In 1995 N. Hitchin constructed explicit algebraic solutions to the Painlevé VI (1/8,-1/8,1/8,3/8) equation starting with any Poncelet trajectory, that is a closed billiard trajectory inscribed in a conic and circumscribed about another conic. In this talk I will show that Hitchin's construction is the Okamoto transformation between Picard's solution and the general solution of the Painlevé VI (1/8,-1/8,1/8,3/8) equation. Moreover, this Okamoto transformation can be written in terms of an Abelian differential of the third kind on the associated elliptic curve, which allows to write down solutions to the corresponding Schlesinger system in terms of this differential as well. This is a joint work with V. Dragovic.

# Past Geometry and Analysis Seminar

The classical Yamabe problem asks to find in a given conformal class a metric of constant scalar curvature. In fully nonlinear analogues, the scalar curvature is replaced by certain functions of the eigenvalue of the Schouten curvature tensor. I will report on quantitative Liouville theorems and fine blow-up analysis for these problems. Joint work with Yanyan Li.

We give a noncommutative analogue of Castelnuovo's classic theorem that (-1) lines on a smooth surface can be contracted, and show how this may be used to construct an explicit birational map between a noncommutative P^2 and a noncommutative quadric surface. This has applications to the classification of noncommutative projective surfaces, one of the major open problems in noncommutative algebraic geometry. We will not assume a background in noncommutative ring theory. The talk is based on joint work with Rogalski and Staffor

An "open de Rham space" refers to a moduli space of meromorphic connections on the projective line with underlying trivial bundle. In the case where the connections have simple poles, it is well-known that these spaces exhibit hyperkähler metrics and can be realized as quiver varieties. This story can in fact be extended to the case of higher order poles, at least in the "untwisted" case. The "twisted" spaces, introduced by Bremer and Sage, refer to those which have normal forms diagonalizable only after passing to a ramified cover. These spaces often arise as quotients by unipotent groups and in some low-dimensional examples one finds some well-known hyperkähler manifolds, such as the moduli of magnetic monopoles. This is a report on ongoing work with Tamás Hausel and Dimitri Wyss.

Sasakian manifolds are odd-dimensional counterparts of Kahler manifolds in even dimensions,

with K-contact manifolds corresponding to symplectic manifolds. It is an interesting problem to find

obstructions for a closed manifold to admit such types of structures and in particular, to construct

K-contact manifolds which do not admit Sasakian structures. In the simply-connected case, the

hardest dimension is 5, where Kollar has found subtle obstructions to the existence of Sasakian

structures, associated to the theory of algebraic surfaces.

In this talk, we develop methods to distinguish K-contact manifolds from Sasakian ones in

dimension 5. In particular, we find the first example of a closed 5-manifold with first Betti number 0 which is K-contact but which carries no semi-regular Sasakian structure.

(Joint work with J.A. Rojo and A. Tralle).

Work of Schoen--Yau in the 70's/80's shows that area-minimizing (actually stable) two-sided surfaces in three-manifolds of non-negative scalar curvature are of a special topological type: a sphere, torus, plane or cylinder. The torus and cylinder cases are "borderline" for this estimate. It was shown by Cai--Galloway in the late 80's that the torus can only occur in a very special ambient three manifold. We complete the story by showing that a similar result holds for the cylinder. The talk should be accessible to those with a basic knowledge of curvature in Riemannian geometry.

After reviewing the main results relating holomorphic Poisson geometry to generalized Kahler structures, I will explain some recent progress in deforming generalized Kahler structures. I will also describe a new way to view generalized kahler geometry purely in terms of Poisson structures.