Abstracts for Higher Geometric Structures along the Lower Rhine XI, March 8-9, 2018

Homotopy coherent commutative multiplications on chain complexes of modules over a commutative ring can be encoded by the action of an $E$-infinity operad. Alternatively, one can model such $E$-infinity differential graded algebras by commutative $I$-dgas, which are strictly commutative objects in diagrams of chain complexes indexed by the category of finite sets and injections $I$. In this talk $I$ will explain how the cochain algebra of a space arises as a commutative $I$-dga in a natural way.

The abstract structure of the Batalin-Vilkovisky formalism is nicely captured in operadic language, so that 1-shifted Poisson algebras encode the algebra of classical theories and BD algebras encode the algebra of quantum theories. We explain this language and then discuss what it means for a Lie algebra (or higher versions) to act on such algebras. Our primary aim is then to extract consequences at the level of factorization algebras for field theories.

Anderson duality sends a spectrum to a spectrum with the "dual homotopy groups". This is a survey talk about recent computations and applications of Anderson duals.