Award to mathematician for best PhD thesis

Manuel Krannich was PhD student at the Department of Mathematical Sciences from the fall of 2015. He defended his PhD thesis in October 2018. He is currently employed as a postdoctoral researcher at the Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, where he has Oscar Randal-Williams as a postdoctoral supervisor.

It is not unusual for Manuel to get awards:

In 2013, at the Karlsruhe Institute of Technology, he received the faculty award for best bachelor's thesis "On the structure of abelian categories". And two years later he received the faculty award for best master's thesis, "Spaces of submanifolds and the classifying space of topological bordism category ".

The award-winning PhD thesis consists of three articles. The first two articles are accepted for publication in Geometry & Topology and Mathematische Annalen respectively.

All three articles deal with diffeomorphism groups of manifold, a classic topic that has seen spectacular progress in recent years due to, among other things, work by Madsen-Weiss and Galatius-Randal-Williams.

Manuel shows a general homological stability result, calculates how the homology of the diffeomorphism group changes under connected sum with exotic spheres, and finally, in collaboration with Jens Reinhold, he studies characteristic classes of smooth bundles over surfaces. Krannich uses a wide range of methods, ranging from classic methods from the 60s to completely modern methods from the last few years.

From the thesis abstract:

“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.”

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