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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.

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.

Independently, Braverman-Kazdhan (2003) and Lafforgue (2013) introduced a new approach to Langlands’s functoriality involving Fourier transformations associated to Langlands transfert morphisms. The Langlands functoriality has an analog over finite fields, which has been proved in full generality by Lusztig. So the Fourier transformation part of the above approach makes sense in that context. In the talk, I will present some results that we have recently obtained with Emmanuel Letellier.

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 FominZelevinsky. One motivation came from the multiplicative structure of upper global basis (or dual canonical basis). We use quiver Hecke algebras to categorify cluster algebras. 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 simple modules.