# Workshop Programme

## for period 2 - 5 April 2013

### Higher Structure 2013: Operads and Deformation Theory

2 - 5 April 2013

Timetable

 Tuesday 02 April 09:00-09:25 Registration 09:25-09:30 Welcome from INI Director, John Toland 09:30-10:30 Tillmann, U (University of Oxford) Homology stability for symmetric diffeomorphisms and their mapping classes Sem 1 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. 10:30-11:00 Morning Coffee 11:00-12:00 Francis, J (Northwestern University) Factorization homology of topological manifolds Sem 1 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. 12:15-13:15 Lunch at Wolfson Court 13:30-14:30 Berglund, A (Stockholm University) Rational homotopy theory of automorphisms of highly connected manifolds Sem 1 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. 14:30-15:00 Afternoon Tea 15:00-16:00 Roytenberg, D (Universiteit Utrecht) A Dold-Kan-type correspondence for superalgebras of differentiable functions and a "differential graded" approach to derived differential geometry Sem 1 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). 16:30-17:30 Tsygan, B (Northwestern University) Operations on Hochschild complexes, revisited Sem 1 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. 17:30-19:00 Wine Reception & Poster session
 Wednesday 03 April 09:30-10:30 Cattaneo, A (Universität Zürich) Classical and quantum Lagrangian field theories on manifolds with boundaries Sem 1 Classical and quantum field theories may be thought of as appropriate functors from (some version of) the cobordism category. At the quantum level this was proposed by Segal as an axiomatization. Incarnations of this exist for nonperturbative topological field theory (by Witten following Atiyah's version of the axioms for TFTs) as well as in one and two dimensional field theories. This talk (based on joint work with Mnev and Reshetikhin) will give an introduction to the classical version and to the Batalin-Vilkovisky version, which forms the starting point for the perturbative quantization. The possibility of including boundaries of boundaries (and so on) naturally yields to a Lurie-type description. Eventually, one might be able to reconstruct perturbative quantum theories on manifolds by gluing simple pieces together. Work in progress on this will be presented. 10:30-11:00 Morning Coffee 11:00-12:00 Severa, P (Université de Genève) Integration of differential graded manifolds Sem 1 I will explain why the Sullivan simplicial set given by a differential non-negatively graded manifold is actually a Kan simplicial manifold. The basic tool is an integral transformation that linearizes the corresponding PDE. By imposing a gauge condition, we can then locally find a finite-dimensional Kan submanifold, which can be seen as a local Lie n-groupoid integrating the dg manifold. These local n-groupoids are (non-uniquely) isomorphic on the overlaps. I will mention many open problems. Based on a joint work in progress with Michal Siran. 12:15-13:15 Lunch at Wolfson Court 13:30-14:30 Segal, G (University of Oxford) Bott periodicity in Algebra Sem 1 Topological K-theory was based on the Bott periodicity theorem, which made it possible to define K-groups in all degrees. Quillen's algebraic K-theory made use of a quite different principle. Nevertheless, Bott periodicity, seemingly so topological, has a variety of purely algebraic manifestations, which shed light on the sense in which an algebraically-defined space has a homotopy type. 14:30-15:00 Afternoon Tea 15:00-16:00 Hess, K (EPFL) The Boardman-Vogt tensor product of operadic bimodules Sem 1 (Joint work with Bill Dwyer.) The Boardman-Vogt tensor product of operads endows the category of operads with a symmetric monoidal structure that codifies interchanging algebraic structures. In this talk I will explain how to lift the Boardman-Vogt tensor product to the category of composition bimodules over operads. I will also sketch two geometric applications of the lifted B-V tensor product, to building models for spaces of long links and for configuration spaces in product manifolds. 16:30-17:30 Moerdijk, I (Radboud Universiteit Nijmegen) Two models for infinity-operads Sem 1 Following the foundational work of Joyal and Lurie, the theory of infinity-categories is now widely being used. There are two ways of extending the theory to "infinity-operads", one of them [CM] based on the theory of dendroidal sets, the other [HA] on the theory of cocartesian fibrations between infinity-categories, and ever since the appearance of the first versions of [HA] it was conjectured that the two theories are equivalent. In this lecture I will present an outline of a proof of this conjecture [HHM]. [CM] D.-C. Cisinski, I. Moerdijk, Dendroidal sets as models for homotopy operads, Journal of Topology 2011 [HA] J. Lurie, Higher Algebra, book available at http://www.math.harvard.edu/~lurie/ [HMM] G. Heuts, V. Hinich, I. Moerdijk, The equivalence of the dendroidal model and Lurie's model for infinity operads, in preparation. 19:30-22:00 Conference Dinner at Christ's College
