Lectures by main speakers

Sabin Cautis

(Columbia University)

Title: Geometric categorification

Talk #1.

Title: Geometric constructions of representations Abstract: We construct some representations of sl(n) using the geometry of partial flag varieties.

Talk #2.

Title: Categorical sl(2) actions Abstract: After reviewing the basics of constructible sheaves we take the simplest example from lecture one and define a categorical sl(2) action on it.

Talk #3.

Title: Further categorication in geometry Abstract: We survey other possible geometric categorifications such as categorical Heisenberg actions on Hilbert schemes.

Aaron Lauda

(Columbia University)

Title: A diagrammatic categorification of quantum sl2

TALKS #1 and #2: We will review the simplest Lie algebra sl2 of traceless 2x2 matrices and its quantum deformation. The Jones polynomial for tangles can be understood in terms of the representation theory of this quantum group. After a review of 2-categories and their related diagrammatics, we then categorify quantum sl2 using a 2-category that can be defined via a planar diagrammatic calculus. The existence of this categorification was conjectured by Igor Frenkel. Categorifications of irreducible representations of quantum sl2 defined using cohomology rings of partial flag varieties will also be discussed.

TALK #3: New developments related to the categorification of quantum sl2 will be covered, including a proof of the main categorification result over the integers (joint with Khovanov, Mackaay, and Stosic), as well as a categorification of the quantum Casimir element associated to sl2 (joint with Khovanov and Beliakova).

Catharina Stroppel

(University of Bonn)

Title: Link homologies from Lie theory

1) Soergel bimodules and Lie theory

2) From Khovanov homology to Khovanov-Lauda algebras

3) Rational Euler characteristics and categorification of 3j-symbols and networks

Dylan Thurston

(Columbia University/Barnard College)

Title: Heegaard Floer homology

Lecture 1: Heegaard Floer homology from grid diagrams Heegaard Floer homology is a homological invariant of 3-manifolds and knots whose Euler characteristic is the Alexander polynomial. It detects knot genus (or more generally the Thurston norm) and fibration, and has many other uses. There is an elegant combinatorial formulation of knot Heegaard Floer homology from grid diagrams, a grown-up version of tic-tac-toe. We will explain this formulation and its connection to counting holomorphic curves.

Lecture 2: A toy model for bordered Heegaard Floer homology Another approach to Heegaard Floer homology for 3-manifolds is to chop up your 3-manifold into elementary pieces along surfaces. We give a "toy model" for this construction, based on a simplified version of grid diagrams, which still illustrates many key features of the full theory.

Lecture 3: General structure of bordered Heegaard Floer homology Bordered Heegaard Floer homology in general assigns invariants to 3-manifolds with parametrized boundary, in such a way that the invariants of closed 3-manifolds can be reconstructed. We explain this theory in general, including how to use it to give another way to compute Heegaard Floer homology in general (with no holomorphic curves required), and the relation to associated 4-manifold invariants.

Ben Webster

(MIT)

Title: Categorifying Reshetikhin-Turaev invariants

Lecture 1: Categorification of tensor products I'll introduce some new diagrammatic algebras (similar to those of Khovanov and Lauda) whose representations categorify the tensor products of simple representations of a semi-simple Lie algebra. I'll describe the simple and standard representations of these algebras, give several examples and show that for sl(n), these algebras are related to category O.

Lecture 2: The ribbon structure on categorifications In this lecture I'll give a brief introduction to ribbon categories and how they arise from quantum groups, followed by a description the functors which categorify the braiding (R-matrix), pairing and ribbon maps between tensor products of simple representations.

Lecture 3: Categorified quantum invariants Finally, I'll describe how these functors can be connected together to construct a knot homology whose Euler characteristic is the quantum invariant for any finite-dimensional representation of any semi-simple Lie algebra.

Contributed talks

Arkhipov, Sergey

University of Toronto

Title: On the invariants of the Braid group action on the category of twisted D-modules on the flag variety. Abstract: We recall the construction of the (weak) action of the braid group on the derived category of modules over the T-invariant differential operators on the base affine space for a simple algebraic group G. We prove that the derived category of modules over the Lie algebra of G is equivalent to the category of invariants for this action. We propose a conjectural construction of the affine braid group action and of its invariants in the case of the corresponding quantum group at a root of unity. The conjecture implies a localization statement for the derived categories of singular blocks in the category of modules over the quantum group.

Beliakova, Anna

University of Zurich

Title: A categorification of the Casimir of quantum sl(2) Abstract: In the category of complexes over the Khovanov-Lauda 2-category, we construct an element, categorifying the Casimir operator of quantum sl(2), and study its properties. It is a joint work with Aaron Lauda and Mikhail Khovanov.

