Entrance hall of the ICMS

Categorification and Geometrisation from Representation Theory

Apr 13, 2009 - Apr 18, 2009

Room 516, Department of Mathematics, University Gardens, University of Glasgow, G12 8QW

Organisers

Name Institution
Brown, Ken A University of Glasgow
Gordon, Iain University of Edinburgh
Reshetikhin, Nicolai University of Berkeley
Rouquier, Raphaël University of Oxford
Stroppel, Catharina University of Bonn

This workshop is in association with the programme on Algebraic Lie Theory at the Isaac Newton Institute for Mathematical Sciences.

For a long time the idea of categorification has been in the background of many ideas in algebraic Lie theory and its connections to geometry. Several hard questions in Lie theory have been solved by translation (often via geometry) into combinatorics. For example, irreducible modules are labelled by combinatorial data and multiplicity formulas can be computed via combinatorially defined polynomials. On the other hand, topological questions are sometimes transferred into combinatorics in order to produce a clean answer: combinatorially defined knot invariants via polynomials; changing of coordinate systems via mutation rules; etc. It is becoming increasingly clear that the connecting principle of many such results in both Lie theory and topology is the idea of categorification. The notion “categorification” goes back to Crane and Frenkel, motivated by mathematical physics, and in particular by the hope to construct higher dimensional topological quantum field theories.

The conference wants to clarify the notion of categorification and its appearance in three different areas of mathematics: algebraic geometry, symplectic geometry and representation theory. The first goal of this workshop is to bring together the international figures working on categorification with connections to representation theory, providing the possibility for people from different fields to meet. Our conference should provide the basis for interactions between the areas of representation theory, symplectic geometry and topology. There are many different strands to the field, and it is vital that as much discussion as possible takes place between researchers.

The conference is intended to be quite specialised around the subject categorification. An Introductory Workshop will provide the necessary background and basis for communication.

20th ARTIN Meeting
The twentieth meeting of Algebra and Representation Theory in the North (ARTIN) will be held in the Department of Mathematics at the University of Glasgow. For further information about this PhD and Postdoc meeting please follow this link to the 20th ARTIN Meeting.

Arrangements

This meeting will be made up of two separate workshops:

Introductory Workshop
Commencing with lunch and registration at 12.00 Monday 13 April, plus full day Tuesday 14 April 2009 (Funded by the Isaac Newton Institute for Mathematical Sciences)

Workshop on Categorification and Geometrisation from Representation Theory
Wednesday 15 April to Saturday 18 April 2009; closing at 13.30 on Saturday 18 April. (Funded by ICMS)

Venue
The workshop will be held in Room 516, Department of Mathematics, University of Glasgow. Location maps - the Department of Mathematics can be found at D4. Lecture Room 516 is equipped with blackboards, data-video projector, overhead projector, overhead visualiser and screens.

Travel
Information about travel to the UK and Glasgow is available here. Follow this link for bus information including the Airlink Services to and from the Airport. If you are travelling by air, the transport sections of the following websites may be useful:Glasgow International Airport and Glasgow Prestwick Airport

Directions from Hillhead Subway Station, Byres Road, to Department of Mathematics, University of Glasgow
Turn left on exit Hillhead Subway Station. Next to the Station there is an Iceland supermarket, turn left here into Ashton Lane. Walk along the lane, bearing right as the lane curves and take the first small flight of steps you see on your left. The steps will bring you out on University Gardens. Turn right at the top of the steps and the Department of Mathematics is the second building on the right.

Accommodation
If you indicated, during on-line registration, that you wish ICMS to arrange your accommodation, a reservation will have been made for you to stay in the Glasgow Pond Hotel, Great Western Road, West End, Glasgow, Scotland, G12 0XP. No further action is required and full details will be emailed to you nearer the time of the workshop. However, if your arrival and departure dates alter in any way, please email Audrey Brown at ICMS immediately.

If you wish to arrange your own accommodation, there may still be rooms available at the Glasgow Pond Hotel and it is within walking distance (approximately 30 minutes walk) from the workshop venue. Alternatively, the Glasgow Tourist Information Office may be able to help you find accommodation.

The hotel is situated next to the Glasgow Gartnavel Hospital which may be used as a landmark. It is well served by buses from the city centre (No 66 and No 20 from Hope Street in the city centre and the No 118 from Buchanan Street Bus Station). Follow this link for bus information including the Airlink Services to and from the Airport. For a route map of buses serving the city centre please click here. For travel within Glasgow City Centre single fares range from £0.85 for a single fare stage (approximately half a mile) up to £1.50. Please note that on all services you pay the driver as you get on. Services operate on an exact fare (no change given) basis so please try to have the correct money ready if you know your fare.

If you have some local knowledge of Glasgow, it is possible to take a train from Glasgow Queen Street to Hyndland Station. It is important to exit Hyndland Station on the Gartnavel side of the track. Follow a footpath leading to the car park of Gartnavel Hospital. After crossing the car park, the Glasgow Pond Hotel is situated at the entrance to the hospital grounds on the Great Western Road. However, please note that signposts for the hotel face the Great Western Road and are not obvious from the car park at Gartnavel.

Directions from Glasgow Pond Hotel to Department of Mathematics, University of Glasgow
Click here for a map showing the route. Exit the Pond Hotel by the main exit and turn right along Great Western Road. After about 25-30 minutes walking distance, turn right on Byres Road. Passing Hillhead Subway Station (on your left) walk along to until you reach University Avenue or Ashton Road on your left. Walk along University Avenue or Ashton Road. Just before the Boyd Orr tower (on your left), turn left and up steps onto University Gardens. Turn right and the Department of Mathematics is the second building on the right.

Meals and Refreshments
A campus map is available here.
The Introductory Workshop will commence with registration and a sandwich lunch at 12.00 on Monday 13 April in the Common Room, ground floor of the Department of Mathematics at University of Glasgow (D4 on the campus map). At 18.00 on Tuesday 14 April, a buffet evening meal will be provided in the Atrium of the Wolfson Medical School Building. The Wolfson Medical School Building (C8 on the campus map) is located on University Avenue, directly behind the Department of Mathematics. Hot food will be served between 18.00 and 19.30. This meal is open to all participants who have attended the Introductory Workshop or are arriving for the Categorification Workshop. It is intended that there will be a Workshop Dinner on Thursday 16 April, meeting at 19.00 in the Café Bar of 1A The Square (A9 on the campus map), with dinner being served at 19.30 in the Ferguson Room of 1A The Square. Morning and afternoon refreshments will be provided throughout both workshops in the Department of Mathematics Common Room. Unless otherwise agreed, there will be no cost to participants for the catering listed above.

Participants are free to explore the restaurants and bars nearby to purchase all other lunches and evening meals.

Financial Matters
Unless otherwise specified in your invitation email, the workshop grant will cover the cost of your bed and breakfast accommodation, tea/coffee Monday to Saturday, lunch on Monday only, the informal buffet evening meal on Tuesday and the Workshop Dinner on Thursday evening.

If we have agreed to pay some of your travel costs, you will have been informed in your invitation email. Reimbursement will take place after the workshop and will involve payment directly into your bank account. At Registration you will be given an expenses claim form and this should be submitted to ICMS, with receipts. It would be helpful if you could bring your bank details to the workshop. In addition to the bank account number, participants from the USA and Canada will require their bank’s routing number, those from the UK will be asked for the bank sort code, and those from Europe and the rest of the world, their IBAN and SWIFT/BIC code. We cannot reimburse any item without a receipt.

Unless otherwise stated in your invitation or further correspondence, there will be a registration fee of 30.00 GBP for the Categorification Workshop. We ask that this is paid in advance by using this credit or debit card payment form. The form should be printed out, completed and faxed back (as email is not a secure way of sending credit card information). The fax number is on the form. If it is not possible for you pay in advance, you may print out the credit/debit card form above and bring the completed form along to Registration. We prefer not to handle cash at Registration.

