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.

10aPainlevé equation1 aGuzzetti, Davide uhttp://hdl.handle.net/1963/652501270nas a2200109 4500008004100000245012700041210007000168260002800238520083700266100002101103856003601124 2012 en d00aSolving the Sixth Painlevé Equation: Towards the Classification of all the Critical Behaviors and the Connection Formulae0 aSolving the Sixth Painlevé Equation Towards the Classification o bOxford University Press3 aThe 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.1 aGuzzetti, Davide uhttp://hdl.handle.net/1963/609300562nas a2200109 4500008004100000245004400041210004400085260001900129520024700148100002100395856003600416 2012 en d00aTabulation of Painlevé 6 transcendents0 aTabulation of Painlevé 6 transcendents bIOP Publishing3 aThe critical and asymptotic behaviors of solutions of the sixth Painlev'e equation PVI, obtained in the framework of the monodromy preserving deformation method, and their explicit parametrization in terms of monodromy data, are tabulated.1 aGuzzetti, Davide uhttp://hdl.handle.net/1963/652000665nas a2200109 4500008004100000245007400041210007100115260001900186520029300205100002100498856003600519 2011 en d00aAn asymptotic reduction of a Painlevé VI equation to a Painlevé III0 aasymptotic reduction of a Painlevé VI equation to a Painlevé III bIOP Publishing3 aWhen 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.1 aGuzzetti, Davide uhttp://hdl.handle.net/1963/512401019nas a2200133 4500008004100000020001800041245004500059210004500104260001000149520064400159653002500803100002100828856003600849 2011 en d a978311027558200aSolving PVI by Isomonodromy Deformations0 aSolving PVI by Isomonodromy Deformations bSISSA3 aThe 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, 2011.10aPainlevé Equations1 aGuzzetti, Davide uhttp://hdl.handle.net/1963/652200688nas a2200109 4500008004100000245008400041210006900125260001000194520031700204100002100521856003600542 2008 en d00aOn the Logarithmic Asymptotics of the Sixth Painleve\' Equation (Summer 2007)0 aLogarithmic Asymptotics of the Sixth Painleve Equation Summer 20 bSISSA3 aWe 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.1 aGuzzetti, Davide uhttp://hdl.handle.net/1963/652100924nas a2200121 4500008004100000245012500041210006900166260004700235520035300282100002100635700002100656856012500677 2007 en d00aThe Asymptotic Behaviour of the Fourier Transforms of Orthogonal Polynomials II: L.I.F.S. Measures and Quantum Mechanics0 aAsymptotic Behaviour of the Fourier Transforms of Orthogonal Pol b2007 Birkh¨auser Verlag Basel/Switzerland3 aWe 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 discussed1 aGuzzetti, Davide1 aMantica, Giorgio uhttp://www.math.sissa.it/publication/asymptotic-behaviour-fourier-transforms-orthogonal-polynomials-ii-lifs-measures-and00691nas a2200109 4500008004100000245006700041210006500108260001000173520034100183100002100524856003600545 2006 en d00aMatching Procedure for the Sixth Painlevé Equation (May 2006)0 aMatching Procedure for the Sixth Painlevé Equation May 2006 bSISSA3 aWe 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.1 aGuzzetti, Davide uhttp://hdl.handle.net/1963/652400854nas a2200133 4500008004100000020002200041245006500063210006000128260003400188520041800222653002300640100002100663856003600684 2004 en d a978-2-85629-229-700aThe elliptic representation of the sixth Painlevé equation.0 aelliptic representation of the sixth Painlevé equation bSociete Matematique de France3 aWe find a class of solutions of the sixth Painlev´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).10aPainlevé equation1 aGuzzetti, Davide uhttp://hdl.handle.net/1963/652900651nas a2200109 4500008004100000245006800041210006400109260001000173520030100183100002100484856003600505 2002 en d00aThe Elliptic Representation of the General Painlevé 6 Equation0 aElliptic Representation of the General Painlevé 6 Equation bSISSA3 aWe 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.1 aGuzzetti, Davide uhttp://hdl.handle.net/1963/652300515nas a2200121 4500008004100000245005900041210005500100260006700155520009100222653002300313100002100336856003600357 2002 en d00aThe Elliptic Representation of the Painleve 6 Equation0 aElliptic Representation of the Painleve 6 Equation bKyoto University, Research Institute for Mathematical Sciences3 aWe review our results on the elliptic representation of the sixth Painleve’ equation10aPainleve equations1 aGuzzetti, Davide uhttp://hdl.handle.net/1963/653001122nas a2200121 4500008004100000245011200041210006900153260003100222520054200253653005000795100002100845856013400866 2001 en d00aOn the Critical Behavior, the Connection Problem and the Elliptic Representation of a Painlevé VI Equation0 aCritical Behavior the Connection Problem and the Elliptic Repres bKluwer Academic Publishers3 aIn this paper we find a class of solutions of the sixth Painlevé 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 Painlevé\r\ntranscendents in the elliptic representation.10aPainleve Equations, Isomonodromy deformations1 aGuzzetti, Davide uhttp://www.math.sissa.it/publication/critical-behavior-connection-problem-and-elliptic-representation-painlev%C3%A9-vi-equation-001140nas a2200121 4500008004100000245008100041210006900122260002700191520059400218653007100812100002100883856011400904 2001 en d00aInverse Problem and Monodromy Data for Three-Dimensional Frobenius Manifolds0 aInverse Problem and Monodromy Data for ThreeDimensional Frobeniu bRIMS, Kyoto University3 aWe 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 Painlevé 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–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.10aFrobenius Manifolds, Painleve Equations, Isomonodromy deformations1 aGuzzetti, Davide uhttp://www.math.sissa.it/publication/inverse-problem-and-monodromy-data-three-dimensional-frobenius-manifolds00383nas a2200097 4500008004100000245010000041210006900141260001800210100002100228856003600249 2000 en d00aInverse problem for Semisimple Frobenius Manifolds Monodromy Data and the Painlevé VI Equation0 aInverse problem for Semisimple Frobenius Manifolds Monodromy Dat bSISSA Library1 aGuzzetti, Davide uhttp://hdl.handle.net/1963/155700604nas a2200133 4500008004100000020001800041245007300059210007000132260002700202520016100229653002300390100002100413856003600434 2000 en d a4-907719-07-800aStokes Matrices for Frobenius Manifolds and the 6 Painlevé Equation0 aStokes Matrices for Frobenius Manifolds and the 6 Painlevé Equat bKobe University, Japan3 aThese notes are a short review on the theory of Frobenius manifolds and its connection to problems of isomonodromy deformations and to Painlev'e equations.10aPainlevé equation1 aGuzzetti, Davide uhttp://hdl.handle.net/1963/654600949nas a2200109 4500008004300000245008100043210006900124260001300193520057600206100002100782856003600803 1999 en_Ud 00aStokes matrices and monodromy of the quantum cohomology of projective spaces0 aStokes matrices and monodromy of the quantum cohomology of proje bSpringer3 an 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.1 aGuzzetti, Davide uhttp://hdl.handle.net/1963/3475