Deformation Spaces

InrecentyearstheHochschildandcycliccomplexandtheiralgebraicstructures have been intensively studied from di?erent perspectives. Some of these algebraic gadgetshavebeenaroundsincetheearlyworkofGerstenhaberonthedeformations of associative algebras, while others, such as cyclic homology, were introduced by Connes in the early development of noncommutative geometry. More recent dev- opments from this perspective include the theory of Hopf cyclic (co)-homology of Hopf algebra. Various algebraic structures of Hochschild and Cyclic (co)-homology, such as Batalin-Vilkoviskyand Gerstenhaber algebras, received a topologicalreincarnation by the works of Chas and Sullivan and other authors on free loop space. These compelling ideas, such as an action of the moduli space of surfaces possibly with various compacti?cations, have been considered in several di?erent settings. The algebraic analogue of these constructions on Hochschild and cyclic complexes (of Frobenius algebras) are usually known under the name of Deligne conjecture. This theory develops parallel to symplectic ?eld theory and Gromov-Witten invariants. As an algebraic theory, this corresponds to a deformation problem over PROPs or properads as opposed to operads, which naturally include genus, or in physics terminology, the correct h-bar terms. The editors had organized two workshops in July 2007 and August 2008 at the Max-Planck-Institut fur ] Mathematik in Bonn with a generous support from the Hausdor? Center. Participants of these workshops were mainly algebraic topo- gist, noncommutative geometers, and specialists in deformations theory. The aim oftheseworkshopswastobringtogetherthemathematicianswhoworkondefor- tions of algebraic and geometric structures and Hochschild and cyclic complexes. As it is clear from the volume, the subject of these activities was in?uenced by physics and the new perspectives that it o?ers."