Abstracts

Manuel Amann: Non-negatively curved vector bundles over cohomogeneity one manifolds

Generalizing homogeneous spaces, it was proved by Grove–Ziller that cohomogeneity one manifolds under certain restrictions provide an important class of spaces which admit metrics of non-negative sectional curvature. On these manifolds we identify conditions under which vector bundles over them (up to suitable stabilizations) admit metrics of non-negative sectional curvature as well—thus providing a certain converse to the soul theorem. We achieve this by relating the bundles to equivariant ones up to stabilization. Beside constructions of bundle metrics, we essentially draw on K-theory computations to obtain the result. Moreover, we use the connection between (rational) K-theory and cohomology in order to link equivariant K-theory to equivariant (singular) cohomology—investigating the latter via rationalhomotopy theory. These methods are also applied in order to answer related open questions concerning the (equivariant) K-theory of homogeneous spaces. This talk reports on joint work with David Gonzalez-Alvaro and Marcus Zibrowius.

Daniele Angella: Formalities for holomorphic manifolds

For compact Kähler manifolds, formality is a fundamental property with strong topological consequences. We try to understand notions of formality for complex non-Kähler manifolds, both in the holomorphic and in the Hermitian settings. The talk is based on collaborations with: Tommaso Sferruzza, Nicoletta Tardini, Adriano Tomassini.

Giovanni Bazzoni: Homotopy Invariants and almost non-negative curvature

Almost non-negative sectional curvature (ANSC) is a curvature condition on a Riemannian manifold, which encompasses both the almost flat and the non-negatively curved case. It was shown in a remarkable paper by Kapovitch, Petrunin and Tuschmann that, modulo some technical details, a compact ANSC manifold is a fiber bundle over a nilmanifold, and that the fiber satisfies a curvature condition only slightly more general than ANSC. In this talk, based on joint work with G. Lupton and J. Oprea, we will discuss such manifolds from the point of view of Rational Homotopy Theory, presenting various invariants of bundles of such type, and proving a (rational) Bochner-type theorem.

Joana Cirici: Interactions of rational homotopy theory and geometry

In this overview talk, intended for non-experts, I will review the main ideas behind minimal models and formality, two notions that are central in rational homotopy theory. I will then explain various results, mostly related to formality, that arise from the interaction of homotopy theory and geometric structures (such as Kähler manifolds, complex manifolds and algebraic varieties).

José Manuel Moreno-Fernández: Iterated suspensions are coalgebras over the little disks operad

In this talk, I will explain how n-fold suspensions are naturally coalgebras over the little n-disks operad. This is a partial Eckmann-Hilton dual to a celebrated theorem of Peter May on n-fold loop spaces. This coalgebra structure allows for defining operations in the homotopy groups of n-fold suspensions, in a dual form to how operations arise in the (co)homology of iterated loop spaces. I will also describe the (rational) homotopy Browder cooperation, an obstruction for extending an n-fold suspension to an (n+1)-fold suspension, and comment on some results on equivariant (discrete) rational homotopy theory that were essential for proving our results. (Joint work with Felix Wierstra)

Marcel Rubió: Structure theorems for cohomology jump loci via L-infinity pairs

I will first present a short introduction to deformation theory, covering Deligne's principle and its implications by Goldman-Millson. I will then define cohomology jump loci and discuss the classical structure theorem by Simpson. The main goal of the talk will be to extend such theorems to singular varieties using L-infinity tools. This is joint work with Nero Budur.

Felix Wierstra: Models for non-simply-connected homotopy theory

In algebraic topology one often tries to model spaces by algebraic objects. This has been done in many interesting cases, but there are almost always strong assumptions on the fundamental group of the spaces, it can be nilpotent at best.

Recently Manuel Rivera and Mahmoud Zeinalian have shown that it is possible to extract the fundamental group of a space from its singular chains. This remarkable discovery makes it potentially possible to model non-simply-connected spaces. In this talk I will describe some work in progress towards modelling non-simply-connected spaces. This is joint work with Manuel Rivera and Mahmoud Zeinalian.

Masoumeh Zarei: Equivariant cohomology of cohomogeneity one Alexandrov spaces

Equivariant cohomology of a G-space X is defined as the cohomology of the Borel construction X_G:=X ×_G EG. The action of G on X is called Cohen–Macaulay if the Krull dimension of the equivariant cohomology of X is equal to its depth as a module over the cohomology of the classifying space. In this talk, I give a characterization of those Alexandrov spaces admittinga cohomogeneity one action of a compact connected Lie group G for which the action is Cohen–Macaulay. This generalizes a similar result for manifolds to the singular setting of Alexandrov spaces where, in contrast to the manifold case, we find several actions which are not Cohen–Macaulay. This is joint work with Manuel Amann.