Applied derived categories
Lead Research Organisation:
Imperial College London
Department Name: Mathematics
Abstract
Derived categories are abstract algebraic objects that package geometric information. The way they do this is inspired by topology -- a more flexible type of geometry which allows more deformations. As such they endow the original geometry with more flexibility and symmetries. They also filter out a little of the geometric information, so two different geometries might lead to the same derived category. The way in which they do this is very interesting, both in mathematics and physics, where derived categories describe topological D-branes .It has become clear in recent years that derived categories are not quite as abstract, mysterious or fearsome as often thought. Extracting the geometry (and invariants of the geometry) turns out to be quite natural in many situations, and the different geometries that can come from the same derive category give new and important points of view that solve previously intractable problems. Their extra symmetries and flexibility make them more useful in many applications.Derived categories bring a different philosophy to problems, suggesting new approaches to them. We propose to bring this new way of thinking to areas of broad areas of geometry, linking many which have not been touched derived categories before. We hope to solve problems and develop new areas of mathematics, helping to make derived categories into standard mathematical tools used all over the subject.
Organisations
Publications
Abuaf R
(2013)
Orthogonal bundles and skew-Hamiltonian matrices
Abuaf R
(2015)
Wonderful resolutions and categorical crepant resolutions of singularities
in Journal für die reine und angewandte Mathematik (Crelles Journal)
Abuaf R
(2018)
Orthogonal Bundles and Skew-Hamiltonian Matrices
in Canadian Journal of Mathematics
Abuaf R
(2015)
Categorical crepant resolutions for quotient singularities
in Mathematische Zeitschrift
Abuaf R
(2016)
Homological Units
in International Mathematics Research Notices
Addington N
(2014)
Hodge theory and derived categories of cubic fourfolds
in Duke Mathematical Journal
Addington N
(2017)
Brauer Groups and Obstruction Problems
Addington N
(2015)
The Pfaffian-Grassmannian equivalence revisited
in Algebraic Geometry
Addington N
(2016)
New derived symmetries of some hyperkähler varieties
in Algebraic Geometry
Addington N
(2016)
On two rationality conjectures for cubic fourfolds
in Mathematical Research Letters
| Description | We have made progress in 3 broad areas: mirror symmetry, enumerative geometry, and Kähler-Einstein metrics. Mirror symmetry is a remarkable conjecture linking different parts of mathematics. We have pursued its consequences and predictions in some of these areas, turning up unexpected structures and results for derived categories and for the classification of certain important building blocks called "Fano manifolds". Enumerative geometry is a venerable subject going back centuries. It aims to count classical configurations like the number of conics through 5 points. We have made progress using the new theory of "stable pairs in the derived category" which is equivalent to older formulations but much more computable. This has allowed us to prove some old conjectures such as the Göttsche conjecture and Katz-Klemm-Vafa conjecture from string theory. Finally, finding the conditions under which an algebraic variety admits a Kähler-Einstein metric (i.e. a solution of Einstein's equations of general relativity) is perhaps the most important problem in complex differential geometry. It has now been completely solved by Donaldson and his collaborators. |
| Exploitation Route | Our results are part of a worldwide effort to advance our understanding of various parts of geometry. Other international groups are already using and building upon our work. |
| Sectors | Other |
| URL | http://arxiv.org/archive/math |
| Description | Royal Society Research Professorship |
| Amount | £1,462,224 (GBP) |
| Funding ID | RP\R1\201054 |
| Organisation | The Royal Society |
| Sector | Charity/Non Profit |
| Country | United Kingdom |
| Start | 03/2020 |
| End | 02/2025 |
| Description | Vafa-Witten invariants of projective surfaces |
| Amount | £707,020 (GBP) |
| Funding ID | EP/R013349/1 |
| Organisation | Engineering and Physical Sciences Research Council (EPSRC) |
| Sector | Public |
| Country | United Kingdom |
| Start | 09/2018 |
| End | 08/2021 |
| Description | Film "Thinking space" |
| Form Of Engagement Activity | A broadcast e.g. TV/radio/film/podcast (other than news/press) |
| Part Of Official Scheme? | No |
| Geographic Reach | National |
| Primary Audience | Public/other audiences |
| Results and Impact | LMS commissioned film followed 8 mathematicians, including PI Richard Thomas and filmed their views on maths, visualisation, intuition etc |
| Year(s) Of Engagement Activity | 2015 |
| URL | https://www.plymouth.ac.uk/whats-on/thinking-space |