 Thursday 04 April 09:30-10:30 Toën, B (Université de Montpellier 2) Symplectic and Poisson structures in the derived setting Sem 1 This is a report on an ongoing project joint with T. Pantev, M. Vaquié and G. Vezzosi. The purpose of this talk is to present the notions of n-shifted symplectic and n-shifted Poisson structures on derived algebraic stacks and to explain their relevance for the study of moduli spaces. I will start by the notion of n-shifted symplectic structures and present some existence results insuring that they exist in many examples. In a second part I will present the notion of n-shifted Poisson structures and explain its relation with the geometry of branes (maps from spheres to a fixed target). To finish I will explain what deformation quantization is in the n-shifted context. 10:30-11:00 Morning Coffee 11:00-12:00 Dotsenko, V (Trinity College Dublin) Givental action is gauge symmetry in a homotopy Lie algebra Sem 1 In this talk, I shall explain the mystery of hypercommutative algebras (=genus 0 cohomological field theories): it has been known for a while that the space of hypercommutative algebra structures on a given vector space possesses an unusually large group of symmetries. I shall explain how that appears naturally as a group of gauge symmetries of Maurer--Cartan elements in a certain homotopy Lie algebra. The existence of that homotopy algebra is still a result of several wonderful coincidences, but somehow I shall explain precisely what coincidences one should be hunting for in order for that sort of phenomena to occur. 12:15-13:15 Lunch at Wolfson Court 13:30-14:30 Manin, YI (Max-Planck-Institut fur Mathematik, Bonn) Computability and Zipf's Law: operadic perspective Sem 1 The classical model of computability is the theory of partial recursive functions. Church's thesis postulates the "universality" of this model, and a vast corpus of other approaches confirms this thesis. Partial recursive functions is the minimal subset of partial functions containing a list of elementary functions and stable wrt another list of basic operations. One part of my talk is dedicated to the operad generated by basic operations, and possibly larger algebras over this operad formalizing also oracle assisted computations. Another part will deal with applications of computability and complexity to the creation of a mathematical model of Zipf's law: empirical probability measure observable on a vast amount of data, starting with distribution of words in texts. 14:30-15:00 Afternoon Tea 15:00-16:00 Pridham, JP (University of Cambridge) Strong homotopy (bi)algebras, homotopy coherent diagrams and derived deformations Sem 1 Spaces of homotopy coherent diagrams or of strong homotopy (s.h.) algebras (for arbitrary monads) can be realised by right-deriving sets of diagrams or of algebras. This description involves a model category generalising Leinster's homotopy monoids. For any monad on a simplicial category, s.h. algebras thus form a Segal space. A monad on a category of deformations then yields a derived deformation functor. There are similar statements for bialgebras, giving derived deformations of schemes or of Hopf algebras. 16:30-17:30 Wahl, N (University of Copenhagen) Higher string topology Sem 1 We given an algebraic and geometric approach to string topology operation parametrized by Sullivan diagrams, producing in particular non-trivial higher degree operations associated to families of surfaces of any genus or any number of boundary components.
 Friday 05 April 09:30-10:30 Getzler, E (Northwestern University) The nerve of a differential graded algebra Sem 1 The nerve of a differential graded algebra is a quasicategory: in this talk, we explain what the quasi-iso-morphisms look like in this quasicategory. If the dg algebra A is a dg Banach algebra concentrated in dimensions i>-n , the nerve of A is a Lie n-stack, that is, a quasicategory enriched in Banach analytic spaces. We show that Kuranishi's approach yields a finite dimensional Lie n-stack parametrizing deformations of a complex of holomorphic vector bundles on a compact complex manifold of length n. This is joint work with Kai Behrend. 10:30-11:00 Morning Coffee 11:00-12:00 Willwacher, T (Harvard University) Deformations of the En-operad Sem 1 We show that the cohomology of the deformation complex of the En operad may be expressed through M. Kontsevich's graph cohomology. 12:15-13:15 Lunch at Wolfson Court 13:30-14:30 Giansiracusa, J (Swansea University) Moduli spaces of geometric structures on surfaces and modular operads Sem 1 It is well-known that the moduli space of Riemann surfaces with nonempty boundary is homotopy equivalent to the moduli space of metric ribbon graphs. We generalize this to a large class of geometric structures on surfaces, including unoriented, Spin, r-spin, and principal G-bundles, etc. For any class of geometric structures that can be described in terms of sections a suitable sheaf of spaces on a surface, we define a moduli space and show that it is homotopy equivalent to a moduli space of graphs with appropriate decorations at the vertices. The construction rests on the contractibility of the arc complex and can be interpreted in terms of derived modular envelopes of cyclic operads. 14:30-15:00 Afternoon Tea 15:00-16:00 Cohen, R (Stanford University) Gauge theory, loop groups, and string topology Sem 1 In this talk I will describe a joint project with John Jones, in which we relate the gauge theory of a principal bundle G --> P --> M to the string topology spectrum of the principal bundle. This spectrum has, as its homology the homology of the corresponding adjoint bundle. In this study G can be any topological group. In the universal case when P is contractible, the adjoint bundle is equivalent to the free loop space, LM, and this spectrum realizes the original Chas-Sullivan string topology homological structure. One of our main results is to identify the group of units the string topology spectrum, and to relate it to the gauge group of the original bundle. We import some of the basic ideas of gauge theory, such as the action of the gauge group on the space of connections, to the setting of principal fiberwise spectra over a manifold, and show how it allows us to do explicit calculations. We also show how, in the universal case, an action of a Lie group on a manifold yields a representation of the loop group on the string topology spectrum. We end by discussing a functorial perspective, which describes a sense in which the string topology spectrum of a principal bundle is the "linearization" of the gauge group.

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