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Timetable (MOSW02)

Representations of surface groups and Higgs Bundles

Monday 14th March 2011 to Friday 18th March 2011

Monday 14th March 2011
09:00 to 09:55 Registration at Mathematical Institute
09:55 to 10:00 Welcome
10:00 to 11:00 Higgs bundles and surface group representations in non-compact real groups
We will describe how Higgs bundles, inherently holomorphic objects, are related to surface group representations in non-compact real Lie groups and how this relationship can be used to answer questions about the corresponding representation varieties. The real group Sp(4,R) will be given special attention.
11:00 to 11:30 Morning Coffee at Mathematical Institute
11:30 to 12:30 Generalizations of parabolic bundles related to Higgs bundles
The first part of the talk will be devoted to work in progress together with O. Biquard and Ó. García-Prada on a version of parabolic Higgs bundles which correspond to (semistable) G-local systems on a punctured Riemann surface, where G is a real semisimple Lie group. In the second part we will talk about ongoing work with M. Logares on a generalization of parabolic bundles which correspond (via a construction which relates them to conic bundles) to Sp(6,R)-local systems on compact Riemann surfaces. Emphasis will be on the stability conditions; a unifying theme will be the interpretation of the local terms in the formula for the parabolic degree in terms of GIT.
13:00 to 14:00 Lunch served at Wadham College
14:00 to 15:00 Free time
15:00 to 16:00 An Algebra of Observables for Cross Ratios
We define a Poisson Algebra called the swapping algebra using the intersection of curves in the disk. We interpret a subalgebra of the fraction swapping algebra -- called the algebra of multifractions -- as an algebra of functions on the space of cross ratios and thus as an algebra of functions on the Hitchin component as well as on the space of SL(n;R)-opers with trivial holonomy. We finally relate our Poisson structure to the Drinfel'd-Sokolov structure and to the Atiyah-Bott-Goldman symplectic structure for classical Teichmüller spaces and Hitchin components.
16:00 to 16:30 Afternoon Tea at Mathematical Institute
16:30 to 17:30 Topology and singularities of free group character varieties
We will discuss some generalities of the geometry, topology and singularities of the the G-character variety of F, that is, the moduli space Hom(F,G)/G of representations of a finitely presented group F into a Lie group G.

Then, we concentrate on the case when G is a complex affine reductive Lie group with maximal compact subgroup K, and F is a free group of rank r. In this situation, it can be proved that Hom(F,K)/K is a strong deformation retract of Hom(F,G)/G; in particular, both spaces have the same homotopy type. In the case G=SL(n,C), one can explicitly describe the singular locus of these character varieties, showing that they have the homotopy type of a manifold only when F or G are abelian, or r+n
17:45 to 18:30 Wine reception at Mathematical Institute
Tuesday 15th March 2011
10:00 to 11:00 BPS states, Donaldson-Thomas invariants, and Hitchin pairs
This is a report on recent work in collaboration with Wu-yen Chuang and Guang Pan relating Donaldson-Thomas theory and BPS to the cohomology of the Hitchin system. A string theoretic construction will be presented establishing a connection between curve counting invariants and the work of Hausel and Rodriguez-Villegas. A generalization to parabolic Hitchin pairs will be also briefly discussed.
11:00 to 11:30 Morning Coffee at Mathematical Institute
11:30 to 12:30 Self-duality of of reducible GL(n) Hitchin fibers
This is a report on a joint work in progress with D.Arinkin. We investigate the self-duality question for the compactified stacky Jacobian of a reducible curve with planar singularities. This is needed for extending the classical limit Langlands duality for GL(n) to the whole moduli of Higgs bundles. I will explain how the self-duality statement extends to a single reduced but possibly reducible curve and will discuss the technical issues that one needs to overcome to prove the statement in families. I will also discuss the way self-duality interacts with stability.
13:00 to 14:00 Lunch served at Wadham College
14:00 to 15:00 Free time
15:00 to 16:00 On the motives of moduli spaces of Higgs bundles
We will explain an approach to the computation of the cohomology of moduli spaces of Higgs bundles on surfaces, that is closely related to the argument of Harder-Narasimhan for moduli spaces of vector bundles and give an application to the middle cohomology of the moduli space of SL_n Higgs bundles. This is joint work with O. Garcia-Prada and A. Schmitt.
16:00 to 16:30 Afternoon Tea at Mathematical Institute
16:30 to 17:30 R Wentworth ([Maryland])
Topology of some representation varieties of surface groups
I will discuss recent generalizations of the techniques of Atiyah and Bott on equivariant Morse theory. These extend results on stable bundles to Higgs bundles and stable pairs. As a consequence, information is obtained on the topology of representation varieties into noncompact Lie groups.
Wednesday 16th March 2011
10:00 to 11:00 Domains of Discontinuity for Anosov Representations and Generalized Teichmüller Spaces
Many representations of surface groups (in particular those belonging to "generalized" Teichmüller spaces) are known to satisfy a strong dynamical property: they are Anosov representations. We shall first explain more fully this notion due to F. Labourie. Secondly we will explain how an Anosov representation $\Gamma \to G$ (for any group $\Gamma$) can be interpreted as the holonomy representation of a geometric structure by constructing a domain of discontinuity with compact quotient for $\Gamma$ into a homogenous $G$-space. At last we shall see to what extent this construction can be used in interpreting the generalized Teichmüller spaces as moduli of geometric structures.

