In 1990, G. Lusztig constructed a new basis of the positive part of the enveloping algebra of a simple Lie algebra, which he
called the canonical basis. Its definition relied on the theory of quantum groups and the geometry of quiver varieties.
In 1993, Berenstein and Zelevinsky formulated a conjecture on the dual of the canonical basis, that might lead to a more
combinatorial description of this remarkable but rather mysterious basis.
In 2001, Fomin and Zelevinsky came up with a more precise conjecture in terms of a new class of rings
called cluster algebras. The notion of a cluster algebra is elementary and combinatorial, and there are many
examples, among which the dual of the positive part of the enveloping algebra of a simple Lie algebra.
Fomin and Zelevinsky conjectured that the dual canonical basis contains all cluster monomials.
This conjecture was proved in 2015 by Kang-Kashiwara-Kim-Oh, using categorification methods based on Khovanov-Lauda-Rouquier
The minicourse will try to give an accessible introduction to the Fomin-Zelevinsky conjecture, whose proof will
be presented by M. Kashiwara.
Hilbert's twenty-first problem (also known as the Riemann-Hilbert problem) asks
for the existence of linear ordinary differential equations with
prescribed regular singularities and monodromy. In higher dimensions, Deligne formulated it as a correspondence between regular meromorphic flat connections and local systems.
In the early eighties, Kashiwara generalized it to a correspondence
between regular holonomic D-modules and perverse sheaves on a
The analogous problem for possibly irregular holonomic D-modules
(a.k.a. the Riemann–Hilbert–Birkhoff problem) has been standing for a
One of the difficulties was to find a substitute target to the
category of perverse sheaves.
In the eighties, Deligne and Malgrange proposed a correspondence
between meromorphic connections and Stokes filtered local systems on a
Recentely, Kashiwara and the speaker solved the problem for general
holonomic D-modules in any dimension.
The construction of the target category is based on the theory of
ind-sheaves by Kashiwara-Schapira and uses Tamarkin’s work on symplectic topology. Among the main ingredients of the proof is the
description of the structure of flat meromorphic connections due to
Mochizuki and Kedlaya.
In the early 90's, Kashiwara defined crystals as limits of bases of representations of quantum groups, and developed their theory in various situations. Subsequent works by Kashiwara, Lusztig, Littelmann, and Berenstein-Fomin-Zelevinsky elucidated the structure of these combinatorial objects. It was later discovered that Kashiwara's crystals also describe certain geometrical situations. For instance, as observed by Braverman-Gaitsgory, they occur in the geometric Satake correspondence. This theory, due to Lusztig, Ginzburg, Beilinson-Drinfeld, Mirković-Vilonen, and Ngô-Polo provides a construction of representations of a reductive group G from the geometry of the affine Grassmannian of the Langlands dual of G. We will study properties of bases that naturally arise in this context. This is joint work with S. Gaussent and P. Littelmann.
In the local classification of differential equations of one complex variable, torsors under a certain sheaf of algebraic groups (the Stokes sheaf) play a central role.
On the other hand, Deligne defined in positive characteristic a notion of skeletons for l-adic local systems on a smooth variety, constructed an algebraic variety parametrizing skeletons and raised the question wether every skeleton comes from an actual l-adic local system.
After some recollections on the Stokes phenomenon, we will explain how to use a variant of Deligne’s skeleton conjecture in characteristic 0 to prove the existence of an algebraic variety parametrizing Stokes torsors. We will show how the geometry of this moduli can be used to prove new finiteness results on differential equations.
Starting from the Riemann - resp. Birkhoff - existence theorem for linear differential equations of one complex variable, I will motivate on the example of hypergeometric - resp. confluent hypergeometric - equations the variant of Hodge theory called 'irregular Hodge theory', originally introduced by Deligne in 1984. I will also explain the interest of this theory in relation with mirror symmetry of Fano manifolds.
The notion of cluster algebras was introduced
by Fomin-Zelevinsky. One motivation came from
the multiplicative structure of
upper global basis (or dual canonical basis).
We use quiver Hecke algebras to categorify
Namely, the category of modules over quiver Hecke algebras has
a structure of monoidal category.
Its Grothendieck group has a cluster algebra structure.
Simple modules correspond to the upper global basis,
and cluster monomials correspond to