# Seminars in geometry/topology

## 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.

## 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.

## November 7, 10:15 - 11:15, room 656, Sentralbygg 2

Martin Frankland (University of Regina): The DG-category of secondary cohomology operations

Abstract: In joint work with Hans-Joachim Baues, we study track categories (i.e., groupoid-enriched categories) endowed with additive structure similar to that of a 1-truncated DG-category, except that composition is not assumed right linear. We show that if such a track category is right linear up to suitably coherent correction tracks, then it is weakly equivalent to a 1-truncated DG-category. This generalizes work of Baues on the strictification of secondary cohomology operations. As an application, we show that the secondary integral Steenrod algebra is strictifiable.

## November 8, 13:00 - 14:00, room 734, Sentralbygg 2

Irakli Patchkoria (University of Aberdeen): On the de Rham-Witt complex of perfectoid rings

Abstract: Perfectoid rings are generalizations of perfect $\mathbb{F}_p$-algebras and have a well-behaved cotangent complex. An example of a perfectoid ring is $O_{\mathbb{C}}$, the ring of integers of the field of p-adic complex numbers $\mathbb{C}$. This talk will concern the algebraic K-theory of perfectoid rings and will generalize Hesselholt's construction of the divided Bott class for $O_{\mathbb{C}}$ which lives in the target of certain trace map on algebraic $K_2$. Our construction of the divided Bott element is purely algebraic and uses only calculations in the de Rham-Witt complex. We will try to introduce and motivate the main players in this talk and make it available for general audience. The talk will be quite algebraic. This is all joint with Christopher Davis.

## November 18, 13:15 - 14:15, room 734, Sentralbygg 2

Hongyi Chu (Max Planck Institute for Mathematics): Homotopy-coherent algebras and polynomial monads

Abstract: In this talk I will introduce a general framework for homotopy-coherent algebraic structures defined by presheaves satisfying Segal-type limit conditions. We will see that this theory recovers many well-known examples for infinity categorical structures such as $(\infty,n)$-categories, $\infty$-operads and $\infty$-properads. The indexing category corresponding to the presheaf associated to a homotopy-coherent algebraic structure is then called an algebraic pattern as it encodes all the algebraic data. Special functors between algebraic patterns will induce Kan extensions which have interesting applications and at the end of the talk we will observe that the category of algebraic patterns and special functors between them is tightly related to the category of monads.