topology@ub

16 February 2022, 10:00, IMUB

Bigradings and complex manifolds
Abstract: The vector space of complex-valued differential forms on a complex manifold comes equipped with a bigrading, two anticommuting differentials and a multiplication for which these are derivations. Abstracting this structure, one arrives at the notions of  'double complex' (without the multiplication) and 'commutative bigraded bidifferential algebra', short: cbba.
I will survey some recent work on these structures, always with applications to complex geometry in mind. In particular, I will present a strong notion of quasi isomorphism and how it allows to define a 'holomorphic' homotopy theory for compact complex manifolds. Throughout, I will also try to present questions I consider interesting, some of which have an answer and some of which have not (yet).

16 February 2022, 11:30, IMUB

Formality and non-zero degree maps
Abstract: Given the existence of a non-zero degree map between closed orientable manifolds, under which conditions do we have that the target is formal if the domain is formal (in the rational homotopy theoretic sense)? I will discuss some scenarios in which this implication holds, both in the topological and holomorphic settings, where in the latter Serre duality plays the role of Poincaré duality and there are several notions of formality.

13 December 2021, 12:00, IMUB

The diagonal of the operahedra
Abstract: The set-theoretic diagonal of a polytope has the crippling defect of not being cellular: its image is not a union of cells. Our goal here is to develop a general theory, based on the method introduced by N. Masuda, H. Thomas, A. Tonks and B. Vallette, in order to understand and manipulate the cellular approximations of the diagonal of any polytope. This theory will allow us to tackle the problem of the cellular approximation of the diagonal of the operahedra, a family of polytopes ranging from the associahedra to the permutohedra, and which encodes homotopy operads. In this way, we obtain an explicit formula for the tensor product of two such operads, with interesting combinatorial properties.

13 December 2021, 15:00, IMUB

Vopěnka's principle in ∞-categories
Abstract: Vopěnka's principle has arisen as a model theoretical statement, provably independent of ZFC set theory. However, there are a number of categorical ways of formulating it, preventing the existence of proper classes of objects with some conditions in presentable categories, and these are what our attention will be focused on. In particular, we will look at analogous statements in the context of ∞-categories and we will ask how these new statements interact with the older ones. Moreover, some of the consequences of Vopěnka's principle on classes of subcategories of presentable categories are investigated and to some extent generalized to ∞-categories. A parallel discussion is undertaken about the similar but weaker statement known as weak Vopěnka's principle.