Stochastic rewriting systems evolving over graph-like structures are a versatile modeling paradigm that covers in particular biochemical reaction systems. In fact, to date rewriting-based frameworks such as the Kappa platform [1] are amongst the very few known approaches to faithfully encode the enormous complexity in both molecular structures and reactions exhibited by biochemical reaction...

After a short introduction to the functorial approach to logical proofs and programs initiated by Lambek in the late 1960s, based on the notion of free cartesian closed category, we will describe a recent convergence with the notion of ribbon category introduced in 1990 by Reshetikhin and Turaev in their functorial study of quantum groups and knot invariants. The connection between proof...

The lambda calculus was invented by Church in the late 1920s, as part of an ambitious project to build a foundation for mathematics around the concept of function. Although his original system turned out to be logically inconsistent, Church was able to extract from it two separate usable systems, with a typed calculus for logic and an untyped calculus for pure computation. Through the work of...

In this seminar we study the constructive loop vertex expansion for stable matrix models with (single trace) interactions of arbitrarily high even order in the Hermitian and real symmetric cases. It relies on a new and simpler method which can also be applied in the previously treated complex case. We prove analyticity in the coupling constant of the free energy for such models in a domain...

The action of subgroups on a product of symmetric groups allows one to enumerate different families of graphs. In particular, bipartite ribbon graphs (with at most edges) enumerate as the orbits of the adjoint action on two copies of the symmetric group (of order n!). These graphs form a basis of an algebra, which is also a Hilbert space for a certain sesquilinear form. Acting on this Hilbert...

One of the many connections between Grassmannians and combinatorics is cohomological: The cohomology ring of a Grassmannian ${\rm Gr}(k,n)$ is a quotient of the ring $S$ of symmetric polynomials in $k$ variables. More precisely, it is the quotient of $S$ by the ideal generated by the k consecutive complete homogeneous symmetric polynomials $h_{n-k}, h_{n-k+1}, \ldots , h_n$. We deform this...

Reflection Equation Algebra is one of the Quantum matrix algebras, associated with a given Hecke symmetry, i.e. a braiding of Hecke type. I plan to explain how to introduce analogs of Hermitian Matrix Models arising from these algebras. Some other applications of the Reflection Equation Algebras will be discussed.

Convex sets can be defined over ordered fields with a non-archimedean valuation. Then, tropical convex sets arise as images by the valuation of non-archimedean convex sets. The tropicalization of polyhedra and spectrahedra are of special in- terest, since they can be described in terms of deterministic and stochastic games with mean payoff. In that way, one gets a correspondence between...

We prove that, for a tropical rational map if for any point the convex hull of Jacobian matrices at smooth points in a neighborhood of the point does not contain singular matrices then the map is an isomorphism. We also show that a tropical polynomial map on the plane is an isomorphism if all the Jacobians have the same sign (positive or negative). In addition, for a tropical rational map we...

We present some bialgebras and their monoid of characters. We entend, to the case of some rings, the well-known theorem (in the case when the scalars form a field) about linear independence of characters. Examples of algebraic independence of subfamilies and identites derived from their groups (or monoids) of characters are *provided. In this framework, we detail the study of one-parameter...

We review simultaneously the essential steps to establish the equation bridging the algebraic structures of converging polyzetas, via their noncommutative generating series put in factorised form MRS. This equation then allows us to describe polynomial relations, homogenous in weight, among these polyzetas, via an identification of local coordinates.

I will talk on homology calculations for the higher cyclic Hochschild complex and on combinatorial description of Lie structure on highly noncommutative words.

It is based on the texts: Arxiv:1906.07134 (J. ALgebra, 2020), preprints IHES M/19/14.

After a brief introductory account, I’ll explain how a quasi-shuffle compatible definition (by no means unique) of multiple zeta values can be given for integer arguments of any sign, through Connes-Kreimer’s Hopf-algebraic renormalization. Finally, I’ll introduce the Ohno-Okuda-Zudilin model of q-analogues for multiple zeta values, describe the algebraic structure which governs it, and...

A generalization of Dirac’s equation is presented, incorporating the three-valued colour variable in a way which makes it intertwine with the Lorentz transformations. We show how the Lorentz-Poincaré group must be extended to accomodate both $SU(3)$ and the Lorentz transformations. Both symmetries become intertwined, so that the system can be diagonalized only after the sixth...