Workshop on Generalised Lie algebras in Derived Geometry 29 May - 2 June 2023 @ Utrecht Speakers Ben Antieau (Northwestern) Greg Arone (Stockholm) Alexander Berglund (Stockholm) Joana Cirici (Barcelona) Benoit Fresse (Lille) Ezra Getzler (Northwestern) Jesper Grodal (Copenhagen) Jeremy Hahn (MIT) Geoffroy Horel (Paris) Nick Kuhn (University of Virginia) Thomas Nikolaus (Münster) Joost Nuiten (Toulouse) Jon Pridham (Edinburgh) Nick Rozenblyum (Chicago) Paolo Salvatore (Rome) Andrew Senger (Harvard) Yuqing Shi (Utrecht) Allen Yuan (Columbia) Organisers Lukas Brantner and Gijs Heuts. Contact: derivedutrecht2023@gmail.com. Lodging All conference participants will be housed at Hotel Ernst Sillem Hoeve. Arrival date: Sunday, the 28th of May. Travel To get to the venue there are several options: + A taxi from Schiphol airport takes about 40min (but can be somewhat costly), a taxi from Utrecht Central Station takes 20min. + Via public transport, the closest train station is Den Dolder. From there it is usually only a 5 minute bus ride, but unfortunately that bus will not run on May 28. We will shuttle back and forth between the station and the Ernst Sillem Hoeve. If you wish to make use of this, please let us know by email and telling us your approximate time of arrival, so that we can organise. Food Food will be provided at the venue, starting with Sunday dinner and ending with Friday lunch. There will be no further talks scheduled Friday afternoon. Schedule Monday Tuesday Wednesday Thursday Friday 09.00 Coffee 09.00 Coffee 09.00 Coffee 09.00 Coffee 09.00 Coffee 09.30 Arone 09.30 Nuiten 09.30 Berglund 09.30 Nikolaus 09.30 Getzler 10.30 Coffee 10.30 Coffee 10.30 Coffee 10.30 Coffee 10.30 Coffee 11.00 Rozenblyum 11.00 Shi 10.45 Cirici 11.00 Yuan 10.45 Hahn 12.00 Lunch 12.00 Lunch 11.45 Horel 12.00 Lunch 11.45 Kuhn 15.00 Fresse 15.00 Pridham 12.45 Lunch 15.00 Senger 12.45 Lunch 16.00 Coffee 16.00 Coffee 16.00 Coffee 16.15 Grodal 16.15 Salvatore 16.15 Antieau 17.15 Gong show Gong show speakers: Miguel Barata, Max Blans, Jiaqi Fu, Ezra Getzler, Sofía Marlesca Aparicio, Sven van Nigtevecht, Noé Sotto. Titles and abstracts Antieau, Integral models for spaces Generalizing and building on the work of Kriz, Ekedahl, Goerss, Lurie, Mandell, Mathew, Mondal, Quillen, Sullivan, Toën and Yuan, I will describe an integral cochain model for nilpotent spacees of finite type. A binomial ring is a lambda-ring in which all Adams operations act as the identity. A derived binomial ring is a derived Λ-ring equipped with simultaneous trivializations of the commuting Adams operations. For example, if X is a space, then ZX, the integral cochains on X, is naturally a derived binomial ring. In work-in-progress, I show that the induced contravariant functor from spaces to derived binomial rings is fully faithful when restricted to nilpotent spaces of finite type. Arone, Lie algebras in the category of orthogonal sequences. An orthogonal sequence (of spectra) is a functor from the category of Euclidean spaces and isomorphisms to the category of spectra. The category of orthogonal sequences is left tensored over the category of symmetric sequences equipped with a composition product. As a result, one can talk about left modules, or algebras, over an operad in the category of orthogonal sequence. There is a bar-cobar duality between algebras (in the category of orthogonal sequences) over the commutative operad and ove rthe spectral Lie operad. We use this to formulate a chain rule in orthogonal calculus, and to show that the derivatives of a functor in orthogonal calculus form an algebra over the spectral Lie operad. Some examples will be discussed. Berglund, Lie algebras and classifying spaces of fibrations. I will discuss some recent results on the rational homotopy theory of classifying spaces of fibrations that feature Lie algebras and the Lie operad in an essential way. Fix a simply connected finite CW-complex X and consider the classifying space, Baut(X), of fibrations with fiber X. The space Baut(X) is far from nilpotent in general, so its rational homotopy type cannot be modeled by a Lie algebra in chain complexes of Q-vector spaces as in Quillen's theory. In joint work with Tomas Zeman, we construct a new type of model that instead uses Lie algebras in chain complexes of algebraic representations of a certain reductive algebraic group. As a corollary, we express the rational cohomology of Baut(X) in terms of the cohomology of arithmetic groups and Lie algebra cohomology. If time admits, I will also discuss how Kontsevich's graph complex associated to the Lie operad makes an appearance in the cohomology of Baut(X) when X is a Poincaré duality complex, generalizing earlier work of Ib Madsen and myself. Cirici, Formality of hypercommutative algebras of Calabi-Yau manifolds. Any Batalin-Vilkovisky algebra with a homotopy trivialization of the BV-operator gives rise to a hypercommutative algebra structure at the cochain level which, in general, contains more homotopical information than the hypercommutative algebra introduced by Barannikov