Blanchet, Christian

IMJ - Université Paris Diderot

Title: Nodal foams and link homology Abstract: We define a cobordism category where objects are trivalent graphs, and morphisms are foams with nodal points, which we call nodal foams. This is used to categorify the Homflypt invariant. For a link we obtain an invariant in the homotopy category of formal complexes build with nodal foams. More generally we produce a 2-functor on the tangle 2-category which categorifies the Homflypt (generic Hecke) representation.

Chbili, Nafaa

College of Science, UAE University

Title: Equivariant Khovanov homology associated with symmetric links Abstract: Let Delta be a trivial knot in the three-sphere. For every Finite cyclic group G of odd order, we construct a G-equivariant Khovanov homology with coefficients in the field F2. This homology is an invariant of links up to isotopy in (S3, Delta). Our theory is functorial in the sense that an oriented cobordism between two links defines a linear map between their homologies, such that isotopic cobordisms define the same map. Another interpretation of the G-equivariant Khovanov homology is given using the categorification of the Kauffman bracket skein module of the solid torus.

Diaz, Rafael

Universidad Sergio Arbolea

Title: Categorification of the Weyl algebra Abstract: I will discuss the categorification of the Weyl algebra and the related quantum and meromorphic Weyl algebras from the viewpoint of the theory of species. Thus we provided a suitable methodology for the study of quantum phenomena with categorical/combinatorial techniques. References: R. Diaz, E. Pariguan, Super, Quantum and Non-Commutative Species, African Diaspora Journal of Mathematics 8 (2009) 90-130 R. Diaz, E. Pariguan, On the q-meromorphic Weyl algebra, Sao Paulo J. Math.Sci. 3, 1 (2009), 281-296 R. Diaz, E. Pariguan, Symmetric quantum Weyl algebras Ann. Math. Blaise Pascal 11 (2004) 187-203

Dumitrescu, Florin

Max Planck Institute for Mathematics, Bonn

Title: On 2-dimensional topological field theories Topological 1-dimensional field theories over a space X are described by vector bundles with connections over X. In this talk, we will give a similar characterization of topological 2-dimensional field theories over X as Frobenius bundles (i.e. vector bundles with fibers Frobenius modules) with connections over LX, the free loop space of X.

Elias, Benjamin
Columbia University

Title: A Diagrammatic Categorification of the Hecke Algebra
Abstract: In the early 90s, Soergel discovered a category of bimodules over a polynomial ring, which is a categorification of the Hecke algebra. In joint work with M. Khovanov, we present this category diagrammatically, providing generators and relations. We mention applications to knot theory and topology, due to joint work with D. Krasner, and work of Vaz and Mackaay.
References:
arXiv:0902.4700
arXiv:0906.4761
arXiv:0909.3495

Hoffnung, Alexander
University of California, Riverside

Title: The Antigravity Method in Cyclotomic Quotients of KLR algebras (joint with Aaron Lauda)
Abstract: We give some background on the relationship between KLR algebras, quantum groups and cyclotomic Hecke algebras. Then show how a combinatorial tool called "antigravity" determines nilpotency in cyclotomic quotients of KLR algebras, proving a conjecture of Brundan and Kleshchev.

References:
1) J. Brundan and A. Kleshchev. Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras. ArXiv:0808.2032, 2008.
2) J. Brundan and A. Kleshchev. Graded decomposition numbers for cyclotomic Hecke algebras. ArXiv:0901.4450, 2009.
3) M. Khovanov and A. Lauda. A diagrammatic approach to categorification of quantum groups I. math.QA/0803.4121. To appear in Represenation Theory, 2008.
4) A. Kleshchev and A. Ram. Homogeneous representations of Khovanov-Lauda algebras. ArXiv:0809.0557, 2008.
5) R. Rouquier. 2-Kac-Moody algebras. arXiv:0812.5023, 2008.

Ito, Noboru
Waseda University
Title: A bicomplex for Khovanov homology of the colored Jones polynomial

Abstract: We discuss the existence of a bicomplex which is a Khovanov-type complex associated with categorification of colored Jones polynomial. This is an answer to the question proposed by A. Beliakova and S. Wehrli. Then the second term of the spectral sequence of the bicomplex corresponds to the Khovanov-type homology. In this talk, we devote ourselves to the definition of this bicomplex.
References:
1. A. Beliakova and S. Wehrli, Canad. J. Math. 60 (2008), 1240--1266.
2. N. Ito, math.GT/0907.5247.
3. M. Khovanov, J. Knot Theory Ramifications 14 (2005), 111--130.