Programme

Introductory Workshop

Monday 13 April

12.00 - 13.00

Sandwich Lunch and Registration

 

SESSION 1

13.00 - 14.00

Joel Kamnitzer (University of Toronto)
Geometry and Categorification I
Categorification via geometric Satake correspondence

14.30 - 15.30

Ian Grojnowski (University of Cambridge) & Kevin McGerty (Imperial College London)
Representation Theory and Categorification I
D-modules and localization

15.30 - 16.00

Tea/Coffee

16.00 - 17.00

Gregor Masbaum (CNRS)
Topology and Categorification I
Introduction to Reshetikhin-Turaev invariants and modular categories

 

Tuesday 14 April

 

SESSION 2

09.00 - 10.00

Sabin Cautis (Rice University and MSRI)
Geometry and Categorification II
Braid group actions in algebraic geometry

10.00 - 10.30

Tea/Coffee

10.30 - 11.30

Ian Grojnowski (University of Cambridge) & Kevin McGerty (Imperial College London)
Representation Theory and Categorification II
Hall algebras, quivers and canonical bases

11.35 - 12.35

Ciprian Manolescu (UCLA/University of Cambridge)
Topology and Categorification II
Categorification via Floer homology and the Seidel-Smith construction

12.35 - 13.55

Lunch break

 

SESSION 3

13.55 - 14.55

Sabin Cautis (Rice University and MSRI)
Geometry and Categorification III
Categorical sl(2) actions on categories of coherent sheaves

15.00 - 16.00

Ian Grojnowski (University of Cambridge) & Kevin McGerty (Imperial College London)
Representation Theory and Categorification III
Further examples

16.00 - 16.30

Tea/Coffee

16.30 - 17.30

Jürgen Fuchs (Karlstad University)
Topology and Categorification III
Modular categories and rational conformal field theory

18.00 - 19.30

Informal buffet meal and registration for Categorification Workshop

 

Categorification Workshop

Wednesday 15 April

08.30 - 09.30

Registration

09.30 - 10.30

Jonathan Brundan (University of Oregon)
The graded Lascoux-Leclerc-Thibon conjecture

10.30 - 11.00

Tea/Coffee

11.00 - 12.00

John Baez (University of California, Riverside)
Categorification and topology

12.00 - 13.30

Lunch break

13.30 - 14.30

Louis Crane (Kansas State University)
Categorification and physical spacetime

14.30 - 15.30

Bernhard Keller (University Paris 7)
The periodicity conjecture via 2-Calabi-Yau categories

15.30 - 16.00

Tea/Coffee

16.00 - 17.00

Mikhail Khovanov (Columbia University)
Categorification of quantum groups

 

Thursday 16 April

09.30 - 10.30

Aaron Lauda (Columbia University)
Categorifying quantum sl2

10.30 - 11.00

Tea/Coffee

11.00 - 12.00

Volodymyr Mazorchuk (Uppsala University)
Representation theory of the symmetric group via categorification

12.00 - 13.30

Lunch break

13.30 - 14.30

Dror Bar-Natan (University of Toronto)
Convolutions on lie groups and lie algebras and ribbon 2-knots

14.30 - 15.30

Jacob Rasmussen (University of Cambridge)
Sutured Floer homology: de (and re) categorification

5.30 - 16.00

Tea/Coffee

16.00 - 17.00

Raphael Rouquier (University of Oxford)
Higher representation theory

19.00

Workshop Dinner

 

Friday 17 April

09.30 - 10.30

Richard Thomas (Imperial College London))
Joyce’s Hall algebra

10.30 - 11.00

Tea/Coffee

11.00 - 12.00

Ulrike Tillmann (University of Oxford)
Higher dimensional cobordism categories and their topology

12.00 - 14.30

Lunch break

14.30 - 15.10

Joel Kamnitzer (University of Toronto)
Categorical geometric skew Howe duality

15.10 - 15.50

Ciprian Manolescu (UCLA/University of Cambridge)
Symplectic instanton homology

15.50 - 16.20

Tea/Coffee

16.20 - 17.00

Gregor Masbaum (CNRS)
Integral lattices in TQFT and integral modular categories

 

Saturday 18 April

09.30 - 10.30

Alexey Bondal (University of Aberdeen)
Categorifying integral polytops geometry

10.30 - 11.00

Tea/Coffee

11.00 - 12.00

Vladimir Baranovski (University of California, Irvine)
Uhlenbeck compactification as a stack

12.00 - 13.00

Anton Kapustin (California Institute of Technology)
Three-dimensional topological field theory and a categorification of the derived category of coherent sheaves

13.00

Close of workshop

 

Presentations for the Introductory Workshop:

Sabin Cautis: Braid group actions in algebraic geometry
We discuss various examples of braid group actions on derived categories of coherent sheaves while paying attention to the geometry involved.

