Research School on Bicategories, Categorification and Quantum Theory

University of Leeds, UK
11-15 July 2022

Introduction

This Research School features four introductory courses on modern categorical algebra and its application to quantum theory. The goal is to provide a forum for graduate students and junior researchers to gain a strong technical foundation in this rich and dynamic area, and to meet some of the leading researchers in the area. There will also be many opportunities for networking with other students and researchers.

Courses

The school will feature four courses given by expert researchers from across the globe. Tutorial sessions given by expert teaching assistants will help participants learn the material. There will be many opportunities for informal interaction and networking.

Prerequisites. To get the most from the courses, participants should already have basic knowledge of category theory (categories, functors, natural transformations, isomorphisms, limits and colimits), and individual courses have further mild prerequisites given below.

Bicategories

Just as monoids have categories as a many-object generalisation, so monoidal categories have a many-object generalisation, which are Benabou's bicategories. This course will study bicategories in their guise as many-object monoidal categories. Particular attention will be paid to the notion of enrichment in a monoidal category or in a bicategory. Besides examining a range of standard examples, we will also discuss how to set up bespoke enrichments using the theory of free presentations of cocomplete monoidal categories and bicategories.

Prerequisites: Basic category theory.
Materials: Key Definitions, Lecture 1, Lecture 2, Lecture 3, Problems
Teaching Assistant: Christina Vasilakopoulou, University of Patras

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Categories like Hilbert Spaces

Hilbert spaces are the mathematical foundation of quantum theory. This course concerns the abstract structure of the monoidal dagger category they form. We identify conceptual ingredients for quantum computation, such as dual objects and Frobenius structures to model entanglement and measurement. Spatiotemporal structure is discussed not only via the graphical calculus, but also in terms of subunits and sheaves.

Prerequisites: Basic ideas about Hilbert spaces (adjoint operator, orthonormal basis, tensor product), basic ideas about point-set and pointless topology (topological space, frame, locales).
Materials: Lecture 1, Lecture 2, Lecture 3, Tutorial 1, Tutorial 1 Answers, Tutorial 2, Tutorial 2 Answers, Tutorial 3, Tutorial 3 Answers, Handout 1, Handout 2, Handout 3
Teaching Assistant: Carmen Constantin, University of Edinburgh and Mansfield College, Oxford.

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Categorification

These lectures will motivate and explain finitary bicategories and their finitary birepresentations, their cell theory, the relationship between birepresentations and coalgebra 1-morphisms via the internal hom construction. This will be supported by explicit small examples and culminate in an extended example considering the categorification of Hecke algebras via Soergel bimodules.

Prerequisites: Basic representation theory of finite-dimensional algebras, for example the first few sections of Alex Bartel's notes.
Materials: Lecture 1, Lecture 2, Lecture 3
Teaching Assistant: James MacPherson, University of East Anglia

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Hopf Algebras

We shall discuss the structure of Hopf algebras, focusing on classification problems in the finite dimensional and, in particular, in the semisimple case. Some of their main features and invariants will be discussed. We shall study the connection with tensor categories through their representation theory and discuss a number of applications and open problems.

Prerequisites: Algebras over fields, modules for algebras.
Materials: Exercises
Teaching Assistant: Aryan Ghobadi, Queen Mary University of London

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Research Lecture

The conference will also feature a special research lecture.

The beauty of braids

Slides

Via different perspectives on braids we indicate how the search for topological invariants leads to crucial developments in representation theory and to speculations on higher topological invariants and TQFTs.

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Programme

All sessions take place in the Roger Stevens Lecture Theatre 22 at the University of Leeds.

Time Monday 11 July Tuesday 12 July Wednesday 13 July Thursday 14 July Friday 15 July
0900-0930 Registration
0930-1100 Bicategories 1 Tutorial 1 Hopf 2 Bicategories 3 Tutorial 3
1100-1130 Break Break Break Break Break
1130-1300 Categorification 1 Bicategories 2 Tutorial 2 Categorification 3 Research Lecture
1300-1400 Lunch Lunch Lunch Lunch Lunch
1400-1530 Hopf 1 Categorification 2 Hopf 3
1530-1600 Break Break Break
1600-1730 Quantum 1 Quantum 2 Quantum 3

