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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 complex manifold.

The analogous problem for possibly irregular holonomic D-modules (a.k.a. the Riemann–Hilbert–Birkhoff problem) has been standing for a long time. One of the difficulties was to find a substitute target to the category of perverse sheaves. In the 80's, Deligne and Malgrange proposed a correspondence between meromorphic connections and Stokes filtered local systems on a complex curve. Recently, 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

Following Mikio Sato’s pioneering work, Masaki Kashiwara developed, starting from his thesis in the seventies, the theory of D-modules. The point of view is to study a system of linear partial differential equations by using the methods of modern algebraic geometry and homological algebra. This change of paradigm led to a new field of research in mathematics, now called Algebraic Analysis. Among his striking results obtained in collaboration or alone, let us cite the involutivity of the characteristic variety of a D-module, the index theorem, the Riemann-Hilbert correspondence for holonomic D-modules in the regular case and recently, in the irregular case. Kashiwara always kept a special interest for representation theory. One of his celebrated results is the proof, obtained in the early eighties in collaboration with Brylinski, of the Kazhdan-Lusztig conjecture (also proven independently by Beilinson and Berstein). This conjecture proposed a mysterious equality between multiplicities in the representation theory of semi-simple Lie algebras and numerical data encoding singularities of Schubert varieties. In their proof the theory of D-modules provided a geometrical bridge connecting the two sides of the equality. Motivated by the advances in the theory of integrable systems in the eighties, in particular the discovery of quantum groups, Kashiwara introduced crystal bases. These objects can be seen metaphorically as the residue of bases of representation of quantum groups when the temperature tends toward zero. There are now an important tool for the combinatorial description of representations of semi-simple Lie algebras, or more generally of Kac-Moody or even Borcherds algebras. They come together with bases called global crystal bases or canonical bases whose study has led to the current important theory of cluster algebras.

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 algebras.

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 complex manifold.

The analogous problem for possibly irregular holonomic D-modules (a.k.a. the Riemann–Hilbert–Birkhoff problem) has been standing for a long time. One of the difficulties was to find a substitute target to the category of perverse sheaves. In the 80's, Deligne and Malgrange proposed a correspondence between meromorphic connections and Stokes filtered local systems on a complex curve. Recently, 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