Sabin Cautis: Categorical sl(2) actions on categories of coherent sheaves
We explain how categorical sl(2) actions can be used to construct equivalences generalizing Seidel-Thomas twists. This is illustrated by an sl(2) action on cotangent bundles of Grassmannians which generalizes the Seidel-Thomas twist on the cotangent bundle of P^1. This is joint work with Joel Kamnitzer and Anthony Licata.

Jürgen Fuchs: Modular categories and rational conformal field theory
I will present a construction of correlation functions in two-dimensional rational conformal field theory. The construction expresses a correlation function as the invariant of a ribbon graph in a three-manifold. This invariant is computed with the help of three-dimensional topological field theory. Besides the modular tensor category C that provides the decoration data for the 3-d TFT, the crucial ingredient is a certain type of Frobenius algebra in C. Some aspects of such algebras will be described as well.

Ian Grojnowski and Kevin McGerty: D-modules and localization
We will discuss the theory of D-modules and how they play a crucial role in representation theory via the famous Beilinson-Bernstein localization theorem.

Ian Grojnowski and Kevin McGerty: Hall algebras, quivers and canonical bases
We will sketch the Hall algebra construction of the positive part, and how its "faisceaus-fonctions" lift yields the canonical basis.

Ian Grojnowski and Kevin McGerty: Further examples
We will discuss further examples of interaction between representation theory and geometric structures.

Joel Kamnitzer: Categorification via geometric Satake correspondence
The geometric Satake correspondence relates the tensor category of representations of a reductive group to the topology of the affine Grassmannian for the Langlands dual group. I will explain a program, developed with Sabin Cautis, to use this correspondence and derived categories of coherent sheaves in order to construct knot homology theories.

Gregor Masbaum: Introduction to Reshetikhin-Turaev invariants and modular categories
This talk will be an introduction to Witten-Reshetikhin-Turaev quantum invariants of 3-manifolds, Topological Quantum Field Theory (TQFT), and Turaev's axiomatization of it in terms of modular categories.

Ciprian Manolescu: Categorification via Floer homology and the Seidel-Smith construction
I will explain the general principles of how Lagrangian Floer homology can be used for categorification, and give a survey of the relevant results in the literature. Examples will include Heegaard Floer homology and Seidel-Smith homology.

 

Presentations for the Categorification Workshop:

Presentations:

Presentation Details
Baez, John
Categorification and topology
View Abstract Down
Baranovsky, Vladimir
Uhlenbeck compactification as a stack
View Abstract Down
Bondal, Alexei
Categorifying integral polytops geometry
View Abstract Down
Brundan, Jonathan
The graded Lascoux-Leclerc-Thibon conjecture
View Abstract Down
Crane, Louis
Categorification and physical spacetime
View Abstract Down
Kamnitzer, Joel
Categorical geometric skew Howe duality
View Abstract Down
Kapustin, Anton
Three-dimensional topological field theory and a categorification of the derived category of coherent sheaves
View Abstract Down
Keller, Bernhard
The periodicity conjecture via 2-Calabi-Yau categories
View Abstract Down
Khovanov, Mikhail
Categorification of quantum groups
View Abstract Down
Lauda, Aaron
Categorifying quantum sl2
View Abstract Down
Manolescu, Ciprian
Symplectic instanton homology
View Abstract Down
Masbaum, Gregor
Integral lattices in TQFT and integral modular categories
View Abstract Down
Mazorchuk, Volodymyr
Representation theory of the symmetric group via categorification
View Abstract Down
Rasmussen, Jacob
Sutured Floer homology: de (and re) categorification
View Abstract Down
Rouquier, Raphaël
Higher representation theory
View Abstract Down
Thomas, Richard
Joyce’s Hall algebra
View Abstract Down
Tillmann, Ulrike
Higher dimensional cobordism categories and their topology
View Abstract Down

