Seminar Algebra und Topologie

Das Seminar findet normalerweise am Mittwoch in I.13.71 (!), 16:30 - 17:30 statt.
Beteiligte Dozenten: Jens Hornbostel, Sascha Orlik, Kay Rülling, Tobias Schmidt, Britta Späth, Matthias Wendt.

Christopher Lazda: Boundedness of the p-primary torsion of the Brauer groups of K3 surfaces

The transcendental Brauer group of a variety X over a field k is the image of its Brauer group inside the Brauer group of the base change of X to a separable closure of k. If X is a K3 surface, and k is finitely generated of characteristic 0, then it was shown by Skorobogatov and Zarhin that this group is finite. If k is finitely generated of characteristic p (and X is again a K3 surface), then later work of Skorobogatov and Zarhin (in the case p =/=2) and Ito (in the case p=2) showed that its prime-to-p torsion subgroup is finite. One cannot in general expect finiteness of p-torsion in characteristic p, however, I will explain how to use Madapusi-Pera's proof of the Tate conjecture for K3 surface to show that one does have such a finiteness result in the case that X is non-supersingular. Combined with known results in the supersingular case, this shows that, in general, the p-torsion will always at least be of finite exponent, that is, annihilated by a fixed power of p. This is joint work with Alexei Skorobogatov.

Non-semisimple quantum invariants for 3-manifolds

Since the advent of Reshetikhin-Turaev invariants for 3-manifolds, the study of quantum invariants and its categorification has witnessed fruitful developments in both topology and algebra. In this talk we step out of the semi-simple setting and see how this allows us to capture richer topological and algebraic data. Early non-semisimple constructions like Hennings invariants and Lyubashenko invariants didn’t receive enough attention due to the fact that once the first Betti number is positive, the invariants always become 0.  More recently, the discovery of modified trace made it possible to construct interesting non-semisimple invariants such as CGP invariants and DGGPR invariants. In my joint work with De Renzi and Martel we constructed a graded Hennings TQFT which incorporates homological information of cobordisms, and has the potential to capture more subtle information of mapping class groups of surfaces. With Runkel and Gainutdinov we are now trying to formulate this TQFT in Lyubashenko’s language with modified trace, leading to a non-semisimple version of Turaev’s homotopy QFT, which should include BCGP TQFT as an example. The algebraic tool that we need for this goal are G-crossed ribbon categories and twisted local modules. We provide a concrete example from unrolled quantum sl2.

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