2-representation theory and categorification
Lead Research Organisation:
University of East Anglia
Department Name: Mathematics
Abstract
While in mathematics it is very often helpful to break complicated problems down into less complicated ones by simplifying and forgetting data, the converse has often proved useful. The term categorification refers to the process of finding more complicated structures which, upon forgetting some information, reproduce the original problem that one wants to study. The more complicated structures often allow us to deduce useful information that was previously inaccessible. For example, an integer number solving a certain equation might a priori be anything, but if we then discover that this certain number, in fact, describes the number of elements in a set (e.g. cats in a household), it cannot be negative.
Categorification in representation theory is usually formulated in terms of certain structures called 2-categories encoding generalised symmetries of other categories, or in other words, 2-representations of 2-categories. To develop the theory of 2-representations of 2-categories (with certain nice properties abstracted from interesting examples) and to apply them to some of the original problems that inspired their definition is the aim of this proposal.
Categorification in representation theory is usually formulated in terms of certain structures called 2-categories encoding generalised symmetries of other categories, or in other words, 2-representations of 2-categories. To develop the theory of 2-representations of 2-categories (with certain nice properties abstracted from interesting examples) and to apply them to some of the original problems that inspired their definition is the aim of this proposal.
Planned Impact
The impact of this project will first and foremost be of academic nature, through knowledge transfer and training the next generation of researchers.
Scientific advances in 2-representation theory will have impact on surrounding areas like geometry, topology and number theory. Quick dissemination through uploading articles on the arXiv upon completion before publishing them in leading peer-reviewed journals, and presenting results in seminars and at conferences will guarantee fast transfer of the newly acquired knowledge. The summer school envisioned as part of this project will provide training for young mathematicians in the techniques required to successfully apply those advances. Similarly, the training of the PDRA will have significant impact on his or her career development.
Scientific advances in 2-representation theory will have impact on surrounding areas like geometry, topology and number theory. Quick dissemination through uploading articles on the arXiv upon completion before publishing them in leading peer-reviewed journals, and presenting results in seminars and at conferences will guarantee fast transfer of the newly acquired knowledge. The summer school envisioned as part of this project will provide training for young mathematicians in the techniques required to successfully apply those advances. Similarly, the training of the PDRA will have significant impact on his or her career development.
Organisations
People |
ORCID iD |
Vanessa Miemietz (Principal Investigator) |
Publications
Mazorchuk V
(2019)
2-categories of symmetric bimodules and their 2-representations
Mazorchuk V
(2020)
2-categories of symmetric bimodules and their 2-representations
in Pacific Journal of Mathematics
Hristova K
(2022)
Basic Hopf algebras and symmetric bimodules
Hristova K
(2023)
Basic Hopf algebras and symmetric bimodules
in Journal of Pure and Applied Algebra
Mackaay M
(2024)
Evaluation birepresentations of affine type A Soergel bimodules
in Advances in Mathematics
Mackaay M
(2022)
Evaluation birepresentations of affine type A Soergel bimodules
Mackaay M
(2021)
Finitary birepresentations of finitary bicategories
in Forum Mathematicum
Mackaay M
(2022)
Kostant's problem for fully commutative permutations
Mackaay M
(2023)
Kostant's problem for fully commutative permutations
in Revista Matemática Iberoamericana
Laugwitz R
(2022)
Pretriangulated 2-representations via dg algebra 1-morphisms
Description | In joint work with Marco Mackaay, Volodymyr Mazorchuk, Daniel Tubbenhauer, Xiaoting Zhang, we recently proved the conjecture we formulated last year on the classification of simple transitive 2-representations for Soergel bimodules in finite Coxeter type in characteristic zero. This answers Objective 3 of the proposal and is published in Proceedings of the London Mathematical Society (joint with MacKaay, Mazorchuk, Tubbenhauer, Zhang). Objective 2 turned out to be the stepping stone for this and was achieved in the paper Finitary birepresentations of finitary bicategories (joint with MacKaay, Mazorchuk, Tubbenhauer, Zhang). Objective 1 was deliberately open-ended, and some progress towards it was made in the paper Basic Hopf algebras and symmetric bimodules (with Katerina Hristova). |
Exploitation Route | The findings are relevant to other research in pure mathematics, and are expected to inspire new research both in various branches of representation theory and at the interfaces to mathematical physics and how-dimensional topology. |
Sectors | Other |
Description | Research School on Bicategories, Categorification and Quantum Theory |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Postgraduate students |
Results and Impact | Research School for graduate students and postdocs in Leeds, organised with funding by the London Mathematical Society |
Year(s) Of Engagement Activity | 2023 |
URL | https://conferences.leeds.ac.uk/bcqt2022/ |
Description | Workshop Representations of monoidal categories and 2-categories |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Other audiences |
Results and Impact | Workshop on Representations of monoidal categories and 2-categories |
Year(s) Of Engagement Activity | 2019 |
URL | https://archive.uea.ac.uk/~byr09xgu/workshop.html |