November 12th to November 19th: Vadim Schechtman is visiting SISSA. 

We have a special week with talks from participants to the lecture series and from James Pascaleff and Vadim Schechtman. Below you find talks and abstracts.

Talks on Tuesday 14th:

1. 2:30 p.m.  Paolo Tomasini (SISSA) -- Equivariant elliptic cohomology and mapping stacks

Abstract: We propose a new construction of equivariant elliptic cohomology for algebraic varieties with actions of tori in terms of functions over a stack of "quasi-constant" maps from an elliptic curve to a quotient stack. Suitably manipulated, these functions recover the equivariant elliptic cohomology of the analytification of the variety. Based on joint work with Nicolò Sibilla.

2. 4 p.m. James Pascaleff (U. Illinois) -- Higher Fukaya categories and 3d HMS

Talks on Thursday 16th: 

3. 2 p.m. Emanuele Pavia (SISSA) -- Local systems of higher categories and higher Koszul duality

Abstract: One of the most celebrated instances of Koszul duality concerns the correspondence between the derived category of the algebra of cochains over a (sufficiently nice) topological space, and the derived category of the algebra of chains over its loop space. This can be interpreted as a vast generalization of the well-known Koszul duality between the symmetric algebra over a vector space and the exterior algebra over its dual. In this talk, I shall describe how the highly sophisticated formalism of homotopy theory and of enriched ∞-categories allows us to categorify this Koszul duality principle. First, I shall revisit the theory of local systems of categories over topological spaces, which naturally comes into picture in the framework of cobordism theory and of the geometric Langlands program. Then, I shall formulate a higher 𝔼n-Koszul duality principle between cochains over a space and chains over its loop space, which generalizes the classical one. Finally, I shall describe a wide class of topological spaces for which such 𝔼n-Koszul duality holds. This talk is based on joint works with J. Holstein and M. Porta, and with J. Pascaleff and N. Sibilla. 

4.  3:15 p.m. Vadim Schechtman (IM Toulouse) -- Chiral Gauss - Manin connection 

Abstract: We will explain how to construct the Gauss - Manin connection on the relative chiral de Rham complex. The construction uses (a chiral version of) the "calculus" formalism introduced  by Boris Tsygan. This is a joint work with Fyodor Malikov and Boris Tsygan.

5. 4:30 p.m. Zhengping Gui (ICTP) -- Trace map on chiral Weyl algebras 

Abstract: We construct a trace map on the chiral homology of chiral Weyl algebra for any smooth Riemann surface. Our trace map can be viewed as a chiral version of the deformed HKR quasi-isomorphism. This also provides a mathematical rigorous construction of correlation function for symplectic bosons in physics. We calculate some examples of trace maps with one insertion and find they are closely related to the variation of analytic torsion for holomorphic bundles on Riemann surfaces.