SCIENCE-pris til matematiker for bedste ph.d.-afhandling

Manuel får prisen for afhandlingen ”On characteristic classes of manifold bundles”, vejledt af Nathalie Wahl. Manuel var ph.d.-studerende ved Institut for Matematiske Fag fra efteråret 2015, tilknyttet Center for Symmetri og Deformation. Han forsvarede sin ph.d.-afhandling i oktober 2018.

Det er ikke første gang, Manuel modtager en pris: I 2013 fik han på Karlsruher Institut für Technologie fakultetets pris for bedste bachelor-speciale, ”On the structure of abelian categories”. Og to år senere fik han prisen for bedste kandidat-speciale, ”Spaces of submanifolds and the classifying space of the topological bordism category”. Et sjældent set hattrick.

Manuel er i dag ansat som postdoc ved Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, hvor han har Oscar Randal-Williams som postdoc-mentor.

Den prisbelønnede ph.d.-afhandling består af tre artikler. De to første er accepteret til publicering i henholdsvis Geometry & Topology og Matematische Annalen.

Alle tre artikler omhandler diffeomorfi grupper af mangfoldigheder, et klassisk emne, som har set en spektakulær fremgang i de senere år på grund af bl.a. arbejde af Madsen-Weiss og Galatius-Randal-Williams.

Manuel viser et generelt homologisk stabilitetsresultat, udregner hvordan homologi af diffeomorfi-gruppen ændrer sig under connected sum med exotic spheres, og endelig, i samarbejde med Jens Reinhold, studerer han karakteristiske klasser af fiberbundne overflader. Krannich bruger et bredt spektrum af metoder, som spænder fra klassiske metoder fra 60erne til helt moderne metoder fra de sidste par år.

Fra afhandlingens resume:

“In Paper A, we compare the ring of characteristic classes of smooth bundles with fibre a closed simply connected manifold M of dimension 2n≠4 to the respective ring resulting from replacing M by the connected sum M#Σ with an exotic sphere Σ. We show that, after inverting the order of Σ in the group of homotopy spheres, the two rings in question are isomorphic in a range of degrees. Furthermore, we construct infinite families of examples witnessing that inverting the order of Σ is necessary.

In Paper B, which is joint with Jens Reinhold, we study smooth bundles over surfaces with highly connected almost parallelisable fibre M of even dimension. We provide necessary conditions for a manifold to be bordant to the total space of such a bundle and show that, in most cases, these conditions are also sufficient. Using this, we determine the characteristic numbers realised by total spaces of bundles of this type, deduce divisibility constraints on their signatures and Â-genera, and compute the second integral cohomology of BDiff(M) up to torsion in terms of generalised Miller–Morita–Mumford classes.

In Paper C, we introduce a framework to study homological stability properties of E2-algebras and their modules, generalising work of Randal-Williams and Wahl in the case of discrete groups. As an application, we prove twisted homological stability results for various families of topological moduli spaces, such as configuration spaces and moduli spaces of manifolds, and explain how these results imply representation stability for related sequences of spaces.”

Emner