Group Photograph

Participants

Robert Allen, University of Bristol Agoston Kaposi, Eötvös Loránd University
Rhiannon Savage, The University of Oxford
Sarah Almateari, University of Leeds Keisuke Hoshino, RIMS, Kyoto University
Nino Scalbi, Instituto Superior Técnico
Brad Ashley, The University Of Sheffield Matthew Koban, University of Toronto Léo Schelstraete, UCLouvain
Nicolas Blanco, University of Birmingham Zoltán Kolarovszki, Eötvös Loránd University
Arber Selimi, Instituto Superior Técnico, Lisbon
Dylan Braithwaite , University of Strathclyde Fernando Liu Lopez, Rice University
Andrew Slattery, University of Leeds
Elena Caviglia, University of Leicester Gabriele Lobbia, Masaryk University
Matteo Spadetto, University of Leeds
Cole Comfort, University of Oxford Leo Lobski, University College London
Benjamin Spencer, Indiana University
Carmen Constantin, University of Edinburgh Martino Lupini, Newcaste University
Miloslav Štepán, Masaryk University
Alexander Corner, Sheffield Hallam University Marco Mackaaij, University of Algarve
Mateusz Stroinski, Uppsala University
Alexander Cowtan, University of Oxford James Macpherson, University of East Anglia
Catherina Stroppel, University of Bonn
Jonathan Davies, University of Nottingham Ioannis Markakis, University of Cambridge
Filippos Ilarion Sytilidis, Oxford University
Will Donovan, Yau MSC, Tsinghua U Maciej Markiewicz, University of Warsaw
Calin Tataru, University of Cambridge
Ali Elokl, North Carolina State University Joseph Martin, University of Sheffield
Mrudul Thatte, Columbia University
Diogo Freire de Andrade, Instituto Superior Técnico Emily McGovern, North Carolina State University
Vincent Thompson, Rice University
Zeinab Galal, University of Leeds Luca Mesiti, University of Leeds
Fiona Torzewska, University of Leeds
Nicola Gambino, University of Leeds Vanessa Miemietz, University of East Anglia Sachin Valera, NYUAD
Richard Garner, Macquarie University Antonio Michele Miti, MPI Bonn
Bram Vancraeynest-De Cuiper, Ghent University
Aryan Ghobadi, Queen Mary Charles Nsukukazifani Msipha, Tshwane University of Technology
Christina Vasilakopoulou, University of Patras
Mahdie Hamdan, Cardiff University Sonia Natale, Universidad Nacional de Córdoba
Pedro Vaz, Université catholique de Louvain
Samuel Hannah, Cardiff University Federico Olimpieri, University of Leeds
Florrie Verity, Australian National University
Marius Henry, University of Cambridge Isaac Packtor, DePaul University
Jamie Vicary, University of Cambridge
Chris Heunen, University of Edinburgh Aberdeen Powell, University of East Anglia
Elia Vincenzi, Università di Roma Tor Vergata
Kengo Hirata, RIMS, Kyoto University Jan Pulmann, University of Edinburgh
Abigail Watkins, Indiana University
Nick Hu, University of Oxford Manar Qadi, University of Leeds
Haiqi Wu, ETH Zürich / University of Cambridge
Edward Jones-Healey, University of Manchester Jack Romo, University of Leeds
Tony Zorman, Technische Universität Dresden
Helena Jonsson, Uppsala university
Chiara Sarti, University of Cambridge

Local Information

Accommodation during the Research School will be provided at Storm Jameson Court, a new high-standard accommodation block with a lounge area and a 24-hour reception service. The cost of accommodation is covered by the registration fee.

During the Research School, the lectures will be in the Roger Stevens Building, and coffee breaks will be in the School of Mathematics.

Further information on travel to Leeds and the campus itself is available here:

Important Dates

  • Pre-Registration Deadline: Friday 29 April 2022
  • Notification of Acceptance: Friday 6 May 2022 Friday 13 May 2022
  • Summer School: 11-15 July 2022

Fees and Financial Support

Registration is now closed.

Registration costs will be as follows. These costs include accommodation, breakfast, lunch, coffee breaks, and the conference dinner which will take place one evening during the school.

  • PhD students: £150
  • Early-Career Researchers (up to 5 years since PhD): £250
  • Senior Researchers: £500

To apply for a place you must pre-register. Some financial support is also available, which may reduce the cost below that stated above. A financial support request can be filled out as part of the pre-registration.

Pre-Registration

Pre-registration is now closed.

Pre-registration allows you to express your interest in attending. Due to space restrictions in the venue, we may not be able to accept all applications. The deadline for pre-registration is April 29. Pre-registration is available using the following online form:

Scientific Committee

If you have a question about the Research School, please get in touch with one of the organizers.

Local Organising Committee

Nicola Gambino, Luca Mesiti, Federico Olimpieri, Andrew Slattery, Matteo Spadetto, Florrie Verity and Zeinab Galal.

Allergens

The lunch at the conference is provided by the university refectory. They can advise on the following allergens as ingredients in each food item: celery, gluten, crustaceans, eggs, fish, lupin, milk, molluscs, mustard, nuts, peanuts, sesame seeds, soya and sulphur dioxide. However, it is a conventional kitchen where all food and drinks are prepared in the same space, and so they cannot guarantee that any dish will not contain traces of any allergen.