Participants

Name Institution
Abdelgadir, Tarig University of Glasgow
Achar, Pramod N Louisiana State University
Ackermann, Bernd Universität Stuttgart
Agerholm, Troels Aarhus University
Amazeen, Grétar University of Edinburgh
Arakawa, Tomoyuki Nara Women's University
Baez, John University of California, Riverside
Baranovsky, Vladimir University of California, Irvine
Barthel, Tobias University of Oxford
Begher, Gerrit University of Freiburg
Bellamy, Gwyn University of Manchester
Belolipetsky, Mikhail Durham University
Berczi, Gergely University of Oxford
Bondal, Alexei Steklov Mathematical Institute
Brendle, Tara University of Glasgow
Brown, Ken A University of Glasgow
Brown1, Audrey ICMS
Brundan, Jonathan University of Oregon
Calderon Martin, Antonio J University of Cadiz
Cautis, Sabin Rice University and MSRI
Chapovalova, Valentina Uppsala University
Collins, Julia University of Edinburgh
Cooney, Nicholas University of Oxford
Crane, Louis Kansas State University
Davison, Ben University of Oxford
Docherty, Pamela University of Edinburgh
Donovan, Will Imperial College London
Douglass, Matthew University of North Texas
Ehrig, Michael University of Cologne
Elias, Ben Columbia University
England, Matthew Heriot-Watt University
Everitt, Brent University of York
Faria Martins, Joao Centro de Matemática da Universidade do Porto
Feigin, Misha University of Glasgow
Felstaine, Eyal Ben Gurion University
Fuchs, Jürgen Karlstad University
Gaussent, Stéphane Institut Elie Cartan Nancy
Geck, Meinolf University of Aberdeen
Goodwin, Simon University of Birmingham
Gordon, Iain University of Edinburgh
Grant, Joseph University of Bristol
Grojnowski, Ian University of Cambridge
Gupta, Neha University of Warwick
Harris, Michael Université de Paris VII Denis Diderot
Henrich, Thilo University of Bonn
Hoffnung, Alex University of California, Riverside
Jantzen, Jens Carsten Aarhus Universitet
Juteau, Daniel CNRS, Univesité de Caen Basse-Normandie
Kamnitzer, Joel University of Toronto
Kapustin, Anton California Institute of Technology
Keller, Bernhard Université Paris Diderot - Paris 7
Khovanov, Mikhail Columbia University
King, Alastair University of Bath
Klein, Florian University of Freiburg
Kolb, Stefan Newcastle University
Köppe, Thomas University of Edinburgh
Korff, Christian University of Glasgow
Kraehmer, Ulrich University of Glasgow
Krasner, Daniel Columbia University
Lauda, Aaron Columbia University
Letellier, Emmanuel University of Caen
Libedinsky, Nicolas Université de Paris VII
Licata, Tony Stanford University/MPI
Mackaay, Marco Universidade do Algarve
Manolescu, Ciprian UCLA / University of Cambridge
Martino, Maurizio University of Bonn
Masbaum, Gregor CNRS
Mautner, Carl University of Texas
Mazorchuk, Volodymyr Uppsala University
McGerty, Kevin University of Oxford
Meachan, Ciaran University of Edinburgh
Miemietz, Vanessa University of Oxford
Mikovic, Aleksandar Lusofona University & GFMUL
Morrison, Scott Microsoft Station Q
Mullaney, Joseph University of Glasgow
O'Hagan, Steven University of Glasgow
Owens, Brendan University of Glasgow
Paasch, Anna-Louise Universität Wuppertal
Parker, Alison University of Leeds
Rasmussen, Jacob University of Cambridge
Röhrle, Gerhard Ruhr-Universität Bochum
Rouquier, Raphaël University of Oxford
Rubinsztein, Ryszard Uppsala University
Rumynin, Dmitriy University of Warwick
Savage, Alistair University of Ottawa
Shipman, Ian University of Chicago
Silantyev, Alexey University of Glasgow
Spencer, Christopher University of Edinburgh
Steiner, Richard University of Glasgow
Stevenson, Daniel University of Glasgow
Stosic, Marko Instituto Superior Tecnico
Stroppel, Catharina University of Bonn
Thiel, Anne-Laure IRMA, Strasbourg
Thomas, Richard Imperial College London
Tillmann, Ulrike University of Oxford
Treumann, David University of Minnesota
Turner, Paul Heriot-Watt University
Turner, Will University of Aberdeen
van Roosmalen, Adam-Christiaan Max-Planck-Institut für Mathematik
Vaz, Pedro CAMGSD - Instituto Superior Técnico
Waldron, Jack University of Oxford
Webster, Ben Massachusetts Institute of Technology
Weist, Thorsten Universität Wuppertal
Willerton, Simon University of Sheffield
Williamson, Geordie University of Oxford
Wilson, Benjamin Université de Paris VII
Winn, Dorothy University of Glasgow
Zuevsky, Alexander National University of Ireland, Galway