Seminar Algebra und Topologie

Das Seminar findet normalerweise am Mittwoch in F.13.11, 16:30 - 17:30 statt.
Beteiligte Dozenten: Jens Hornbostel, Sascha Orlik, Kay Rülling, Tobias Schmidt, Britta Späth, Matthias Wendt.

Subramanian: Smith Theory Revisited

The Sullivan conjecture is one of the landmark results in unstable homotopy theory, relating genuine fixed points of a finite p-group action to homotopy fixed points (after p-completion). In this talk, I will explain a new approach to the Sullivan conjecture for C_p-actions using perfect commutative \mathbb{F}_p-algebras.
The key idea is that the tilting functor provides an inverse to the Tate construction in the perfect setting, allowing us to recover the cochains on the genuine fixed-point space from Borel-equivariant cochains. Combining this with Mandell's theorem and the Segal conjecture gives a short proof of the Sullivan conjecture.
This is joint work with Robert Burklund.

Ayoub: On the classicality of the motivic Galois group

The motivic Galois group is naturally a derived affine group scheme. The usual motivic conjectures imply that this group is actually classical, i.e., its Hopf algebra is concentrated in degree zero. I will discuss some work in progress aiming at proving this classicality property.
 

Saunier: About categories of splitting non-commutative motives

The goal of this talk is to explain a different perspective on the category of splitting (=additive) non-commutative motives Mot^splt; this category was first defined by Blumberg--Gepner--Tabuada as the target of the initial splitting invariant out of the category CatEx of stable (infinity-)categories.  We will prove that Mot^splt is naturally the stable envelope of an additive category, which can be realized as a localisation of CatEx and then, we will explain a few concrete consequences of this result. This is motivated by results by Ramzi--Sosnilo--Winges and Efimov for localizing invariants, but it also has a different direction: one of our application simplifies the proof of Barwick--Glasman--Mathew--Nikolaus of the polynomial functoriality of K-theory. In particular, we can also prove a hermitian analogue of our results and in work in progress with Fabian Hebestreit, hope to prove that GW also is (canonically) functorial in polynomial functors of Poincaré categories.

Pieropan: Log geometry and lifting rational points

Determining the image of the set of rational points under a morphism of varieties is a very natural and difficult question. The case where the morphism is geometrically surjective has been studied extensively. From Campana's theory of orbifolds, it follows that over a number field, the image of the set of rational points is contained in the set of Campana points for the orbifold base of the morphism. This first approximation of the image of the set of rational points is refined by Abramovich's theory of firmaments. This talk presents a proof of a claim by Abramovich about lifting firm points under toroidal morphisms in joint work with Herr, Mehidi and Poiret.

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