# Timetable (GDOW02)

## Higher Structure 2013: Operads and Deformation Theory

Tuesday 2nd April 2013 to Friday 5th April 2013

 09:00 to 09:25 Registration 09:25 to 09:30 Welcome from INI Director, John Toland 09:30 to 10:30 U Tillmann (University of Oxford)Homology stability for symmetric diffeomorphisms and their mapping classes A central object of study in geometry and topology are diffeomorphism groups of compact manifolds. Nevertheless, these remain generally poorly understood objects. For surfaces homology stability has played a crucial role in advancing our understanding (Madsen-Weiss). We will present some homology stability results for 'symmetric' diffeomorphisms for manifolds of arbitrary dimension and put them into context. INI 1 10:30 to 11:00 Morning Coffee 11:00 to 12:00 J Francis (Northwestern University)Factorization homology of topological manifolds I'll describe factorization homology and some classes of computations therein which relate Koszul duality, Lie algebra homology, and configuration space models of mapping spaces. Parts of this work are joint with David Ayala, Kevin Costello, and Hiro Lee Tanaka. INI 1 12:15 to 13:15 Lunch at Wolfson Court 13:30 to 14:30 A Berglund (Stockholm University)Rational homotopy theory of automorphisms of highly connected manifolds I will talk about joint work with Ib Madsen on the rational homotopy type of classifying spaces of various kinds of automorphisms of highly connected manifolds. The cohomology of the classifying space of the automorphisms of a g-fold connected sum of S^d x S^d stabilizes degreewise as g tends to infinity. For diffeomorphisms, the stable cohomology has been calculated by Galatius and Randal-Williams. I will discuss recent results on the stable cohomology for homotopy equivalences and for block diffeomorphisms. Curiously, the calculation in these cases involves certain Lie algebras of symplectic derivations that have appeared before in Kontsevich's work on the cohomology of outer automorphisms of free groups. INI 1 14:30 to 15:00 Afternoon Tea 15:00 to 16:00 D Roytenberg (Universiteit Utrecht)A Dold-Kan-type correspondence for superalgebras of differentiable functions and a "differential graded" approach to derived differential geometry The commutative algebra appropriate for differential geometry is provided by the algebraic theory of $C^\infty$-algebras -- an enhancement of the theory of commutative algebras which admits all $C^\infty$ functions of $n$ variables (rather than just the polynomials) as its $n$-ary operations. Derived differential geometry requires a homotopy version of these algebras for its underlying commutative algebra. We present a model for the latter based on the notion of a "differential graded structure" on a superalgebra of differentiable functions, understood -- following Severa -- as a (co)action of the monoid of endomorphisms of the odd line. This view of a differential graded structure enables us to construct, in a conceptually transparent way, a Dold-Kan-type correspondence relating our approach with models based on simplicial $C^\infty$-algebras, generalizing a classical result of Quillen for commutative and Lie algebras. It may also shed new light on Dold-Kan-type co rrespondences in other contexts (e.g. operads and algebras over them). A similar differential graded approach exists for every geometry whose ground ring contains the rationals, such as real analytic or holomorphic. This talk is partly based on joint work with David Carchedi (arXiv:1211.6134 and arXiv:1212.3745). INI 1 16:30 to 17:30 B Tsygan (Northwestern University)Operations on Hochschild complexes, revisited Algebraic structures on Hochschild cochain and chain complexes of algebras are long known to be of considerable interest. On the one hand, some of them arise as standard operations in homological algebra. On the other hand, for the ring of functions on a manifold, Hochschild homology, resp. cohomology, is isomorphic to the space of differential forms, resp. multivector fields, and many algebraic structures from classical calculus can be defined (in a highly nontrivial way) in a more general setting of an arbitrary algebra and its Hochschild complexes. Despite an extensive amount of work on the subject, it appears that large parts of the algebraic structure on the Hochschild complexes of algebras and their tensor products are still unknown. I will review the known results (Kontsevich-Soibelman, Dolgushev, Tamarkin and myself, Keller) and then outline some provisional results and conjectures about new operations. The talk will not assume any prior knowledge beyond the basic facts about complexes and (co)homology. INI 1 17:30 to 19:00 Wine Reception & Poster session