### Le 22-01-2018 : Theodosios Douvropoulos (IRIF,Paris VII - ERC CombiTop)

**Geometric techniques in Coxeter-Catalan combinatorics***A problem that goes back to Hurwitz and the 19th century is to enumerate (reduced) factorizations of the long cycle (12\cdots n)\in S_n into factors from prescribed conjugacy classes. As it happens, and this is a common theme in combinatorics, this question of the symmetric group has a meaningful analog for the other reflection groups as well: The long cycle is replaced by a Coxeter element c.<br /> <br /> Bessis gave a beautiful geometric interpretation of such factorizations by using a variant of the Lyashko-Looijenga (LL) map, a finite morphism coming from Singularity theory. In that setting, there is a natural bijective correspondence ("Trivialization Theorem") between points in a generic fiber of the LL-map, and reduced reflection factorizations of c. This was fundamental in Bessis' dual braid presentation of the generalized braid groups B(W), but relies on a numerological coincidence that is still only proven case-by-case!<br /> <br /> We review the important geometric properties of the LL map and present new results obtained by further analysis of its local behavior. These include enumerating wider classes of factorizations, as well as counting factorizations with prescribed symmetries (in fact, we prove various cyclic sieving phenomena). We also suggest a uniform approach towards the proof of the Trivialization Theorem.*### Le 29-01-2018 : Hidekazu Furusho (Nagoya University)

**A Betti counterpart of the harmonic coproduct (I)***In earlier work, we proved that Racinet's double shuffle group is the stabilizer of the harmonic coproduct defined on a subalgebra of the free algebra over two generators relative to a certain action on this algebra. This leads to the construction of a family of new coproducts on the same algebra, depending on a scalar parameter, which are related with one another<br /> by scaling transformations. The double shuffle torsor can then be described as the set of elements taking<br /> the harmonic coproduct to the new coproduct. We identify the new coproduct with an explicit coproduct of a<br /> suitable subalgebra of the algebra of the free group with two generators. The proof relies on an interpretation of the<br /> harmonic coproduct in terms of infinitesimal braid Lie algebras, which is implicit in the unpublished work of Deligne<br /> and Terasoma from 2005.*### Le 05-02-2018 : Benjamin Enriquez (Université de Strasbourg)

**A Betti counterpart of the harmonic coproduct (II)***In earlier work, we proved that Racinet's double shuffle group is the stabilizer of the harmonic coproduct defined on a subalgebra of the free algebra over two generators relative to a certain action on this algebra. This leads to the construction of a family of new coproducts on the same algebra, depending on a scalar parameter, which are related with one another<br /> by scaling transformations. The double shuffle torsor can then be described as the set of elements taking<br /> the harmonic coproduct to the new coproduct. We identify the new coproduct with an explicit coproduct of a<br /> suitable subalgebra of the algebra of the free group with two generators. The proof relies on an interpretation of the<br /> harmonic coproduct in terms of infinitesimal braid Lie algebras, which is implicit in the unpublished work of Deligne<br /> and Terasoma from 2005.*### Le 12-02-2018 : Louis-Hadrien Robert (University of Hamburg)

**tba**### Le 19-02-2018 : Jitendra Bajpai (University of Goettingen)

**Arithmeticity and Thinness of hypergeometric groups***The monodromy groups of hypergeometric differential equations of type $_nF_{n-1}$ are often called hypergeometric groups. These are subgroups of $GL_n$ . Recently, Arithmeticity and Thinness of these groups have caught a lot of attention. In the talk, a gentle introduction and recent progress to the theory of hypergeometric groups will be presented.*### Le 09-04-2018 : Daniel Juteau (Universite Paris 7)

**tba**### Le 16-04-2018 : Alexandre Bouayad (University of Cambridge)

**tba**### Le 14-05-2018 : Clément Dupont (Université de Montpellier)

**tba**### Le 28-05-2018 : Simon Riche (Université Clermont Auvergne )

**TBA**