Abstract:
In this paper we study the symplectic and Poisson geometry of moduli spaces of flat connections over quilted surfaces. These are surfaces where the structure group varies from region to region in the surface, and where a reduction (or relation) of structure occurs along the boundaries of the regions. Our main theoretical tool is a new form moment-map reduction in the context of Dirac geometry. This reduction framework allows us to use very general relations of structure groups, and to investigate both the symplectic and Poisson geometry of the resulting moduli spaces from a unified perspective. The moduli spaces we construct in this way include a number of important examples, including Poisson Lie groups and their Homogeneous spaces, moduli spaces for meromorphic connections over Riemann surfaces (following the work of Philip Boalch), and various symplectic groupoids. Realizing these examples as moduli spaces for quilted surfaces provides new insights into their geometry.

Abstract:
We consider canonical symplectic structure on the moduli space of flat ${\g}$-connections on a Riemann surface of genus $g$ with $n$ marked points. For ${\g}$ being a semisimple Lie algebra we obtain an explicit efficient formula for this symplectic form and prove that it may be represented as a sum of $n$ copies of Kirillov symplectic form on the orbit of dressing transformations in the Poisson-Lie group $G^{*}$ and $g$ copies of the symplectic structure on the Heisenberg double of the Poisson-Lie group $G$ (the pair ($G,G^{*}$) corresponds to the Lie algebra ${\g}$).

Abstract:
We discuss Witten's formulas for the symplectic volumes of moduli spaces of flat connections on 2-manifolds from the viewpoint of Hamiltonian cobordism as introduced by Ginzburg-Guillemin-Karshon.

Abstract:
Let $G$ be a simple complex Lie group, $\alg{g}$ be its Lie algebra, $K$ be a maximal compact form of $G$ and $\alg{k}$ be a Lie algebra of $K$. We denote by $X\rightarrow \overline{X}$ the anti-involution of $\alg{g}$ which singles out the compact form $\alg{k}$. Consider the space of flat $\alg{g}$-valued connections on a Riemann sphere with three holes which satisfy the additional condition $\overline{A(z)}=-A(\overline{z})$. We call the quotient of this space over the action of the gauge group $\overline{g(z)}=g^{-1}(\overline{z})$ a \emph{hyperbolic} moduli space of flat connections. We prove that the following three symplectic spaces are isomorphic: 1. The hyperbolic moduli space of flat connections. 2. The symplectic multiplicity space obtained as symplectic quotient of the triple product of co-adjoint orbits of $K$. 3. The Poisson-Lie multiplicity space equal to the Poisson quotient of the triple product of dressing orbits of $K$.

Abstract:
We study moduli spaces of flat connections on surfaces with boundary, with boundary conditions given by Lagrangian Lie subalgebras. The resulting symplectic manifolds are closely related with Poisson-Lie groups and their algebraic structure (such as symplectic groupoid structure) gets a geometrical explanation via 3-dimensional cobordisms. We give a formula for the symplectic form in terms of holonomies, based on a central extension of the gauge group by closed 2-forms. This construction is finally used for a certain extension of the Morita equivalence of quantum tori to the world of Poisson-Lie groups.

Abstract:
We introduce a symplectic structure on the space of connections in a G-principal bundle over a four-manifold and the Hamiltonian action on it of the group of gauge transformations which are trivial on the boundary. The symplectic reduction becomes the moduli space of flat connections over the manifold. On the moduli space of flat connections we shall construct a hermitian line bundle with connection whose curvature is given by the symplectic form. This is the Chern-Simons prequantum line bundle. The group of gauge transformations on the boundary of the base manifold acts on the moduli space of flat connections by an infinitesimally symplectic way. This action is lifted to the prequantum line bundle by its abelian extension.

Abstract:
We give simple explicit formulas for deformation quantization of Poisson-Lie groups and of similar Poisson manifolds which can be represented as moduli spaces of flat connections on surfaces. The star products depend on a choice of Drinfe\v{l}d associator and are obtained by applying certain monoidal functors (fusion and reduction) to commutative algebras in Drinfe\v{l}d categories. From a geometric point of view this construction can be understood as a quantization of the quasi-Poisson structures on moduli spaces of flat connections.

Abstract:
Starting from a Heegaard splitting of a three-manifold, we use Lagrangian Floer homology to construct a three-manifold invariant, in the form of a relatively Z/8-graded abelian group. Our motivation is to have a well-defined symplectic side of the Atiyah-Floer Conjecture, for arbitrary three-manifolds. The symplectic manifold used in the construction is the extended moduli space of flat SU(2)-connections on the Heegaard surface. An open subset of this moduli space carries a symplectic form, and each of the two handlebodies in the decomposition gives rise to a Lagrangian inside the open set. In order to define their Floer homology, we compactify the open subset by symplectic cutting; the resulting manifold is only semipositive, but we show that one can still develop a version of Floer homology in this setting.

Abstract:
We consider analytic curves $\nabla^t$ of symplectic connections of Ricci type on the torus $T^{2n}$ with $\nabla^0$ the standard connection. We show, by a recursion argument, that if $\nabla^t$ is a formal curve of such connections then there exists a formal curve of symplectomorphisms $\psi_t$ such that $\psi_t\cdot\nabla^t$ is a formal curve of flat invariant symplectic connections and so $\nabla^t$ is flat for all $t$. Applying this result to the Taylor series of the analytic curve, it means that analytic curves of symplectic connections of Ricci type starting at $\nabla^0$ are also flat. The group $G$ of symplectomorphisms of the torus $(T^{2n},\omega)$ acts on the space $\E$ of symplectic connections which are of Ricci type. As a preliminary to studying the moduli space $\E/G$ we study the moduli of formal curves of connections under the action of formal curves of symplectomorphisms.

Abstract:
We prove that, for any n, there are simply-connected four-manifolds which admit n-tuples of symplectic forms whose first Chern classes have pairwise different divisibilities in integral cohomology. It follows that the moduli space of symplectic forms modulo diffeomorphisms on such a manifold has at least n connected components.