Solstices 2015

Journées Solstice d'été 2015

Colloque International

Noeuds, représentations et catégorification

Institut de Mathématiques de Jussieu - Paris Rive Gauche

18, 19 juin 2015

Université Paris-Diderot, Bâtiment Sophie Germain - 75013 Paris
Salle 2011 (2ème étage)

Métro : Bibliothèque François Mitterrand (ligne 14)
Autobus : Porte de France (lignes 62, 89)
Tramway : Avenue de France (ligne T3b)

Jeudi 18 juin

pause café
9h15-9h30	Accueil
9h30-10h30	Daniel TUBBENHAUER (Université Catholique de Louvain)
	From dualities to diagrams
	The Temperley-Lieb algebra has its origin in the study of sl₂-modules: Rumer, Teller and Weyl showed (more or less) already in the 30ties that the Temperley-Lieb algebra can be seen as a diagrammatic realization of the representation category of sl₂-modules - providing a topological (and fun!) tool to study the latter. In this talk I try to explain how one can proof such a realization. Our main tool is "a machine that takes dualities and produces diagrammatic categories". In particular, we show explicitly how this "machine" works if one feeds it with q-Howe duality - which produces diagrammatic presentations of categories of gl_n-modules akin to the Temperley-Lieb calculus. As an application, I give a diagrammatic version of a symmetry of HOMFLY-PT polynomials. In principal, everything in this talk is amenable to categorification, but we have to stay in the uncategorified world for the moment.
11h00-12h00	Joshua SUSSAN (Medgar Evers College)
	Categorical Heisenberg and related structures
	A graphical category whose Grothendieck group contains a Heisenberg algebra was constructed by Khovanov. By considering certain complexes in the corresponding homotopy category we obtain a conjectural boson-fermion correspondence on the level of categories. We will also study a super version of Khovanov's category in order to obtain a twisted Heisenberg algebra.
pause déjeuner
14h15-15h15	Pedro VAZ (Université Catholique de Louvain)
	Flavours of odd categorification
	In this talk I will start by giving an overview of some constructions in odd categorification from the odd Khovanov homology of Oszvath-Szabo-Rasmussen to the odd categorification of quantum sl(2) of Ellis-Khovanov-Lauda. In the last part of the talk I will explain a generalization of the odd NilHecke algebra of Ellis-Khovanov-Lauda.
15h30-16h30	Hanno BECKER (Universität Bonn)
	A Khovanov-Homology like approximation to Khovanov-Rozansky Homology
	In this talk I will explain how to associate to planar oriented link diagram a bicomplex of graded vector spaces built from the structure maps of the Frobenius algebra Q[x]/(x^k), similar to the complex used in the construction of Khovanov-homology. The vertical-then-horizontal homology of this bicomplex constitutes an approximation to Khovanov-Rozansky sl(k) homology in a sense I will explain, and the proof of which I will sketch. As an example, we consider the trefoil, in which case the approximation already agrees with Khovanov-Rozansky sl(k) homology.'
pause café
17h00	Fête annuelle de Sophie Germain (Amphi Turing)

Vendredi 19 juin

pause café
9h30-10h30	Antonio SARTORI (Albert-Ludwigs-Universität Freiburg)
	Mixed skew Howe duality and categorification of link invariants
	I will present a generalization of skew Howe duality via spider categories which describe tensor products of gl(m\|n) representations and their duals, extending the work of Cautis, Kamnitzer and Morrison. This provides a unified framework for defining gl(m\|n) link invariants and the HOMFLY-PT polynomial and should give insight for a possible categorification. (This is joint work with H. Queffelec.)
11h00-12h00	Radmila SAZDANOVIC (North Carolina State University)
	Categorification of orthogonal polynomials
	We develop a diagrammatic categorification of the polynomial ring Z[x] and explain how it generalizes to categorification of orthogonal polynomials, including Chebyshev polynomials and the Hermite polynomials. We find in these diagrammatic categorifications a new emanation of the Bernstein--Gelfand--Gelfand reciprocity. This is joint work with Mikhail Khovanov.
pause déjeuner
14h15-15h15	Ruslan MAKSIMAU (Université Paris Diderot - IMJ-PRG)
	Categorical actions, KLR algebras and affine category O
	We study the parabolic category O for affine gl(n) at level -e-n. There are functors E, F : O --> O defining a categorical action of affine sl(e) on O. Our goal is to identify the Koszul dual functors to E and F. In order to do this, we need to decompose the functors E and F. To decompose them, we prove a general fact about categorical actions. More precisely, we prove that a category with a categorical action of affine sl(e+1) contains a subcategory with a categorical action of affine sl(e). To prove this statement, we construct an isomorphism between the KLR algebra associated with the e-cycle and a subquotient of the KLR algebra associated with the (e+1)-cycle.
15h30-16h30	Nicolai RESHETIKHIN (University of California, Berkeley)
	Degenerate, or super-integrability
	The notion of Liouville integrability is well known in Hamiltonian mechanics and symplectic geometry. The main theme of this talk is the degenerate integrable systems. On a symplectic manifold of dimension 2n such system has k Poisson commuting integrals and 2n-k integrals of motion which generate a Poisson subalgebra in the algebra of functions on the symplectic manifold. Example related to Poisson Lie groups and representation theory will be given.
16h30	Café

Comité d'organisation
BLANCHET Christian (Projet Topologie et Géométrie Algébriques, IMJ, Université Paris-Diderot)
CHARBONNEL Jean-Yves (Projet Groupes, Représentations et Géométrie, IMJ, Université Paris-Diderot)
DUDAS Olivier (Projet Groupes, Représentations et Géométrie, IMJ, Université Paris-Diderot)

Secrétariat
LAVOLLAY Claire - prenom.nom@imj-prg.fr - poste 791 70 - bureau 8032

Ce colloque a reçu l'aide financière de : Institut de Mathématiques de Jussieu - Paris Rive Gauche, IMJ-PRG : projet Topologie et Géométrie Algébriques, IMJ-PRG : projet Groupes, Représentations et Géométrie.