Norwegian Topology Meeting

The conference is supported by a grant from the Trond Mohn foundation.

Speakers

• Fernando Abellán (NTNU)
• Drew Heard (NTNU)
• Alice Hedenlund (NTNU)
• Hadrian Heine (Oslo)
• Achim Krause (Oslo)
• Clover May (NTNU)
• Marius Verner Bach Nielsen (NTNU)
• Torgeir Aambø (NTNU)

Schedule

Location

All talks will be held in room KJL1 in the Kjelhuset building on NTNUs Gløshaugen campus in Trondheim.

Abstracts

(∞,2)-Topoi and Descent
Fernando Abellán (NTNU)
(Joint with Louis Martini.) This goal of this talk is to introduce the notion of a Grothendieck (∞,2)-topos as a presentable (∞,2)-category satisfying a categorified version of the descent axiom for (∞,1)-topoi of Rezk-Lurie, which we call fibrational descent. As the name indicates, fibrational descent axiomatizes the structure of internal fibrations in an (∞,2)-category and it is closely related to the straightening-unstraightening equivalence of Grothendieck-Lurie. After presenting the main definition, I will give an overview of several different ways of characterising 2-topoi, which includes a 2-dimensional version of Giraud's theorem. Moreover, I will show how the theory of internal categories in an (∞,1)-topos (as develop by Martini and Wolf) can be embedded into our formalism as (∞,1)-localic 2-topoi. If time permits, I will explain how to construct a version of the Yoneda embedding in an (∞,2)-topos and a theory of partially lax Kan extensions.

The spectrum of excisive functors
Drew Heard (NTNU)
We will explain a recent computation of the Balmer spectrum of d-excisive functors from finite spectra to spectra. We will spend the majority of the time trying to explain exactly what this means. This is joint work with Arone, Barthel, and Sanders.

Twisted spectra
Alice Hedenlund (NTNU)
In the 90s, Cohen-Jones and Segal asked the question of whether various types of Floer homology theories could be upgraded to the homotopy level by constructing stable homotopy types encoding Floer data. They also sketched how one could construct these Floer homotopy types as (pro)spectra in the situation that the infinite-dimensional manifold involved is “trivially polarized”. It has since been realized that the correct home for Floer homotopy types, in the polarized situation, is twisted spectra. This is a generalization of parametrized spectra that one can roughly think of as sections of bundles of categories whose fibre is the category of spectra. In this talk, I will give an introduction to twisted spectra and how to construct them formally within the ∞-categorical framwork. In particular, I will cover the connection between twisted spectra and modules over Thom spectra, as well as the 6-functor formalism one obtains from looking at the total category of twisted spectra over a fixed space (letting the twist vary). This is joint work with T. Moulinos.

A directed version of homotopy colimits
Hadrian Heine (Oslo)
In my talk I will discuss a notion of lax colimit and lax limit for diagrams of weak (∞, ∞)-categories thought of as directed homotopy types. I will demonstrate that this notion is appropriate to perform directed analogues of classical constructions of homotopy theory like joins, suspensions, loop spaces and homotopy fibers, and therefore may be thought of as a directed version of homotopy colimits and homotopy limits. This is joint work with David Gepner.

Algebraic K-theory and prismatic cohomology
Achim Krause (Oslo)
In this talk I want to give an overview over recent work with Antieau and Nikolaus, about computing algebraic K-theory of Z/pⁿ.

Classifying modules of equivariant Eilenberg–MacLane spectra
Clover May (NTNU)
Classically, since Z/p is a field, any module over the Eilenberg–MacLane spectrum HZ/p splits as a wedge of suspensions of HZ/p itself. Equivariantly, cohomology and the module theory of G-equivariant Eilenberg–MacLane spectra are much more complicated. For the cyclic group G=C_p and the constant Mackey functor Z/p, there are infinitely many indecomposable HZ/p-modules. Previous work together with Dugger and Hazel classified all indecomposable HZ/2-modules for the group G=C₂. The isomorphism classes of indecomposables fit into just three families. By contrast, we show for G=C_p with p an odd prime, the classification of indecomposable HZ/p-modules is wild. This is joint work in progress with Grevstad.

On the Geometrization of Synthetic Spectra
Marius Verner Bach Nielsen (NTNU)
Using ideas from spectral algebraic geometry, we produce a geometric setting for studying deformations of spectral sequences. One particular example of such a deformation is the category of synthetic spectra. We will end this talk by comparing synthetic Spectra with quasi-coherent sheaves on a certain geometric stack.

Deformations of chromatic homotopy theory
Torgeir Aambø (NTNU)
In recent years there has been significant interest in the deformation theory of stable ∞-categories. This has led to advances in computational results, as well as an increased structural understanding of these categories. In this talk we will survey some deformations relevant to the fundamental building blocks of stable homotopy theory — E(n)-local and K(n)-local spectra — and look at their interactions.