This is a joint work with Anna Wienhard.
11:00 to 11:30 Morning Coffee at Mathematical Institute
11:30 to 12:30 D Toledo ([Utah])
Convexity Properties of Energy on Teichmüller Space
Let M be a closed surface of genus at least two, N a manifold of non-positive Hermitian curvature (the Siu-Sampson condition) and fix a homotopy class of maps from M to N (or a representation of the fundamental group of M in the group of isometries of N). For each complex structure J on M there is a harmonic map f:M->N (or an equivariant harmonic map of the universal covers). In situations where this map is unique it depends smoothly on J and its energy E defines a smooth function on the Teichmüller space of M. We prove that this function is plurisubharmonic, and study conditions when it is strictly plurisubharmonic.

This result was suggested by Gromov as an alternative way of developing and strengthening the Siu-Sampson rigidity theory. Indications of these applications will be given as time permits.
13:00 to 14:00 Lunch served at Wadham College
14:00 to 15:00 Free time
15:00 to 16:00 Local rigidity for complex hyperbolic lattices
I will explain how Hodge theory can be used to prove local rigidity results for complex hyperbolic lattices.
16:00 to 16:30 Afternoon Tea at Mathematical Institute
16:30 to 17:30 Linear coverings of complex projective manifolds
This talk will survey the methods and applications of our joint work with Katzarkov Pantev and Ramachandran arxiv/0409.0693.
Thursday 17th March 2011
10:00 to 11:00 A Iozzi (ETH Zürich)
Surfaces and bounded cohomology
We introduce the notion of causal representation of a surface group and relate it to that of maximal representation and of tight homomorphism. When the target is SL(2,R) we show that these are hyperbolizations. In the process we define and study the bounded fundamental class of a compact surface (with or without boundary) and establish a result characterizing it among all bounded classes. We relate this to the winding number of Chillingsworth and to work of Calegari on stable commutator length.
11:00 to 11:30 Morning Coffee at Mathematical Institute
11:30 to 12:30 M Burger (ETH Zürich)
Causal representations of surface groups
In this talk we will present a structure theorem concerning causal representations; in particular we will discuss the rationality of the Toledo invariant in the non compact case and explain its relation to the characterisation of non tube type domains in terms of the hermitian triple product.
13:00 to 14:00 Lunch served at Wadham College
14:00 to 15:00 Free time
15:00 to 16:00 Asymptotics in TQFT
We will via the geometric quantization of moduli spaces of flat connections discuss various asymptotic properties of the associated representations of the mapping class groups.
16:00 to 16:30 Afternoon Tea at Mathematical Institute
16:30 to 17:30 S Choi & K Choi ([Korea Advanced Institute of Science and Technology])
Deforming convex real projective 3-orbifolds
A convex real projective 3-orbifold is the quotient orbifold of a convex domain in $RP^3$ by a discrete group of projective automorphisms in $PGL(4, R)$. Hyperbolic 3-orbifolds form a subclass. The convex real projective 3-manifolds were begun to be studied by Cooper, Long, and Thistlethwaite. We will summarize some of the recent results on deforming convex real projective structures on 3-dimensional orbifolds, including those of Benoist, myself, Marquis, Lee, Hodgson, Cooper, Tillman, and so on. In particular, a numerical study of real projective structures on Coxeter orbifolds is included. Finally, we discuss open problems in this area. Our topic is related to understanding the deformations of $SL(4,R)$-representations of discrete groups.
18:45 to 19:30 Wine reception at Wadham College (sponsored by Oxford University Press)
19:30 to 21:00 Conference dinner at Wadham College
Friday 18th March 2011
10:00 to 11:00 P Boalch ([ENS])
Irregular connections, Dynkin diagrams and fission
I'll survey some results (both old and new) related to the geometry of moduli spaces of irregular connections on curves. If time permits this will include: 1) new nonlinear geometric braid group actions, 2) new complete hyperkahler manifolds (including some gravitational instantons) [in work with O. Biquard], and 3) new ways to glue Riemann surfaces together to obtain (symplectic) generalisations of the complex character varieties of surface groups.
11:00 to 11:30 Morning Coffee at Mathematical Institute
11:30 to 12:30 Fibrations on the moduli of parabolic connections on P^1 minus 4 points
This reports on joint work with Frank Loray and Masa-Hiko Saito. Given a connection with parabolic structure, one can look at the limit as $t\rightarrow 0$ in Hitchin's twistor space. The limit is a $C^*$-fixed Higgs bundle. Breaking up the moduli space according to the isomorphism class of the limit leads to a decomposition in locally closed subvarieties. In the case of rank $2$ connections on $P^1-\{ t_1,t_2,t_3,t_4\}$ we are able to show that the subvarieties are closed. They are the fibers of fibrations, depending on the parabolic weights, which are already known: appearing for example in work of Arinkin and Lysenko, and of Iwasaki, Inaba, Saito. Katz's middle convolution is one of Okamoto's symmetries exchanging the different types of fibrations.
13:00 to 14:00 Lunch served at Wadham College
14:00 to 17:30 Free afternoon
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons