Vorträge des SFB 1442

Abstract: It is a curious fact of life in geometric topology, that the classification of closed manifolds by surgery theory becomes easier as one passes from smooth to piecewise linear and finally to topological manifolds. It was long conjectured that an even cleaner statement should be expected in the somewhat arcane world of homology manifolds of the title, which ought to fill the role of some "missing manifolds". This was finally proven by Bryant, Ferry, Mio and Weinberger in the 90's in the form a surgery sequence for homology manifolds, building on an earlier theorem of Ferry and Pedersen that any homology manifold admits a euclidean normal bundle. In the talk I will try to explain this surgery sequence, and further that its existence is incompatible with the result of Ferry and Pedersen. The latter is therefore incorrect and/or the proof of the former incomplete.

We will discuss a topological version of Kesten's theorem and its connections with the asymptotic properties of group actions on the orbits of an amenable countable Borel equivalence relation. The talk is based on a joint work with Kate Juschenko and Friedrich Martin Schneider.

Ricci curvature is ubiquitous in mathematics: it appears in Hamilton's Ricci flow (a key tool in Perelman's resolution of the Poincaré conjecture), as well as in Einstein's equations of general relativity. Understanding its interplay with the global shape of Riemannian manifolds has been one of the key broad themes in geometric analysis since its early developments. While this interplay is well understood for manifolds with dimensions less than or equal to 3, several questions remain in dimension 4. After a gentle introduction to Ricci curvature, I will discuss joint work with Elia Bruè and Alessandro Pigati, in which we prove that any Riemannian 4-manifold with nonnegative Ricci curvature and Euclidean volume growth looks like a cone over a spherical space form at infinity. I will provide all the background needed for the precise statement, explain in which sense it is optimal, and explain why one might expect it to be true.

Abstract: Glasman proved that the category of d-excisive endofunctors on spectra is equivalent to a category of Mackey functors. In this talk, I will sketch a new proof of this together with various enhancements. Vaguely speaking, the method is to proceed by first completing a dictionary between genuine equivariant homotopy theory and Goodwillie calculus as suggested in recent work of Arone-Barthel-Heard-Sanders and then stratifying the problem accordingly. If time permits, we will also see other applications of this dictionary. This reports on work-in-progress joint with Tobias Barthel and Nikolai Konovalov.

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