Résumés

Nathan Geer : Modified traces: from algebra to topology

In the last few years, C. Blanchet, F. Costantino, M. De Renzi, B. Patureau, N. Reshetikhin, V. Turaev and myself (in various collaborations) have developed a theory of renormalized quantum invariants of links and 3-manifolds which lead to TQFTs.  These talk will start out by giving an overview of this work.  During the rest of my time, I will describe one of the main tools used to construct these renormalized invariants: the so called modified traces (or m-traces).    These m-traces are a generalization of the categorical trace in a pivotal linear category. They were first introduced by J. Kujawa, B. Patureau and myself.  Recently, m-traces have been studied by a host of others including A. Beliakova, J. Berger, C. Blanchet, A. Gainutdinov, P. Ha and I. Runkel.  As I will explain there are several examples in representation theory where the usual trace are zero, but these m-traces are non-zero.  Such examples include the representation theory of the Lie algebra sl(2) over a field of positive characteristic, Lie superalgebras over the complex numbers and quantum groups at a root of unity.  

Matthew Hogancamp : Khovanov-Rozansky homology and Hilbert schemes of points

 In my lectures I will discuss a special deformation of Khovanov-Rozansky homology constructed last year in joint work with E. Gorsky.  Our deformation appears to be especially well adapted for approaching questions relating Khovanov-Rozansky homology (both the deformed and original theories) with Hilbert schemes of points in C^2.  Most importantly, I will discuss the ``link splitting’’ properties of our invariant (which are inspired from the Batson-Seed picture) and how the full-twist braid mediates the connection with Hilbert schemes.  So far this is the most exciting application of the deformed KR homology theory.  One of the novel features of our deformed theory is that for braids (rather than closed link diagrams) the invariant is a certain ``curved deformation’’ of the Rouquier complex associated to that braid.  That is to say, one must begin to consider ``curved’’ complexes, in which d^2 is no longer zero.  I plan to explain the role of these curved complexes in link homology and, through examples, illustrate how to work with them in practice.

François Costantino : Invariants of 3-manifolds from pivotal categories

After reviewing some basic options on finite dimensional Hopf algebras and pivotal categories, I will recall how different invariants of 3-manifolds can be build using these rich algebraic objects. Then I will outline a recent construction (joint with N. Geer, B. Patureau-Mirand and V. Turaev), which shows how to use unimodular (non necessarily semi-simple) pivotal categories with modified traces, and in particular unimodular Hopf algebras, to produce invariants of 3-manifolds from Heegaard decompositions. If time permits I will also outline how a more recent construction (work in progress) allowing to extend this to general (i.e. non unimodular) pivotal categories.

Abel Lacabanne : Categorification of Z-modular data associated with complex reflection groups

The aim of this talk is to introduce the notion of a Z-modular datum and some examples associated with families of unipotent characters of complex reflection groups. The categorical counterpart of a N-modular datum is the notion of modular categories. We will explain how slightly degenerate pivotal braided fusion categories naturally gives rise to a Z-modular datum, and how supercategories also appear and should be thought as the categorical counterpart of Z-modular data.
We will finally give a categorification of a Z-modular datum associated with the complex refleciton group G(d,1,n(n+1)/2) by constructing a braided fusion category out of representations of the Drinfeld double of the positive part of Uq(sl(n+1))$, where q is a 2d-th root of unity.

Louis-Hadrien Robert : A foamy categorification of the Alexander polynomial

(joint with Emmanuel Wagner)
The Alexander polynomial, has been categorified using symplectic geometry: this is Heegaard--Floer homology. In this talk I will speak about an another approach to categorify this polynomial. The idea is to see the Alexander polynomial as a member of the big family of quantum invariants. I'll show how one can use foams to construct this new categorification (conjecturally isomorphic to the Heegaard--Floer).