Seminars in geometry/topology

August 26, 13:15 - 14:15, room 734, Sentralbygg 2

Barbara Giunti (University of Pavia): Parametrised chain complexes as a model category

Abstract: Persistent homology has proven to be a useful tool to extract information from data sets. Homology, however, is a drastic simplification and in certain situations might remove too much information. This prompts us to study parametrised chain complexes. Dwyer and Spalinski proved that the category of chain complexes allows a model category structure. Following their example, we showed that also the category of parametrised chain complexes is a model category, for some distinguished classes of morphisms. In this seminar, I will present this result and show why it is useful. In particular, I will present two types of invariants, very natural in any model category, that apply to any parametrised chain complex, while the standard decomposition of persistent modules can be applied only on a restricted set of parametrised chain complexes.

August 5, 13:15 - 14:15, room 734, Sentralbygg 2

Benjamin Collas (University of Bayreuth): Moduli Stacks of Curves: Arithmetic and Motives

Abstract: Thanks to the arithmetic of their Knudsen-Mumford stratification, the tower of moduli stacks of curves is a key object in the study of geometric Galois theory and of the Tannakian category of mixed Tate motives. The goal of this talk is to introduce a similar perspective in terms of the stack stratification of the spaces. As a motivation, we will first present how the first cyclic stack inertia strata are endowed with a Galois action of Tate-type, then how Artin-Mazur and Morel-Vovedsky simplicial and homotopical theories provide a fruitful context for some "stack" motivic decomposition and Tannakian results that reflect this arithmetic result. In genus 0, these approach leads in particular to an interpretation of the mixed Tate motivic Galois group as loop-group, and to the definition of computable (hidden) periods of stack nature.

August 12, 13:15 - 14:15, room 734, Sentralbygg 2

Tomer Schlank (Hebrew University, Jerusalem): Ambidexterity in the T(n)-Local Stable Homotopy Theory

Abstract: Chromatic homotopy is the study of the \(\infty\)-category of Spectra trough a filtration by so-called "chromatic primes" The pieces of this filtration (monochromatic layers), that is- The K(n)-local (stable \(\infty\)-)categories \(Sp_{K(n)}\) enjoy many remarkable properties. One example is the vanishing of the Tate construction due to Hovey-Greenlees-Sadofsky. The vanishing of Tate construction can be considered as a natural equivalence between the colimits and limits in \(Sp_{K(n)}\) parametrized by finite groupoids. Hopkins and Lurie proved a generalization of this result where finite groupoids are replaced by arbitrary \(\pi\)-finite \(\infty\)-groupoids.

There is another possible sequence of (stable \(\infty\)-)categories who can be considered as "monochromatic layers", Those are the T(n)-local \(\infty\)-categories \(Sp_{T(n)}\). For the \(Sp_{T(n)}\) the vanishing of the Tate construction was proved by Kuhn. We shall prove that the analog of Hopkins and Lurie's result in for \(Sp_{T(n)}\). Our proof will also give an alternative proof for the K(n)-local case. This is joint work with Shachar Carmeli and Lior Yanovski.