Speaker: Sakie Suzuki
Title: The universal quantum invariant and colored ideal triangulations
Abstract: The Drinfeld double of a finite dimensional Hopf algebra is a quasi-triangular Hopf algebra with the canonical element as the universal R-matrix, and one can obtain a ribbon Hopf algebra by adding the ribbon element. The universal quantum invariant of framed links is constructed using a ribbon Hopf algebra. In that construction, a copy of the universal R-matrix is attached to each crossing, and invariance under the Reidemeister III move is shown by the quantum Yang-Baxter equation of the universal R-matrix.
On the other hand, the Heisenberg double of a finite dimensional Hopf algebra has the canonical element (the S-tensor) satisfying the pentagon relation. In this talk we reconstruct the universal quantum invariant using the Heisenberg double, and extend it to an invariant of equivalence classes of colored ideal triangulations of 3-manifolds up to colored moves. In this construction, a copy of the S-tensor is attached to each tetrahedron, and invariance under the colored Pachner (2,3) moves is shown by the pentagon relation of the S-tensor.
Speaker: Daniel Lopez
Abstract: Modular categories are semisimple monoidal categories satisfying certain finiteness and nondegeneracy conditions. These categories turn to be a central object in low dimensional topology, as Reshetikhin and Turaev showed in the early 90s that one can obtain 3-manifold invariants from any modular category. The main goal of this talk (based on an article of L. Funar) is to give an infinite family of pairs of non-homeomorphic 3-manifolds, namely, certain torus bundles, which are not distinguished by their Reshetikhin-Turaev invariants for any modular category. We will show that any pair of matrices (A,B) in SL(2,Z), non-conjugate in GL(2,Z), but conjugate in any SL(2,Z/NZ) produces such an example.
Speaker: Daniel Lopez
Title: A surgery presentation of the cobordism category with an application to the classification of (2+1)-TQFTs
Abstract: In this talk, we will present a generators/relations presentation of the oriented cobordism category in terms of surgery, based on a yet unpublished work of András Juhász (see
arXiv:1408.0668v5). We will use this to classify (2+1)-TQFTs in terms of a new algebraic structure, called J-algebras, which roughly speaking are graded Frobenius algebras with an involution and a splitting together with mapping class group representations satisfying certain relations.
Speaker: Cristina Anghel
Title: Modified 3-manifold invariants from the Lie super-algebra sl(2|1)
Abstract: In this talk we will present 3-manifold invariants constructed from the Lie super algebra sl(2|1) at roots of unity. In 2011, N. Geer, B. Patureau and V. Turaev defined a construction that having as a starting point any pivotal category with additional properties, gives invariants for links in 3-manifolds. They used a modified quantum dimension on objects and the corresponding modified 6j-symbols in a state-sum setting, in order to obtain topological invariants. Following this line, we will present 3-manifold invariants constructed using the representation theory of the quantum enveloping algebra of the Lie super algebra sl(2|1) at roots of unity. We show that there exists a modified quantum dimension on the category of representations, and we use a purification of a certain subcategory by negligible morphisms in order to have the right algebraic structure that leads to the invariants. (This is a joint work with Nathan Geer.)
Speaker: Marco De Renzi
Title: Invariants de 3-variétés de Hennings-Kauffman-Radford
Speaker: Christian Blanchet
Title: Algèbres de Hopf enrubannées de dimension finie, l'exemple de sl(2) quantique - deuxième partie
Speaker: Léo Benard
Title: Algèbres de Hopf enrubannées, l'exemple de sl(2) quantique
Abstract: On commencera par définir, exemples à l'appui, les algèbres de Hopf enrubannées, puis on montrera qu'une certaine déformation de l'algèbre enveloppante de sl(2) en est une. L'exposé sera voué à être introductif, aucune connaissance particulière du sujet ne sera présupposée.
Speaker: Daniel Lopez
Title: Link invariants from ribbon Hopf algebras
Abstract: In this talk, we will define ribbon Hopf algebras, which are Hopf algebras endowed with an extra element giving solutions to the Yang-Baxter equation. Following ideas of R. Lawrence and T. Ohtsuki, we use these Hopf algebras to construct link invariants.
Speaker: Clinton Reece
Title: Turaev-Viro invariants for unrolled quantum sl(2)
Speaker: Christian Blanchet
Title: Non-semisimple TQFTs: the graded BCGP functor
Speaker: Ramanujan Santharoubane
Title: Finite dimensionality of BCGP TQFTs via skein calculus
Speaker: Marco De Renzi
Title: Universal Construction and surgery axioms
Abstract: I will start the construction of the BCGP TQFTs. The first step will be the definition of the decorated cobordism category which will be used to carry out the Universal Construction. I will also introduce the modified surgery axioms which are satisfied by CGP invariants and which will be used in the following in order to figure out the properties of TQFT vector spaces. Every step will be compared to the classical case of simisimple WRT TQFTs.
Speaker: Ramanujan Santharoubane
Title: Torsion de Reidemeister, polynôme d'Alexander et TQFT non semi-simple en r=2
Abstract: Nous verrons comment la spécialisation en r=2 des invariants de Costantino, Geer et Patureau donne : 1) Le polynôme d'Alexander-Conway pour des entrelacs. 2) La torsion de Reidemeister-Turaev pour les 3-variétés closes. Cet exposé sera basé sur l'article de C. Blanchet, F.Costantino, N. Geer et B. Patureau intitulé "NON SEMI-SIMPLE TQFTS, REIDEMEISTER TORSION AND KASHAEV'S INVARIANTS".
Speaker: Marco De Renzi
Title: CGP invariants of closed 3-manifolds
Abstract: I will present some of the results contained in the paper
Quantum Invariants of 3-Manifolds via Link Surgery
Presentations and Non-Semi-Simple Categories by Costantino, Geer and Patureau. We will construct two families of invariants of closed
3-manifolds indexed by a natural parameter r>1. These invariants are built out of the non-semisimple category of representations of the
unrolled quantum group U_q^H(sl_2) at a 2r-th root of unity we saw in the first talk. The secondary invariants conjecturally extend the original
Reshetikhin-Turaev invariants for the small quantum group \tilde{U}_q(sl_2).
The use of richer categories pays off as these non-semisimple invariants are strictly finer than the original semisimple ones: indeed they can
be used to recover the classification of lens spaces, which Reshetikhin-Turaev invariants could not always distinguish.
Reference: a small survey I wrote on the subject can be found here.
Speaker: Cristina Anghel
Title: The modified link invariants for representations of unrolled quantum sl(2)
Speaker: Nathan Geer
Title: Unrolled quantum sl(2)
Abstract: In this talk I will discuss a particular quantization of the Lie algebra sl(2). I will start by giving the definition of the
algebra. We will see that the finite dimensional modules over this algebra from a ribbon category, where the simple modules are indexed by the
complex numbers. This category is not semi-simple. However, I will prove that generically the tensor product of two simple modules is
semi-simple.