Libedinsky, Nicolas
Université Paris 12

Title: New bases of some Hecke algebras via Soergel bimodules
Abstract: We will introduce Soergel bimodules, a categorification of the Hecke algebra of a Coxeter system. We will explain its relation with representation theory, combinatorics and knot theory. Then we will introduce some new natural bases of some Hecke algebras coming from Soergel bimodules, satisfying positivity properties. The corresponding bimodules are an "aproximation" of the indecomposable Soergel bimodules.

Lowrance, Adam
University of Iowa

Title: Turaev genus, Khovanov homology, and knot Floer homology
Abstract: The Turaev surface of a knot diagram is a certain Heegaard surface on which the knot has an alternating projection. The Turaev genus of a knot is the minimum genus of any Turaev surface for that knot. In this talk, I will give a connection between the Turaev surface and the spanning tree complexes for Khovanov and knot Floer homology. This leads to several lower bounds for the Turaev genus of a knot.

 
Mikovic, Aleksandar
Lusofona University and GFMUL

Title: Categorification of BF theories

Abstract: BF theories are theories of flat connections on a principal bundle, and their action functionals are important for understanding the knot and spin network invariants. By replacing the Lie group with a 2-Lie group, one can formulate a 2-BF theory as a theory of flat 2-connections on the corresponding principal bundle. We describe a 2-BF theory action functional, which will be usefull for understanding the categorification of Vassiliev invariants and for constructing 2-knot and spin foam invariants.
References:
[1] A. Mikovic and J. F. Martins, "2-BF theory gauge-invariant action", preprint
[2] J. C. Baez and J. Huerta, "An invitation to higher gauge theory", arXiv:1003.4485

Morton, Jeffrey
University of Western Ontario

Title: 2-Linearization in Physics and Topology
Abstract: Groupoidification in the sense of Baez and Dolan gives a way of reflecting structures in linear algebra in the context of groupoids and spans of groupoids, equipped with a functor back into vector spaces. A categorification of this construction, "2-Linearization", gives a 2-functor into 2-vector spaces (abelian categories enriched in the category of vector spaces), which comes from the direct and inverse image functors between the categories of Vect-valued presheaves on the groupoids, and relies on the ambidextrous adjunction between these. These can be used to categorify some simple physical models, such as topological quantum field theory, and the harmonic oscillator. I will describe these constructions and applications.
References:
Baez, Hoffnung, Walker: Groupoidification Made Easy
Morton: 2-Vector Spaces and Groupoids; Extended TQFT, Gauge Theory, and 2-Linearization; Categorified Algebra and Quantum Mechanics

Pfeiffer, Hendryk
The University of British Columbia

Title: Combinatorial characterization of fusion categories
Abstract: Before trying to categorify a fusion category, one ought to better understand its tensor product. There are two kinds of fusion categories. If C is the category of modules over a Hopf algebra (no quotient taken!), it has an `easy\' tensor product. If C is not, then it has a \'difficult\' tensor product. In order to produce a quantum 3-manifold invariant that is stronger than a homotopy type invariant, it seems that one needs a spherical or a modular category with a difficult tensor product. The purpose of this talk is to make the structure of the tensor product fully transparent.
I show that every multi-fusion category C can be characterized as the category of finite-dimensional comodules of a quotient of the path algebra of a quiver GxG. Here G is a finite directed graph that depends on the fusion rules and on the choice of a generator of C. The path algebra k(GxG) is a Weak Bialgebra, and the quotient is modulo two types of relations. The first enforce that the tensor powers of the generator have the appropriate endomorphism algebras, thus providing a Schur-Weyl dual picture. The second type of relations removes suitable group-likes in order obtain a finite-dimensional Weak Hopf Algebra whose category of comodules is the desired fusion category with all the additional structure. As an example, I show the modular categories associated with U_q(sl_2) for suitable roots of unity in this picture.
References: arXiv:0912.0342, arXiv:0806.2903, arXiv:0711.1402.

Putyra, Krzysztof
Columbia University

Title: A 2-category of dotted cobordisms and a universal odd link homology
Abstract: There are two kinds of Khovanov-type link homology theories: even introduced by M.Khovanov in 1999 and odd constructed by P.Ozsvath, Z.Szabo and J.Rasmussen in 2007. Both are derived from the cube of resolutions of a link diagram. However, the odd version is not given by a Frobenius algebra but only by a projective functor. In 2008, I rewrote the construction using chronological cobordisms, i.e. cobordisms equipped with special projections onto a unit interval (in fact it is a 2-category). On a side, I obtained a homology theory that specifies to both even and odd link homologies. Recently, I have found also a chronological version of dotted cobordisms and the neck-cutting relation, what simplifies the whole construction and gives in some sense a universal theory. Some implications are:
-non-existence of odd vesion of Lie theory
-there is only one dot in the odd theory over a field

