Hyperkahler manifolds and nonabelian Hodge theory of (irregular) curves

Short survey based on talk given at the Institut Henri Poincare January 17th 2012, during program on surface groups. The aim was to describe some background results before describing in detail (in subsequent talks) the results of [Boa11c] related to wild character varieties and irregular mapping class groups.Comment: 16 pages, 2 figures, 3 table

G-bundles, isomonodromy and quantum Weyl groups

First an `irregular Riemann-Hilbert correspondence' is established for meromorphic connections on principal G-bundles over a disc, where G is any connected complex reductive group. Secondly, in the case of poles of order two, isomonodromic deformations of such connections are considered and it is proved that the classical actions of quantum Weyl groups found by De Concini, Kac and Procesi do arise from isomonodromy (and so have a purely geometrical origin). Finally a certain flat connection appearing in work of De Concini and Toledano Laredo is derived from isomonodromy, indicating that the above result is the classical analogue of their conjectural Kohno-Drinfeld theorem for quantum Weyl groups.Comment: 30 pages, 1 figure (proof of Theorem 5 simplified

The fifty-two icosahedral solutions to Painleve VI

The solutions of the (nonlinear) Painleve VI differential equation having icosahedral linear monodromy group will be classified up to equivalence under Okamoto's affine F4 Weyl group action and many properties of the solutions will be given. There are 52 classes, the first ten of which correspond directly to the ten icosahedral entries on Schwarz's list of algebraic solutions of the hypergeometric equation. The next nine solutions are simple deformations of known PVI solutions (and have less than five branches) and five of the larger solutions are already known, due to work of Dubrovin and Mazzocco and Kitaev. Of the remaining 28 solutions we will find 20 explicitly using (the author's correction of) Jimbo's asymptotic formula. Amongst those constructed there is one solution that is 'generic' in that its parameters lie on none of the affine F4 hyperplanes, one that is equivalent to the Dubrovin--Mazzocco elliptic solution and three elliptic solutions that are related to the Valentiner three-dimensional complex reflection group, the largest having 24 branches.Comment: 28 pages, 2 tables, final version, to appear in Crelle's journal (minor corrections, added two solutions and remarked that the remaining 8 solutions may be obtained via quadratic transformations

Poisson varieties from Riemann surfaces

Short survey based on talk at the Poisson 2012 conference. The main aim is to describe and give some examples of wild character varieties (naturally generalising the character varieties of Riemann surfaces by allowing more complicated behaviour at the boundary), their Poisson/symplectic structures (generalising both the Atiyah-Bott approach and the quasi-Hamiltonian approach), and the wild mapping class groups.Comment: 33 pages, 3 figure

Through the analytic halo: Fission via irregular singularities

This article is concerned with moduli spaces of connections on bundles on Riemann surfaces, where the structure group of the bundle may vary in different regions of the surface. Here we will describe such moduli spaces as complex symplectic manifolds, generalising the complex character varieties of Riemann surfaces.Comment: 16 pages, 2 figures. Applied and extended in arXiv:1111.6228 (v2: gluing symbol fixed