The holonomy group at infinity of the Painleve VI Equation

We prove that the holonomy group at infinity of the Painleve VI equation is virtually commutative.Comment: 21 pages, 3 figures, Added references, Corrected typo

Painlev\'e I and exact WKB: Stokes phenomenon for two-parameter transseries

For more than a century, the Painlev\'e I equation has played an important role in both physics and mathematics. Its two-parameter family of solutions was studied in many different ways, yet still leads to new surprises and discoveries. Two popular tools in these studies are the theory of isomonodromic deformation that uses the exact WKB method, and the asymptotic description of transcendents in terms of two-parameter transseries. Combining methods from both schools of thought, and following work by Takei and collaborators, we find complete, two-parameter connection formulae for solutions when they cross arbitrary Stokes lines in the complex plane. These formulae allow us to study Stokes phenomenon for the full two-parameter family of transseries solutions. In particular, we recover the exact expressions for the Stokes data that were recently found by Baldino, Schwick, Schiappa and Vega and compare our connection formulae to theirs. We also explain several ambiguities in relating transseries parameter choices to actual Painlev\'e transcendents, study the monodromy of formal solutions, and provide high-precision numerical tests of our results.Comment: 71 pages, 16 figures, 5 tables and 4 appendices. v2: Minor changes (rewrites, typos, added references

Open Problems for Painlevé Equations

In this paper some open problems for Painlevé equations are discussed. In particular the following open problems are described: (i) the Painlevé equivalence problem; (ii) notation for solutions of the Painlevé equations; (iii) numerical solution of Painlevé equations; and (iv) the classification of properties of Painlevé equations

Supersymmetric Field Theories and Isomonodromic Deformations

The topic of this thesis is the recently discovered correspondence between supersymmetric gauge theories, two-dimensional conformal field theories and isomonodromic deformation problems. Its original results are organized in two parts: the first one, based on the papers [1], [2], as well as on some further unpublished results, provides the extension of the correspondence between four-dimensional class S theories and isomonodromic deformation problems to Riemann Surfaces of genus greater than zero. The second part, based on the results of [3], is instead devoted to the study of five-dimensional superconformal field theories, and their relation with q-deformed isomonodromic problems

Exponential asymptotics for discrete Painlevé equations

In this thesis we study Stokes phenomena behaviour present in the solutions of both additive and multiplicative difference equations. Specifically, we undertake an asymptotic study of the second discrete Painlevé equation (dPII) as the independent variable approaches infinity, and consider the asymptotic behaviour of solutions of the q-Airy equation and the first q-Painlevé equation in the limits q□(→┴ ) 1 and n□(→┴ ) ∞. Exponential asymptotic methods are used to investigate Stokes phenomena and obtain uniform asymptotic expansions of solutions of these equations. In the first part of this thesis, we obtain two types of asymptotic expansions which describe vanishing and non-vanishing type solution behaviour of dPII. In particular, we show that both types of solution behaviour can be expressed as the sum of an optimally-truncated asymptotic series and an exponentially subdominant correction term. We then determine the Stokes structure and investigate Stokes behaviour present in these solutions. From this information we show that the asymptotic expansions contain one free parameter hidden beyond-all-orders and determine regions of the complex plane in which these asymptotic descriptions are valid. Furthermore, we deduce special asymptotic solutions which are valid in extended regions and draw parallels between these asymptotic solutions to the tronquée and tri-tronquée solutions of the second Painlevé equation. In the second part of this thesis, we then extend the exponential asymptotic method to q-difference equations. In our analysis of both the q-Airy and first q-Painlevé equations, we find that the Stokes structure is described by curves referred to as q-spirals. As a consequence, we discover that the Stokes structure for solutions of q-difference equations separate the complex plane into sectorial regions bounded by arcs of spirals rather than traditional rays

Geometry and Analytic Theory of Semisimple Coalescent Frobenius Structures: an Isomonodromic approach to Quantum Cohomology and Helix structures in Derived Categories

In this Thesis we study geometrical and analytic aspects of semisimple points of Frobenius manifolds presenting a phenomenon of coalescence of canonical coordinates. Particular attention is given to the isomonodromic description of these resonances as well as to their (still conjectural) relationships with the derived geometry of Fano varieties

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described