Abstracts

• Domenico Fiorenza
  BV algebras, they do not really exist
  
  Given a differential graded Gerstenhaber algebra (\mathcal{G}, d, \cdot ,\{,\}) and a bigraded homotopy Cartan calculus for it, the choice of a Poincaré duality element enhances the Gerstenhaber algebra structure on H(\mathcal{G}) to a BV algebra structure. Every BV algebra can be realized, essentially in a tautological way, as an instance of homotopy Cartan calculus with Poincaré duality. Possibly more interestingly, homotopy Cartan calculus has a transparent origin in the homotopy theory for differential graded Lie algebras. Also, the dependence of the BV-Laplacian on the choice of the Poincaré duality element provides a justification for the typically observed phenomenon that the BV bracket is "more canonical" than the BV-Laplacian. Classical examples of BV algebras from homotopy Cartan calculus with Poincaré duality include the divergence of multivector fields on Riemannian manifolds and the Ginzburg-Menichi BV algebra structure on Hochschild cohomology. Possibly less known is the fact that Ševera's derivation of the BV algebra structure on the Gerstenhaber algebra of functions on an odd symplectic manifold is an instance of homotopy Cartan calculus. No surprise, Poincaré duality elements in this setting are precisely compatible half-densities. Based on joint works with Eugenia Boffo and Niels Kowalzig.
• Connor Malin
  Formal geometric Lie algebras: a homotopical perspective on complete Lie algebras
  
  Mandell's Theorem asserts that the category of finite-type nilpotent and p-complete spaces fully faithfully embeds into the category of E_\infty-algebras over an algebraically closed field of characteristic p. There are major obstructions to proving Mandell's theorem using L_\infty-algebras since Koszul duality is very degenerate when restricted to the E_\infty-algebras associated to spaces in characteristic not equal to 0. Roca i Lucio has given an interesting point-set solution to this problem by transferring the model structure on E_\infty-algebras to a certain category of complete L_\infty-algebras. In this talk, I will give a homotopical solution by completing the \infty-category of spectral Lie algebras to a certain category of "Formal Geometric Lie algebras". We will give a construction of a space's "tangent formal geometric Lie algebra" over the sphere spectrum and demonstrate that it agrees with a completion of the cotangent complex of the space's spherical chains.
• Tommaso Rossi
  Koszul duality: old and new results
  
  Koszul duality can be seen as a bridge connecting two seemingly unrelated objects, and has various manifestations in algebra and topology. An important example of this is the duality between commutative and Lie algebras, which is crucial to connect Sullivan's and Quillen's approach to rational homotopy theory. In this talk we will review the basics of the theory and its main applications. Then we will focus on the duality between the hypercommutative and gravity operad, two operads related to the moduli spaces of genus zero curves. At the homological level, this duality was proved by Getzler. At the chain level, this is part of a joint work with Paolo Salvatore.