Conference titles and abstracts

Patrick Dehornoy ; Braid ordering: history and connection with knots

We shall review some of the many aspects of the standard braid order, including its origins connected with some bizarre questions of set theory. A special emphasis will be put on the connections with knot theory and recent promising developments in this direction.

Bernard Leclerc; On the dual canonical basis (old and new)

Around 1991, Lusztig, motivated by earlier works of Gelfand and Zelevinsky, introduced a "canonical basis" of the coordinate ring C[N] of the maximal unipotent subgroup N of a simple algebraic group G. This basis was in fact the specialization as q tends to 1 of the dual canonical basis of the quantum enveloping algebra U_q(n) (where n = Lie(N)). Berenstein and Zelevinsky started studying the multiplicative properties of the dual canonical basis of U_q(n), and proposed an exciting conjecture in 1993. In the case of G = SL(n), this conjecture leads to some interesting q-commuting properties of quantum minors, studied in a joint paper by Zelevinsky and myself. As a by-product, we conjectured the purity of a certain simplicial complex. This conjecture has now two different proofs: by Danilov, Karzanov, and Koshevoy (2009), and by Oh, Postnikov, and Speyer (2011). After several efforts for proving particular cases of the conjecture (by Caldero, Reineke, and Nazarov, Thibon and myself), I have given counter-examples to it in 2003. Still, the conjecture proved to be very useful as one of the main incentives for the introduction by Fomin and Zelevinsky of the new concept of cluster algebra. In the talk, I'll try to review this long story, and if time allows, to mention some recent results and conjectures about the relations between dual canonical bases and cluster algebras.