Women in Homotopy Theory and Algebraic Geometry

In string topology one studies the algebraic structures of the chains of the free loop space of a manifold by defining operations on them. Recent results show that these operations are parametrized by certain graph complexes that compute the homology of compatifications of the Moduli space of Riemann surfaces. Finding non-trivial homology classes of these compactifications is related to finding non-trivial string operations. However, the homology of these complexes is largely unknown. In this talk I will describe one of these complexes: the chain complex of Sullivan diagrams. I will describe two stabilization maps for Sullivan diagrams one with respect to genus and one with respect to punctures and describe how some components of this complex have homological stability with respect to these maps. I will also give some computational results for small genus and number of punctures.

The HKR theorem gives us an important equivalece between the Hochschild homology groups and the groups in the chain complex used to calculate deRham cohomology, for (rational) commutative differential graded algebras. Kantorovitz-McCarthy realized this as a statement about the layers of the Taylor tower of a certain functor. In joint work with Bauer, Johnson and McCarthy, we use the recently developed unbased calculus to promote this to a statement for ring spectra, conjectured originally by Waldhausen, and show it is equivalent to another proposed combinatorial model. In this talk, I will have time to say broadly the relationship between the deRham complex and functor calculus, and why the development of the unbased calculus was a necessary step to extend this further, as well as list some future directions which are more algebro-geometric.

As is well known, due to higher ramification, there are no Lefschetz theorems for the fundamental group in positive characteristic. However, there is a weak form for smooth l-adic sheaves (used by Deligne and Drinfeld for varieties over finite fields in their solution to Deligne's conjecture in Weil II), and for overconvergent isocrystals with Frobenius structure for varieties over a finite field (joint with Abe). If time permits, we shall also present Deligne's conjectures for smooth l-adic sheaves for varieties over an algebraically closed field.

The tropical vertex group of Kontsevich and Soibelman is generated by formal symplectomorphisms of the 2-dimensional algebraic torus. Based on the tropical vertex group, Gross, Pandharipande and Siebert introduced an interesting Gromov-Witten theory on weighted projective planes which admits a very special expansion in terms of tropical counts. I will describe a refinement or "q-deformation" of this expansion, motivated by wall-crossing ideas, using Block-Göttsche invariants. This leads naturally to the definition of a class of putative q-deformed curve counts. We prove that this coincides with another natural q-deformation, provided by a result of Reineke and Weist in the context of quiver representations, when the latter is well defined. Joint work with Jacopo Stoppa.

In this talk, we focus our attention on Differential Graded Lie Algebras and their role in deformation theory. In particular, we investigate the infinitesimal deformations of pairs (X,D), where D is a smooth divisor in a smooth projective variety X. Using this technique we are able to prove the unobstructedness of infinitesimal deformations in some cases.

This is joint work with K.Hess, E.Riehl and B.Shipley. In this talk I will introduce a class of accessible model structures on locally presentable categories, which includes, but is more general than, combinatorial model structures. An accessible model structure is particularly good if one wants to left or right induce it along an adjunction - by a theorem of Burke and Garner the induced weak factorization systems always exist, so one needs to check only a compatibility condition. If it holds then the resulting model structure is again accessible. One example of an accessible model structure is the Hurewicz model structure on Ch_R (the category of unbounded chain complexes over a ring R), which can be induced to many categories of interest, like algebras, coalgebras, comodules, comodule algebras, coring comodules and bialgebras. I will discuss ideas behind some of the proofs for induced model structures and give examples.

In the eighties Mukai proved that the singularities of the moduli space of sheaves on a K3 surface are contained in the locus of strictly semistable sheaves, that is not empty just if the polarization is non generic or if the Mukai vector is non primitive. In the first case, Kaledin, Lehn and Sorger conjectured that the dg-algebra that controls deformations of sheaves on a K3 is formal. This would give a complete description of the singularities of the moduli space. The conjecture was proved in some cases by Kaledin-Lehn and Zhang. In a work in progress with Manfred Lehn we aim to complete the proof and to extend our ad hoc construction to a general proof.

There is a general theory of K-theory for infinity categories introduced by Blumberg, Gepner, Tabada and Barwick, which is related with the K-theory defined by Waldhausen in 1985. Morel and Voevodsky proved the representation of the K-theory in the classical algebraic geometry. In this talk, for the K-theory of a certain infinity category, we give the object in the infinity category which represents the K-theory.

TMF and its variants can be viewed as homology theories arising from elliptic curves and capturing structural information about stable homotopy groups of spheres. In an ongoing joint work with Lennart Meier, we translate the question about the structure of TMF_0(7) into a representation theory problem related to modular forms of level 7. Through computations on this side, we study splittings of TMF_0(7) into easier pieces.

In stable homotopy theory Galois extensions of commutative ring spectra are rare. In joint work with Dundas and Lindenstrauss we investigate relative topological Hochschild homology as a detecting device for ramification of maps between commutative ring spectra. Typical cases arise as connective covers of Galois extensions which behave with respect to some aspects like extensions of rings of integers in number fields. I will describe examples like maps between versions of topological K-theory and topological modular forms.

Let G be a finite group. Elmendorf showed that the homotopy category of G-topological spaces is equivalent to the homotopy category of contravariant functors from category of finite G-sets to topological spaces. We would like to explain the analog for equivariant infinite loop spaces using the notion of Γ-spaces. In this talk, we will discuss the notion of Mackey functors in Γ-spaces and relate it to special equivariant Γ-spaces.

Derived algebraic geometry and derived symplectic geometry in the sense of Pantev-Toen-Vaquié -Vezzosi allows for a reinterpretation/analog of the classical AKSZ construction for certain sigma-models. After recalling this procedure I will explain how it leads to a fully extended oriented TFT in the sense of Lurie with values in a higher category whose objects are n-shifted symplectic derived stacks and (higher) morphisms are (higher) Lagrangian correspondences. It is given by taking mapping stacks with a fixed target building and describes "semi-classical TFTs". This is joint work in progress with Damien Calaque and Rune Haugseng.

Operads have been explored in the 1970s in connection with n-fold loop spaces and infinite loop spaces, the latter being of particular interest as they give rise to generalised cohomology theories. In the 1990s, influenced by ideas from mathematical physics and field theory in particular, operads experienced a renaissance.
We will prove that operads satisfying a certain homology stability condition detect infinite loop spaces. We present here a general framework which applies in many cases including the operad of Riemann surfaces that plays a central role in Segal's axiomatic approach to conformal field theory.
This is a joint work with M. Basterra, I. Bobkova, K. Ponto, and S. Yeakel which started as a WIT project.

The classical Hodge decomposition for Hochschild homology allows to express Hochschild homology of a commutative algebra in terms of higher order André-Quillen homology in characteristic zero. Teimuraz Pirashvili showed that a similar decomposition exists for Hochschild homology of higher order n, and that the terms occuring in these decompositions only depend on the parity of n, hence allowing calculations of higher order Hochschild homology via knowledge of the Hodge summands for n=1 or n=2. A generalization of higher order Hochschild homology to algebras which are commutative only up to coherent higher homotopies is E_n-homology. In this talk, I will recall the construction of the spectral sequence yielding the classical Hodge decomposition. I will introduce higher order Hochschild homology and discuss the Hodge decomposition in this case. Finally, we will see how to obtain a Hodge decomposition spectral sequence for E_n-homology of E_infinity algebras.