nstarting from the behaviors of PIII transcendents.}, doi = {10.1088/1751-8113/44/21/215203}, url = {http://hdl.handle.net/1963/5124}, author = {Davide Guzzetti} } @inbook {2011, title = {Solving PVI by Isomonodromy Deformations}, booktitle = {Painlevé equations and related topics : proceedings of the international conference, Saint Petersburg, Russia, June 17-23, 2011 / Aleksandr Dmitrievich Briuno; Alexander B Batkhin. - Berlin : De Gruyter, [2012]. - p. 101-105}, number = {arXiv:1106.2636;}, year = {2011}, note = {12 pages, 1 figurethis paper has been}, publisher = {SISSA}, organization = {SISSA}, abstract = {The critical and asymptotic behaviors of solutions of the sixth Painlev\\\'e\r\nequation, an their parametrization in terms of monodromy data, are\r\nsynthetically reviewed. The explicit formulas are given. This paper has been\r\nwithdrawn by the author himself, because some improvements are necessary.\r\nThis is a proceedings of the international conference \"Painlevé Equations and Related Topics\" which was taking place at the Euler International Mathematical Institute, a branch of the Saint Petersburg Department of the SteklovInstitute of Mathematicsof theRussian Academy of Sciences, in Saint Petersburg on June 17 to 23, 2011.}, keywords = {Painlevé Equations}, isbn = {9783110275582}, url = {http://hdl.handle.net/1963/6522}, author = {Davide Guzzetti} } @article {2008, title = {On the Logarithmic Asymptotics of the Sixth Painleve\' Equation (Summer 2007)}, journal = {J.Phys.A: Math.Theor. 41,(2008), 205201-205247}, number = {arXiv:0801.1157;}, year = {2008}, note = {This paper appeared as a preprint in August 2007. It is published in Journal of Physics A: Mathematical and Theoretical, Volume 41, Issue 20, 6 May 2008, p. 205201-205247. It was on the archive in January 2008 (arXiv:0801.1157). This version does not differ from the published one except for two facts: 1)the addition of subsection 8.2, which proves that tr(M0Mx) = -2 for solutions y(x) \~{} a (ln x)n , n = 1, 2, x {\textrightarrow} 0. 2). The title of the journal article is : The logarithmic asymptotics of the sixth Painlev{\'e} equation}, publisher = {SISSA}, abstract = {We study the solutions of the sixth Painlev\'e equation with a logarithmic\r\nasymptotic behavior at a critical point. We compute the monodromy group\r\nassociated to the solutions by the method of monodromy preserving deformations\r\nand we characterize the asymptotic behavior in terms of the monodromy itself.}, doi = {10.1088/1751-8113/41/20/205201}, url = {http://hdl.handle.net/1963/6521}, author = {Davide Guzzetti} } @article {2007, title = {The Asymptotic Behaviour of the Fourier Transforms of Orthogonal Polynomials II: L.I.F.S. Measures and Quantum Mechanics}, journal = {Ann. Henri Poincar{\textasciiacute}e 8 (2007), 301{\textendash}336}, year = {2007}, publisher = {2007 Birkh{\textasciidieresis}auser Verlag Basel/Switzerland}, abstract = {We study measures generated by systems of linear iterated functions,\r\ntheir Fourier transforms, and those of their orthogonal polynomials. We\r\ncharacterize the asymptotic behaviours of their discrete and continuous averages.\r\nFurther related quantities are analyzed, and relevance of this analysis\r\nto quantum mechanics is briefly discussed}, doi = {10.1007/s00023-006-0309-1}, author = {Davide Guzzetti and Giorgio Mantica} } @article {2006, title = {Matching Procedure for the Sixth Painlev{\'e} Equation (May 2006)}, journal = {Journal of Physics A: Mathematical and General, Volume 39, Issue 39, 29 September 2006, Article numberS02, Pages 11973-12031}, number = {arXiv:1010.1952;}, year = {2006}, note = {This paper appeared in May 2006. I put it on the archive now, with more that four years of delay, for completeness sake. The paper is published in J.Phys.A: Math.Gen. 39 (2006), 11973-12031, with some modifications.}, publisher = {SISSA}, abstract = {We present a constructive procedure to obtain the critical behavior of\r\nPainleve\' VI transcendents and solve the connection problem. This procedure\r\nyields two and one parameter families of solutions, including trigonometric and\r\nlogarithmic behaviors, and three classes of solutions with Taylor expansion at\r\na critical point.}, doi = {doi:10.1088/0305-4470/39/39/S02}, url = {http://hdl.handle.net/1963/6524}, author = {Davide Guzzetti} } @proceedings {2004, title = {The elliptic representation of the sixth Painlev{\'e} equation.}, year = {2004}, publisher = {Societe Matematique de France}, abstract = {We find a class of solutions of the sixth Painlev{\textasciiacute}e equation corresponding\r\nto almost all the monodromy data of the associated linear system; actually, all data\r\nbut one point in the space of data. We describe the critical behavior close to the\r\ncritical points by means of the elliptic representation, and we find the relation among\r\nthe parameters at the different critical points (connection problem).