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Timetable (ALTW03)

Categorification and Geometrisation from Representation Theory

Monday 13th April 2009 to Friday 17th April 2009

Monday 13th April 2009
13:00 to 14:00 Geometry and categorification 1 Categorification via geometric Satake correspondence
14:30 to 15:30 Representation theory and categorification I D-modules and localisation
16:00 to 17:00 Topology and categorification I Introduction to Reshetikhin-Turgev invariants and modular categories
Tuesday 14th April 2009
10:30 to 11:30 Representation theory and categorification II Hall algebras, quivers, and canonical bases
11:35 to 12:35 Categorification via Floer homology and the Seidel-Smith construction
13:55 to 14:55 Geometry and categorification II Categorical sl(2) actions on categoris of coherant sheaves
15:00 to 16:00 Representation theory and categorification III Further examples
16:30 to 17:30 Topology and categorification III Modular categories and rational conformat field theory
Wednesday 15th April 2009
09:30 to 10:30 The graded Lascoux-Leclerc-Thibon conjecture
In type A, some new algebras introduced recently by Khovanov-Lauda and Rouquier give rise to remarkable Z-gradings on group algebras of symmetric groups and more generally on various cyclotomic Hecke algebras. There are also graded versions of Specht modules for these algebras. In recent work joint with Kleshchev we have computed the q-decomposition numbers of these graded Specht modules. The result, which gives a graded version of the Lascoux-Leclerc-Thibon conjecture, is best expressed in the language of categorification.
11:00 to 12:00 Categorification and topology
The relation between n-categories and topology is clarified by a collection of hypotheses, some of which have already been made precise and proved. The "homotopy hypothesis" says that homotopy n-types are the same as n-groupoids. The "stabilization hypothesis" says that each column in the periodic table of n-categories stabilizes at a certain precise point. The "cobordism hypothesis" gives an n-categorical description of cobordisms, while the "tangle hypothesis" does the same for tangles and their higher-dimensional relatives. We shall sketch these ideas, describe recent work by Lurie and Hopkins on the cobordism and tangle hypotheses, and, time permitting, say a bit about how these ideas are related to other lines of work on categorification.
13:30 to 14:30 L Crane ([Kansas State])
Categorification and physical spacetime
Categorification was originally invented to solve problems in Mathematical Physics, namely construction of TQFTs and more fundamentally of the quantum theory of gravity. I will discuss the original motivations and more recent developments.
14:30 to 15:30 The periodicity conjecture via 2-Calabi-Yau categories
The periodicity conjecture for pairs of Dynkin diagrams was formulated at the beginning of the nineties in mathematical physics, in work of Zamolodchikov, Ravanini-Tateo-Valleriani and Kuniba-Nakanishi. We outline a proof based on the machinery of Fomin-Zelevinsky's cluster algebras and their (additive) categorification via 2-Calabi-Yau categories.
16:00 to 17:00 M Khovanov ([Columbia])
Categorification of quantum groups
I'll go over a joint work with Aaron Lauda on categorifications of positive halves of quantum universal enveloping algebras as well as categorification of the BLM form of quantum sl(n).
Thursday 16th April 2009
09:30 to 10:30 Categorifying quantum sl2
Igor Frenkel introduced the idea that the quantum enveloping algebra of sl(2) could be categorified at generic q using its canonical basis. In my talk I will describe a realization of Frenkel's proposal using a diagrammatic calculus. If time permits I will also explain joint work with Mikhail Khovanov on how this construction can be generalized to quantum sl(n).
11:00 to 12:00 Representation theory of the symmetric group via categorification
The action of projective functors on the regular block of the BGG category O for the Lie algebra sl(n) categorifies the right regular representation of the symmetric group S(n). I will try to describe how one can use representation-theoretic properties of this action to deduce some results about representations of S(n), in particular, about simple modules, induced modules, certain filtrations, and Wedderburn's decomposition of the group algebra.
13:30 to 14:30 Convolutions on lie groups and lie algebras and ribbon 2-knots
14:30 to 15:30 Sutured Floer homology: de (and re) categorification
Sutured Floer homology is an invariant of sutured manifolds introduced by Andras Juhasz. In this talk I'll explain how to first decategorify this invariant to produce an invariant that is much like the Alexander polynomial; and then recategorify to give a definition of sutured Floer homology related to the bordered Floer homology of Lipshitz, Ozsvath and Thurston. This is joint work with Andras Juhasz and Stefan Friedl.
16:00 to 17:00 Higher representation theory
We have introduced a 2-category associated with a Kac-Moody algebra (the type A case goes back to joint work with Joe Chuang and a close version of the positive half has been introduced independently by Khovanov and Lauda). We will discuss the 2-representation theory, ie, actions of this 2-category on categories (additive, abelian, triangulated, dg...). We will present a unicity result for simple integrable 2-representations and Jordan-Holder series. We will explain the realisation of simple 2-representations as categories of sheaves on quiver varieties and deduce the description of classes of indecomposable projective modules as canonical basis elements.
Friday 17th April 2009
09:30 to 10:30 R Thomas ([ICL])
Joyce's Hall algebra
I appear to be at the wrong conference, but I do know that some representation theorists like Ringel-Hall algebras, so I’ll describe a bit of Joyce’s stacky Hall algebra and how one can use it.
11:00 to 12:00 Higher dimensional cobordism categories and their topology
Cobordism categories are at the foundations of topological quantum field theory. We will discuss how to define a strict higher dimensional version of the cobordism category, associate a topological space to them, and explain how these spaces relate to classical spaces in cobordism theory as studied by Thom and others in the middle of the last century.
14:30 to 15:10 Categorical geometric skew Howe duality
I will explain a construction of the categorification of the braiding for tensor products of minuscule representations of sl(n). Our construction uses skew Howe duality and the machinery of categorical sl(2) actions. This is joint work with Sabin Cautis and Anthony Licata.
15:10 to 15:50 Symplectic instanton homology
Floer's instanton homology was originally defined as an invariant of integral homology 3-spheres. The Atiyah-Floer Conjecture claims that there should be a symplectic counterpart to instanton theory, based on Lagrangian Floer homology. Starting from a Heegaard decomposition of a 3-manifold, I will explain one way to make sense of the symplectic side of the Atiyah-Floer conjecture, for arbitrary 3-manifolds. This is joint work with Chris Woodward.
Saturday 18th April 2009
09:30 to 10:30 Categorifying integral polytops geometry
The subjcet of combinatorics of integer polytopes is known to be related to geometry of toric varieties. Recently, this relation has received a categorical interpretation as an equivalence of derived categories. We will discuss this equivalence and related results.
10:30 to 11:00 Uhlenbeck compactifications as a stack
I will explain how the Uhlenbeck compactification of vector bundles on a smooth projective surface can be defined as a functor of families (i.e. as an algebraic stack). I will also explain how Hecke correspondences which modify a vector bundle along a divisor on a surface, can be extended to the Uhlenbeck compactification. This construction is related to the conjectural higher dimensional Geometric Langlands program
12:00 to 13:00 Three-dimensional topological field theory anda categorification of the derived category of coherent sheaves
The Rozansky-Witten model is a 3d topological sigma-model whose target space X is a complex symplectic manifold. I will describe the 2-category structure on the set of its boundary conditions and show that it is a categorification of the derived category of coherent sheaves on X. In the special case when X is a cotangent bundle to a complex manifold Y, this 2-category is closely related to the 2-category of derived categorical sheaves over Y introduced by Toen and Vezzosi. I will also explain a surprising connection between a categorification of deformation quantization and complex symplectic geometry.
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons