Derivators and chromatic localisation

Lead Research Organisation: University of Sheffield
Department Name: Mathematics and Statistics

Abstract

This project is part of chromatic homotopy theory, which is part of algebraic topology, which is part of pure mathematics. Topology is the study of multidimensional spaces which can be wrapped and folded in complex ways. Chromatic homotopy is one set of technical tools that can be applied to this study. Bousfield localisation is one of the relevant tools. In the past people have typically used only one Bousfield localisation at a time, or occasionally two different localisations in sequence. In this project we study what can happen if we use a much larger set of combinations of localisations applied one after another.

We will explore the use of derivators as a foundation for the theory of Bousfield localisation. This will help to deal with the fact that localisations are only defined up to weak equivalence, whereas the theory of homotopy limits needs diagrams that commute on the nose. As an application, we will analyse the monoid of endofunctors of the stable homotopy category generated by localisation with respect to finite wedges of Morava K-theories. This will shed light on the Chromatic Splitting Conjecture of Hopkins, the transchromatic theories of Stapleton and Torii, and various related matters. The analysis will involve some combinatorial topology of finite categories to prove that certain homotopy limits can be replaced by homotopy limits of much smaller diagrams.

Publications

10 25 50

Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509735/1 01/10/2016 30/09/2021
1948903 Studentship EP/N509735/1 01/10/2017 31/03/2021 Nicola Bellumat
 
Description The starting objective of my research was to prove that combining together particular tools called localisations we get only a finite number of substantially different combinations. I managed to prove this claim after rephrasing it in a more rigorous way, which allowed me to use the technical language of derivators at its fullest potential. I and my supervisor, prof. Neil Strickland, collected our findings in a paper whose URL is included below.
To explain the significance of our findings: our field of research is dedicated to the study of abstract objects called spectra. These objects are too complicated to be examined directly, thus a common technique to study them is to apply a localisation to them to get simplified versions. To use a metaphor, it is like to look at an object through multiple different lenses: each lens twists the image revealing particular details but obfuscating others. To collect the most data we have to examine the object through all the lenses and piece together the partial details collected. If a localisation corresponds to a particular type of lens then composition of localisations is like overlapping the associated lenses in a precise order: the deformation of each lens combine together to create a new distorted image which could expose new details. There is a finite set of particular localisations we are interested in and we want to know if combining them we get only finitely many compositions: the point is that even if there are only finitely many types of lenses we can juxtapose how many copies we want of the same type, so it is not trivial to see that the possible deformations are only finitely many. Therefore our results ensure that the possible compositions we have to take in account to give the full picture of the examined object are not infinite. We also provided a specific formula which can be used to prove when two combinations coincide i.e. give essentially the same composition.
Exploitation Route My work is highly abstract and of exclusively academic interest. Future contributions will be surely in the research of abstract mathematics: I think that transchromatic homotopy theory or the proof of the splitting conjecture are the areas most likely to benefit from my results. Especially, in the latter the iterated localisation of the sphere spectrum and how this is related to the single localisations are the main points of the conjecture.
Sectors Other
URL https://arxiv.org/abs/1907.07801
 
Description Poster for the Faculty of Science Graduate School Showcase 2019 of the University of Sheffield 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Undergraduate students
Results and Impact On 20th March 2019 the University of Sheffield held the Faculty of Science Graduate School Showcase. This is an exhibition directed mainly to undergraduate students, but also open to the general public, where research groups can communicate to non-specialized public their results. Every 2nd year PhD student is required to present a poster illustrating his work or just concepts of the discipline he is working on. I was asked to participate and I submitted a poster introducing the basic ideas of derivator theory. As stated in the summary of my award, derivator theory is the abstract language I used to state my results regarding the iterated localizations. I had to write down the poster and have it printed and expose it at the dedicated stand. I also presented it to two members of the judging panel for the Outstanding Poster prize. They asked me how I tried to explain this abstract and highly specialized topic to a broad audience and how to convey the main ideas via diagrams and images. I was also available to answer any question of the public.
The event was documented and reported on all the news webpages of the University.
Year(s) Of Engagement Activity 2019
URL https://www.sheffield.ac.uk/faculty/science/news/poster-showcase-graduate-school-1.837353