}, keywords = {Painlev{\'e} equation}, isbn = {978-2-85629-229-7}, url = {http://hdl.handle.net/1963/6529}, author = {Davide Guzzetti} } @article {2002, title = {The Elliptic Representation of the General Painlev{\'e} 6 Equation}, journal = {Communications on Pure and Applied Mathematics, Volume 55, Issue 10, October 2002, Pages 1280-1363}, number = {arXiv:math/0108073;}, year = {2002}, note = {60 pages; Latex; 3 figures. The statements of theorems have been\r\n simplified}, publisher = {SISSA}, abstract = {We study the analytic properties and the critical behavior of the elliptic\r\nrepresentation of solutions of the Painlev\\\'e 6 equation. We solve the\r\nconnection problem for elliptic representation in the generic case and in a\r\nnon-generic case equivalent to WDVV equations of associativity.}, doi = {10.1002/cpa.10045}, url = {http://hdl.handle.net/1963/6523}, author = {Davide Guzzetti} } @proceedings {2002, title = {The Elliptic Representation of the Painleve 6 Equation}, number = {RIMS Kokyuroku;vol. 1296}, year = {2002}, publisher = {Kyoto University, Research Institute for Mathematical Sciences}, abstract = {We review our results on the elliptic representation of the sixth Painleve{\textquoteright} equation}, keywords = {Painleve equations}, url = {http://hdl.handle.net/1963/6530}, author = {Davide Guzzetti} } @article {2001, title = {On the Critical Behavior, the Connection Problem and the Elliptic Representation of a Painlev{\'e} VI Equation}, journal = {Mathematical Physics, Analysis and Geometry 4: 293{\textendash}377, 2001}, year = {2001}, publisher = {Kluwer Academic Publishers}, abstract = {In this paper we find a class of solutions of the sixth Painlev{\'e} equation appearing in\r\nthe theory of WDVV equations. This class covers almost all the monodromy data associated to\r\nthe equation, except one point in the space of the data. We describe the critical behavior close to\r\nthe critical points in terms of two parameters and we find the relation among the parameters at\r\nthe different critical points (connection problem). We also study the critical behavior of Painlev{\'e}\r\ntranscendents in the elliptic representation.}, keywords = {Painleve Equations, Isomonodromy deformations}, doi = {10.1023/A:1014265919008}, author = {Davide Guzzetti} } @article {2001, title = {Inverse Problem and Monodromy Data for Three-Dimensional Frobenius Manifolds}, journal = {Mathematical Physics, Analysis and Geometry 4: 245{\textendash}291, 2001}, year = {2001}, publisher = {RIMS, Kyoto University}, abstract = {We study the inverse problem for semi-simple Frobenius manifolds of dimension 3 and we\r\nexplicitly compute a parametric form of the solutions of theWDVV equations in terms of Painlev{\'e} VI\r\ntranscendents. We show that the solutions are labeled by a set of monodromy data. We use our parametric\r\nform to explicitly construct polynomial and algebraic solutions and to derive the generating\r\nfunction of Gromov{\textendash}Witten invariants of the quantum cohomology of the two-dimensional projective\r\nspace. The procedure is a relevant application of the theory of isomonodromic deformations.}, keywords = {Frobenius Manifolds, Painleve Equations, Isomonodromy deformations}, doi = {10.1023/A:1012933622521}, author = {Davide Guzzetti} } @article {2000, title = {Inverse problem for Semisimple Frobenius Manifolds Monodromy Data and the Painlev{\'e} VI Equation}, number = {SISSA;101/00/FM}, year = {2000}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1557}, author = {Davide Guzzetti} } @inbook {2000, title = {Stokes Matrices for Frobenius Manifolds and the 6 Painlev{\'e} Equation}, booktitle = {Rokko Lectures in Mathematics, Vol 7 [Issue title: Perspective of Painleve equations], (2000), pages : 101-109}, number = {Rokko lectures in mathematics;vol. 7}, year = {2000}, publisher = {Kobe University, Japan}, organization = {Kobe University, Japan}, abstract = {These notes are a short review on the theory of Frobenius manifolds and its connection to problems of isomonodromy deformations and to Painlev{\textquoteright}e equations.}, keywords = {Painlev{\'e} equation}, isbn = {4-907719-07-8}, url = {http://hdl.handle.net/1963/6546}, author = {Davide Guzzetti} } @article {1999, title = {Stokes matrices and monodromy of the quantum cohomology of projective spaces}, journal = {Comm. Math. Phys. 207 (1999) 341-383}, number = {arXiv.org;math/9904099v1}, year = {1999}, publisher = {Springer}, abstract = {n this paper we compute Stokes matrices and monodromy of 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 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.}, doi = {10.1007/s002200050729}, url = {http://hdl.handle.net/1963/3475}, author = {Davide Guzzetti} }