Abstract:
Let G be a Lie group acting by diffeomorphisms on a manifold M and consider the image of T[1]G and T[1]M, of G and M respectively, in the category of differential graded manifolds. We show that the obstruction to lift the action of T[1]G on T[1]M to an action on a R[n]-bundle over T[1]M is measured by the G equivariant cohomology of M. We explicitly calculate the differential graded Lie algebra of the symmetries of the R[n]-bundle over T[1]M and we use this differential graded Lie algebra to understand which actions are hamiltonian. We show how split Exact Courant algebroids could be obtained as the derived Leibniz algebra of the symmetries of R[2]-bundles over T[1]M, and we use this construction to propose that the infinitesimal symmetries of a split Exact Courant algebroid should be encoded in the differential graded Lie algebra of symmetries of a R[2]-bundle over T[1]M. With this setup at hand, we propose a definition for an action of a Lie group on an Exact Courant algebroid and we propose conditions for the action to be hamiltonian.

Abstract:
A pointed fusion category is a rigid tensor category with finitely many isomorphism classes of simple objects which moreover are invertible. Two tensor categories $C$ and $D$ are weakly Morita equivalent if there exists an indecomposable right module category $M$ over $C$ such that $Fun_C(M,M)$ and $D$ are tensor equivalent. We use the Lyndon-Hochschild-Serre spectral sequence associated to abelian group extensions to give necessary and sufficient conditions in terms of cohomology classes for two pointed fusion categories to be weakly Morita equivalent. This result may permit to classify the equivalence classes of pointed fusion categories of any given global dimension.

Abstract:
Chen and Ruan's orbifold cohomology of the symmetric product of a complex manifold is calculated. An isomorphism of rings (up to a change of signs) $H_{orb}^*(X^n/S_n;\complex) \cong H^*(X^{[n]};\complex)$ between the orbifold cohomology of the symmetric product of a smooth projective surface with trivial canonical class $X$ and the cohomology of its Hilbert scheme $X^{[n]}$ is obtained, yielding a positive answer to a conjecture of Ruan.

Abstract:
In this paper we construct an explicit geometric model for the group of gerbes over an orbifold $X$. We show how from its curvature we can obtain its characteristic class in $H^3(X)$ via Chern-Weil theory. For an arbitrary gerbe $\LL$, a twisting $\Korb{\LL}(X)$ of the orbifold $K$-theory of $X$ is constructed, and shown to generalize previous twisting by Rosenberg, Witten, Atiyah-Segal and Bowknegt et. al. in the smooth case and by Adem-Ruan for discrete torsion on an orbifold.

Abstract:
The purpose of this paper is to introduce the notion of loop groupoid associated to a groupoid. After studying the general properties of the loop groupoid, we show how this notion provides a very natural geometric interpretation for the twisted sectors of an orbifold, and for the inner local systems introduced by Ruan by means of a natural generalization of the concept holonomy of a gerbe.

Abstract:
In this paper we introduce the Cheeger-Simons cohomology of a global quotient orbifold. We prove that the Cheeger-Simons cohomology of the orbifold is isomorphic to its Beilinson-Deligne cohomology. Furthermore we construct a string connection (\`{a} la Segal) from a global gerbe with connection over the loop orbifold, refining the corresponding differential character.

Abstract:
We define the category of manifolds with extended tangent bundles, we study their symmetries and we consider the analogue of equivariant cohomology for actions of Lie groups in this category. We show that when the action preserves the splitting of the extended tangent bundle, our definition of extended equivariant cohomology agrees with the twisted equivariant de Rham model of Cartan, and for this case we show that there is localization at the fixed point set, \`a la Atiyah-Bott.

Abstract:
In this expository paper written for physicists and geometers we introduce the notions of TQFT and of orbifold. Then we survey the construction of TQFT's originating from orbifolds such as Chen-Ruan theory and Orbifold String Topology.

Abstract:
In this paper we present a model to calculate the stringy product on twisted orbifold K-theory of Adem-Ruan-Zhang for abelian complex orbifolds. In the first part we consider the non-twisted case on an orbifold presented as the quotient of a manifold acted by a compact abelian Lie group. We give an explicit description of the obstruction bundle, we explain the relation with the product defined by Jarvis-Kaufmann-Kimura and, via a Chern character map, with the Chen-Ruan cohomology, and we explicitely calculate the stringy product for a weighted projective orbifold. In the second part we consider orbifolds presented as the quotient of a manifold acted by a finite abelian group and twistings coming from the group cohomology. We show a decomposition formula for twisted orbifold K-theory that is suited to calculate the stringy product and we use this formula to calculate two examples when the group is $(\integer/2)^3$.