References:
- M.Khovanov, A categorification of the Jones polynomial, http://arxiv.org/abs/math/9908171
- D.Bar-Natan, Khovanov's homology for tangles and cobordisms, http://arxiv.org/abs/math/0410495
- P.Ozsvath, Z.Szabo, J.Rasmussen, Odd Khovanov homology, http://arxiv.org/abs/0710.4300
- K.Putyra, Cobordisms with chronology and a generalisation of the Khovanov homologies, Masters Thesis, Krakow 2008 (to appear on arXiv soon)

Russell, Heather
Louisiana State University
Title: Springer varieties and tangle homology
Abstract: We will discuss connections between certain classes of Springer varieties and homological tangle invariants.
References:
[1] The Bar-Natan skein module of the solid torus and the homology of (n,n) Springer varieties, Geom Dedicata (2009) 142: 71-89, arXiv:0805.0286v1.
[2] Springer representations on the Khovanov Springer varieties, with Julianna S. Tymoczko (submitted), arXiv: 0811.0650v1.

Stosic, Marko
Instituto de Sistemas e Robotica, IST, Lisbon

Title: Categorification of colored HOMFLY polynomial

In this talk we define explicit 2-functor between the diagrammatic 2-category of Soergel
bimodules and the gl(n) extension of the Khovanov and Lauda categorification of
U_q(sl(n)). This also sets the canonical sign choice in the Khovanov-Lauda 2-category,
which coincides with their corrected signed version. Moreover, this functor enables the
categorification of q-Schur algebras and allows a new approach to the categorification of
colored HOMFLY polynomial.

Wagner, Emmanuel
Université de Bourgogne

Title: On link homology theories from extended cobordisms
Abstract: The purpose of this talk will be the study of algebraic structures leading to link homologies. The originally used structures of Frobenius algebra and/or TQFT are modified in two directions. First, we refine 2-dimensional cobordisms by taking into account their embedding into the three space. Secondly, we extend the underlying cobordism category to a 2-category, where the usual relations hold up to 2-isomorphisms. The corresponding abelian 2-functor is called an extended quantum field theory (EQFT). We show that the Khovanov homology, the nested Khovanov homology, extracted by Stroppel and Webster from Seidel-Smith construction, and the odd Khovanov homology fit into this setting.

Wehrli, Stephan
Université Paris 7

Title: Oriented Khovanov homology and a new categorification of the colored Jones polynomial
Abstract: In this talk, I will review some functoriality properties of Blanchet's oriented model for Khovanov homology [1]. By using an action of the symmetric group on the oriented Khovanov homology of the n-cable of a knot, I will also give a new description of Khovanov's categorification of the nonreduced n-colored Jones polynomial [2].
References:
[1] C. Blanchet, An oriented model for Khovanov homology,
http://www.math.jussieu.fr/~blanchet/Articles/oriented_model.pdf
[2] M. Khovanov, Categorifications of the colored Jones polynomial, J. Knot theory and its Ramifications 14 (2005) no.1, 111-130, http://xxx.lanl.gov/abs/math/0302060

Williamson, Geordie
University of Oxford

Title: HOMFLYPT homology and mixed Hodge structures
Abstract: I will explain joint work with Ben Webster, which gives a construction of HOMFLYTPT homology in terms of the cohomology of certain complex algebraic varieties, together with their mixed Hodge structures (in the sense of Deligne). Among other things, this makes precise Khovanov\'s observation that Hochschild homology categorififies the Markov trace.
This talk will be based on the three papers:
"The geometry of Markov traces"
"A geometric construction of colored HOMFLYTPT homology"
"A geometric model for Hochschild homology of Soergel bimodules"
all joint with Ben Webster, and all available from the arxiv (or my homepage).

Yonezawa, Yasuyoshi
Graduate School of Mathematics, Nagoya University

Title: Quantum (sl_n, \\wedge V_n) link invariant and matrix factorizations.
Abstract: Khovanov and Rozansky defined a link homology for a specialized HOMFLY-PT polynomial using matrix factorizations. I explain a generalization of the work of Khovanov and Rozansky for the quantum (sl_n, \\wedge V_n) link invariant, which is a generalization of the specialized HOMFLY-PT polynomial.
References;
1.M. Khovanov; L. Rozansky, Matrix factorizations and link homology, Fund. Math. 199 (2008), no.1, 1-91.
2.Y. Yonezawa, Quantum (sl_n, \\wedge V_n) link invariant and matrix factorizations, arXiv:0906.0220.