and Kontsevich on cohomology. In this talk, I will explain how to use the purity of mixed Hodge structures to show that the canonical hypercommutative algebra defined on any compact Calabi-Yau manifold is formal. This is joint work with Geoffroy Horel. Fresse, Mapping spaces for dg Hopf cooperads and homotopy automorphisms of the rationalization of En-operads. I will report on a joint work with Thomas Willwacher, whose first aim is to define a simplicial enrichment on the category of differential graded Hopf cooperads (the category of dg Hopf cooperads for short). This simplicial enrichment is compatible with model structures, and therefore, gives define a good model of mapping spaces in the category of dg Hopf cooperads. We use the definition of our simplicial enrichment to upgrade results of the literature about the homotopy automorphism spaces of dg Hopf cooperads by dealing with simplicial monoid structures. To be specific, the rational homotopy theory of operads implies that the homotopy automorphism spaces of dg Hopf cooperads can be regarded as models for the homotopy automorphism spaces of the rationalization of operads in topological spaces (or in simplicial sets). We precisely prove, using our simplicial enrichment, that the spaces of Maurer-Cartan forms on the Kontsevich graph complex Lie algebras are homotopy equivalent, in the category of simplicial monoids, to the homotopy automorphism spaces of the rationalization of the operads of little discs. Getzler, Homological perturbation theory for L∞ algebras and comparison between cubical and simplicial sets . We give a streamlined account of a theorem of Berglund and Loday-Valette that constructs explicit an quasi-isomorphism between a curved pronilpotent L∞ algebra and the canonical induced curved L∞ algebra structure on a deformation retract of the underlying complex. As an application, we prove that the simplicial and cubical approaches to integration of curved pronilpotent L∞ algebras yield isomorphic results. Grodal, The cohomology of loop groups and finite groups of Lie type via string topology. Classifying spaces of loop groups LG(C), for G a connected split reductive group scheme over Z, and finite groups of Lie type G(Fq) might a priori seem like very different objects. E.g., one is rationally trivial but the other is not (if G is non-trivial). Nevertheless it has calculationally been observed that their mod ell cohomology tend to agree as long as q is congruent to 1 mod ell. In joint work with Anssi Lahtinen, that combines string topology à la Chas–Sullivan with the theory of ell-compact groups, we provide a structural relationship between these two cohomologies, allowing us to establish a structured isomorphism in very many cases. I’ll discuss this isomorphism, where it comes from, and what extra structure it carries. Multiplications? Brackets? Operations? One could also ask if one could be viewed as a deformation of the other? Heine, A model for restricted L∞-algebras. By a result of Fresse restricted Lie algebras over a field are algebras for the monad underlying an adjunction between vector spaces and hopf algebras over that field, where the left adjoint takes the tensor algebra and the right adjoint takes the primitive elements. We extend the adjunction of tensor algebra/ primitive elements to any stable presentably symmetric monoidal infinity-category and discuss a model for restricted L∞-algebras via algebras for the monad underlying this adjunction. We relate restricted ∞-algebras to other sorts of Lie algebras, among which are spectral Lie algebras, partition Lie algebras and simplicial restricted Lie algebras. Hahn, Configuration spaces and Lie algebras. I will discuss some approaches to the Morava E-theory of loop spaces of spheres, including joint work with Brantner, Knudsen, Senger, and Zhang. Horel, Binomial rings and homotopy theory. Abstract: In a famous paper, Sullivan showed that the rational homotopy theory of finite type nilpotent spaces can be encoded in a fully faithful manner by mapping it to the homotopy category of commutative differential graded algebras over the rational numbers. For integral homotopy theory, a result of Mandell shows that it is faithfully captured by the integral cochains equipped with their E-infinity structure. This functor is however not full. I will explain a way of fixing this problem inspired by work of Toën, using cosimplicial binomial rings instead of E∞ differential graded algebras Kuhn, A canonical filtration of right modules over an operad, topological Andre-Quillen homology, and the Lie operad. More than 20 years ago I wrote a paper with some observations about models for the tensor product of based space and an augmented (or nonunitial) commutatative ring spectrum. In particular, I noted that these came with an interesting canonical filtration. Stabilizing, I noticed this gave a filtration of the then very new Topological Andre-Quillen homology, with subquotients involving spectra appearing in Goodwillie calculus. These were soon after shown by Michael Ching to be the spectra making up the Lie cooperad. A while later, but only recently published, Berhens and Rezk used what appeared to be my filtration of TAQ in their work on stable models for unstable vn periodic homotopy, but gave their own construction. In my talk I will try to put all of this in context. There is a canonical filtration on right modules over an operad, and, specialized to the operad Com, appropriate examples fed into appropriate bar constructions yields the filtrations I describe above. And yes, the Behrens-Rezk filtration agrees with mine: we are basically using two different explicit cofibrant replacements for the same module. Nikolaus, Algebraic K-Theory in geometric topology. While algebraic K-theory has many applications and uses in modern algebra and arithmetic, its origins actually lie in geometric topology through Whithead's work on simple homotopy theory. We will review this in modern language, to elucidate the nature of simple homotopy types. This will also lead to parametrized versions of results of West and Chapman through the use of Efimov-K-Theory. The key is a K-theoretic model of assembly maps which is of independent nature and should have many more applications in the furture. Nuiten, PD operads and partition Lie algebras. Partition Lie algebras have recently been introduced by Brantner and Mathew as certain homotopy theoretic refinements of dg-Lie algebras that control deformation problems in positive characteristic. In this talk, I will try to explain how partition Lie algebras (and other types of Lie algebras) can be understood as certain types of algebras over a PD operad, a refinement of the usual notion of an operad. A version of Koszul duality for such PD operads gives rise to Lie-algebraic descriptions of various types of formal problems appearing in derived algebraic geometry. Based on joint work with Lukas Brantner and Ricardo Campos. Pridham, The role of Lie algebroids in deformation quantisation of derived stacks. Poisson structures and their quantisations only satisfy etale functoriality. For Artin stacks or Lie groupoids, this can be resolved by working with a site of Lie algebroids. Rozenblyum Relative Lie algebras and algebroids. A natural generalization of the notion of a Lie algebra is that of a Lie algebroid. From the point of view of derived geometry, a Lie algebroid is a nilpotent extension of a global stack. I will explain how a theory of "global Lie algebras" allows to describe these in more algebraic terms. As an application, we obtain a very general result relating D-modules and equivariant ind-coherent sheaves on the formal derived loop space of a stack. This is based on joint work with Gaitsgory, and more recent work with Brav. Salvatore, A new approach to the Goodwillie-Sinha spectral sequence. We present a new combinatorial approach to the Goodwillie-Sinha spectral sequence for the homology of the space of long knots in R3 that makes it more accessible to computations. This is joint work with my student Andrea Marino. Senger, The mod (p,v1) K-theory of Z/pn . I will explain how to compute the mod (p,v1) K-theory of Z/pn. This is joint work with Jeremy Hahn and Ishan Levy. Shi, Higher enveloping algebras in monochromatic layers . The universal enveloping algebra functor assigns to a Lie algebra a unital associative algebra, characterised by the property that it sends free Lie algebras to free associative algebras. Ben Knudsen generalises this construction to a functor from the infinity category of spectral Lie algebras to the infinity category of augmented spectral En-algebras for any natural number n, known as the higher enveloping algebra functor. Recall that an E_1-algebra is an associative algebra up to coherent homotopies. The monochromatic layer of height h is the localisation of the infinity category of spectra, where vh-periodic equivalences are inverted, generalising the rational localisation of spectra. In my talk, I will first introduce the construction of higher enveloping algebras, using a different approach than that of Knudsen. We show that the infinity category of spectral Lie algebras at monochromatic height h embeds fully faithfully to a inverse limit of the infinity category of En algebras at height h. If time permits, I will also talk about an application of this to the costablisation of the unstable monochromatic layers. This is joint work with Gijs Heuts Yuan, Strict units of group rings. Just as an ordinary commutative ring has a group of units, one can associate to any commutative ring spectrum R a topological abelian group of "strict units". These strict units are a basic invariant of R, for instance controlling the types of roots one can adjoint to R. Despite this, there have not been many calculations of strict unit groups. I will survey some results in the area and report on work in progress, joint with Shachar Carmeli and Thomas Nikolaus, on calculating strict units of group rings. Acknowledgments The organisers thank the European Research Council Starting Grant "Chromatic homotopy theory of spaces" for funding. Conference photo
