# Projective structures and bending properties of surface diffeomorphisms

Ferry Kwakkel

## Date:

Friday, 13 May, 2016 - 14:30

We consider the Teichmueller space of hyperbolic and convex projective
structures on a closed surface and introduce a notion of bending of a diffeomorphism
between two surfaces, each endowed with a projective structure. These bending properties give rise to a notion of distance on Teichmueller space, and we study
Teichmueller space through this distance by considering the geometric
bending properties diffeomorphisms display.

# Explicit Schoen surfaces

Carlos Rito

## Date:

Friday, 29 April, 2016 - 14:30

## Venue:

Room 0.06

Chad Schoen (2007) used deformations to construct a family of smooth complex algebraic surfaces S with invariants \chi=2, q=4 and K^2=16 with some remarkable properties. Then C. Ciliberto, M. Mendes Lopes and X. Roulleau (2015) showed that the canonical map of S is a double covering of a degree 8 surface X in P^4 with 40 double points. In this talk I will explain how to construct the Schoen surfaces explicitely, by computing equations for the surfaces X.

# Closed and transitive homeomorphism groups of a surface

Ferry Kwakkel

## Date:

Friday, 15 April, 2016 - 14:30

## Venue:

Room 0.06

In this talk, we discuss the problem of classifying the closed and transitive homeomorphism groups of a closed surface. We survey a number of such groups on each surface, and we give several non-trivial relations between such groups. We mainly focus on the case of the sphere, where we provide a structural result about basic properties of such groups that contain the rotation group.

# Mirror symmetry for Cartan branes

Ana Peón

## Date:

Wednesday, 9 March, 2016 - 14:30

## Venue:

Room 0.31

Let G be a complex reductive Lie group, and consider Higgs(G) the moduli space of G-Higgs bundles. The choice of a maximal torus T<G defines a BBB brane Higgs(T)⊂Higgs(G) (the Cartan brane). According to Kapustin--Witten,  this corresponds under mirror symmetry to a BAA brane in Higgs(LG), where LG denotes the Langlands dual group. In this talk I will explain what this BAA should be for rank two Higgs bundles.

# The SL(3,C)-character variety of the figure eight knot

Vicente Muñoz

## Date:

Friday, 26 February, 2016 - 11:15

## Venue:

Room 0.30

Character varieties parametrize representations of the fundamental group of a variety into complex Lie groups. Of special interest for 3-manifold topology is the case of the complement of a knot in the 3-sphere. We shall study the case of the character variety of the figure eight knot for the groups SL(3,C), GL(3,C) and PGL(3,C), giving explicit equations. This character variety has five components of dimension 2, one consisting of totally reducible representations, another one consisting of partially reducible representations, and three components of irreducible representations.

# Mixed Hodge polynomials of abelian character varieties

## Speaker:

Carlos Florentino

## Date:

Friday, 26 February, 2016 - 10:00

## Venue:

Room 0.30

Character varieties are spaces of representations of finitely presented groups into real reductive Lie groups. In some cases, these can be interpreted as moduli spaces of G-Higgs bundles over a Kähler manifold M, spaces which reflect both algebraic properties of G and complex analytic properties of M. In particular, their topology is a very important subject of research.

# Schottky principal bundles over Riemann surfaces

Ana Casimiro

## Date:

Friday, 12 February, 2016 - 15:30

## Venue:

Room 0.04

The (Schottky) uniformization of Riemann surfaces motivated the search of a parametrization of holomorphic bundles. Semistable bundles over a fixed Riemann surface X are obtained from a unitary representation of the fundamental group of X, as was proven by Narasimhan and Seshadri for vector bundles, and by Ramanathan for principal bundles (for reductive groups over C).

# Seidel's morphism of toric 4-manifolds

Sílvia Anjos

## Date:

Friday, 22 January, 2016 - 15:30

## Venue:

Room 0.04

We explain how to calculate, in some particular cases, the Seidel representation of $\pi_1({\rm Ham}(M,\omega))$  in the units of the quantum homology ring,  where ${\rm Ham}(M,\omega)$ denotes the group of Hamiltonian symplectomorphisms of a closed symplectic manifold $(M, \omega)$. This is very difficult to calculate in general. However, following the work of D. McDuff and S.

# Hard Lefschetz theorem for Vaisman manifolds

## Speaker:

Antonio De Nicola (Coimbra)

## Date:

Friday, 27 November, 2015 - 15:30

## Venue:

0.04

It is well known that in any compact Kaehler manifold the exterior multiplication by a suitable power of the symplectic form induces isomorphisms between the de Rham cohomology spaces in complementary degrees. This is the celebrated Hard Lefschetz Theorem.