# Previous Activities

**1 Dec 2015, 13:15-?, Room 16 in house 5, SU. **Emil Sköldberg (NUI Galway): Some monomial ideals with multiplicative resolutions. I will describe some classes of monomial ideals I for which the minimal resolution of k[x1,…,xn]/I has a structure of a DGA. The results are obtained by using discrete Morse theory.

**24 Nov 2015, 13:15-15:00, Room 3418, KTH. **Benoit Fresse (Université Lille): Rational homotopy and intrinsic formality of E_n-operads. The theory of E_n-operads has considerably developed since a decade. Let us mention, among other applications, the second generation of proofs of the Kontsevich formality theorem, based on the formality of E_2-operads, which has hinted the existence of an action of the Grothendieck-Teichmüller group on moduli spaces of deformation-quantization of Poisson algebras, and the description of the Goodwillie-Weiss approximations of embedding spaces in terms of functions spaces on structures associated to E_n-operads.

The main purpose of this talk is to explain an intrinsic formality statement which gives a characterization of E_n-operads in terms of their cohomology. Recall that the homology of an E_n-operad is identified with the operad governing graded Poisson algebra structures of degree n-1. We can apply the Sullivan realization functor to the dual object of this operad of Poisson algebras of degree n-1 in order to retrieve an operad in topological spaces.

The intrinsic formality theorem reads as follows: "If an operad in topological spaces has the same rational homology as an E_n-operad, for some n>2, and is additionally equipped with an involutive isomorphism that mimes the action of a hyperplane reflection in the case n|4, then this operad is rationally weakly-equivalent to the operad in topological spaces which we canonically associate to the operad of Poisson algebras of degree n-1."

The proof relies on methods of obstruction theory and on the interpretation of obstruction cycles in terms of the homology of certain graph-complexes. To conclude the talk, I will explain that these graph-complexes also determine the rational homotopy type of mapping spaces between E_n-operads.

**10 Nov 2015, 13:15-15:00, Room 3418, KTH. **Constanze Roitzheim (University of Kent): Homological Localisation of Model Categories. Bousfield localisation with respect to generalised homology theories of either spaces or spectra has proved to be a powerful tool in topology. We present a feasible version of homological localisation for general model categories and show some examples and applications.

**3 Nov 2015, 13:15-15:00, Room 16 in house 5, SU. **Kathryn Hess (EPFL): From precalculus to calculus in algebraic topology. Joint work in progress with Brenda Johnson. The many theories of "calculus" introduced in algebraic topology over the past couple of decades--e.g., Goodwillie's calculus of homotopy functors, the Goodwillie-Weiss manifold calculus, the orthogonal calculus, and the Johnson-McCarthy cotriple calculus--all have a similar flavor, though the objects studied and exact methods applied are not the same. We have constructed a common, relatively simple category-theoretic framework, which we call precalculus, into which all of the above-mentioned examples fit and which naturally leads us to define new flavors of calculus as well.

In this talk I will define the theory of precalculi, then explain how to derive a full-blown calculus from a precalculus. I will also describe

the precalculi underlying each of the standard calculi, then indicate promising directions in which to look for new and useful calculi arising from precalculi.

**27 Oct 2015, 13:15-14:15, Room 3418, KTH. **Eric Finster (École Polytechnique): The Generalized Blakers-Massey Theorem. I will describe a generalization of the classical Blakers-Masssey theorem to the setting of what I call a factorization hierarchy in an arbitrary infinity-topos E. This consists of an N-indexed family of factorization systems on E satisfying some compatibility conditions and finite generation hypotheses. Applied to the factorization systems consisting of n-truncated and n-connected maps, we recover the classical Blakers-Massey theorem. Moreover, we sketch a proof of the fact that the factorzation systems of n-excisive and n-reduced maps in the Goodwillie calculus also satisfy these axioms, proving a conjecture of Goodwillie.

**20 Oct 2015, 10:15-12:00, Room 16 in house 5, SU. **Thomas Kragh (Uppsala): will discus several of the fundamental concepts and theorems in symplectic geometry/topology, and their relations to algebraic topology and algebraic geometry.

**13 Oct 2015, 13:15-14:15, Room 3418, KTH. **David Sprehn (Copenhagen): Cohomology of finite general linear groups. I will introduce the problem of computing the mod-p cohomology of GLn(k) for k the finite field of order pr, then describe how to construct a new class in (lowest possible) degree r(2p-3), and show it’s nonzero (when n <= p) by restricting to a subgroup of commuting regular unipotent matrices. I’ll first explain how the number r(2p-3) comes out of the invariant theory of finite fields. Lastly, I’ll describe what’s necessary to generalize the result to finite groups of Lie type.

**6 Oct 2015, 11:00-12:00, Room 16 in house 5, SU. **Kai Cieliebak (University of Augsburg): This talk reports on joint work with K.Fukaya, J.Latschev, and E.Volkov (in progress). Its aim is to describe the common algebraic structure arising in string topology, symplectic field theory, and higher genus Lagrangian Floer theory. This structure is an infinity version of an involutive Lie bialgebra. After a brief discussion of the homotopy theory of these structures, I will present a construction associating such a structure to every DGA using Feynman type sums over ribbon graphs. When applied to the de Rham algebra of a closed simply connected manifold this construction yields a chain-level version of string topology, though the precise relation is still not understood.

**29 Sept 2015, 13:15-15:00, Room 3418, KTH. **Stephanie Ziegenhagen: A spectral sequence for calculating E_n-homology. The talk is based on joint work with Birgit Richter. I will introduce a spectral sequence that calculates E_n-homology, discuss examples and show how the spectral sequence gives a new interpretation of the Hodge decomposition.

**22 Sept 2015, 13:15-15:00, Room 16 in house 5, SU. **Greg Arone will entertain us.