Geometry and Analysis Seminar

A 2-connected, rational homotopy 7-sphere is classified up to diffeomorphism by three invariants: its (finite) 4th cohomology group, its q-invariant and its Eells-Kuiper invariant.  The q-invariant is a quadratic refinement of the linking form and determines the homeomorphism type, while the Eells-Kuiper invariant then pins down the diffeomorphism type.  In this talk, I will discuss the diffeomorphism classification of a certain family of non-negatively curved, 2-connected, rational homotopy 7-spheres, discovered by Sebastian Goette, Krishnan Shankar and myself, which contains, in particular, all $S^3$-bundles over $S^4$ and all exotic 7-spheres.

I will discuss an ascending filtration on the Chow group of zero-cycles on a smooth projective variety obtained roughly by considering the successive kernels of the iterates of some modified diagonal embedding of the variety. This filtration is particularly relevant in the case of abelian varieties and of hyper-Kähler varieties, where it is expected to be opposite to the conjectural Bloch-Beilinson filtration. In the case of abelian varieties, it can in fact be described explicitly in terms of the Beauville decomposition, while in the case of hyper-Kähler varieties, I conjecture (and prove in some cases) that it coincides with a filtration introduced earlier by Claire Voisin. As a by-product we obtain in joint work with Olivier Martin a criterion involving second Chern classes for two effective zero-cycles on a moduli space of stable objects on a K3 surface to be rationally equivalent, generalising a result of Marian-Zhao.

A conjecture of Holzegel, Schmelzer and Warnick states that there is a Ricci flow with surgery connecting the two Ricci flat metrics Taub-Bolt and Taub-NUT. We will present some recent progress towards proving this conjecture. This includes showing for the first time the existence of a Ricci flow with surgery with local topology change $\mathbb{CP}^2\setminus\{ \mathrm{pt}\} \rightarrow \mathbb{R}^4$.

As discovered by Perelman, the study of ancient Ricci flows which are $\kappa$-noncollapsed is a crucial prerequisite to understanding the singularity behaviour of more general Ricci flows. In dimension three, these so-called "$\kappa$-solutions" have been fully classified through the groundbreaking work of Brendle, Daskalopoulos, and Šešum. Their classification result can be extended to higher dimensions, but only for those Ricci flows that have uniformly positive isotropic curvature (PIC), as well as weakly-positive isotropic curvature of the second type (PIC2); it appears the classification result fails with only minor modifications to the curvature assumption. Indeed, with the alternative assumption of non-negative curvature operator, a rich variety of new examples emerge, as recently constructed by Buttsworth, Lai, and Haslhofer; Haslhofer himself has conjectured that this list of non-negatively curved $\kappa$-solutions is now exhaustive in dimension four. In this talk, we will discuss some recent progress towards resolving Haslhofer's conjecture, including a compactness result for non-negatively curved $\kappa$-solutions in dimension four, and a symmetry improvement result for bubble-sheet regions. This is joint work with Anusha Krishnan and Timothy Buttsworth.