For a natural number k, Cooke introduced k-stage euclidean rings as a generalization of classical Euclidean rings. His setting was entirely commutative. Later Leutbecher introduced them in a noncommutative settings but his aim was also studying commutative rings.

# Seminars

Finite and profinite monoids have proved to be a powerful tool in the study of regular languages. In particular, understanding the interplay between the topological and algebraic structures of these objects has been crucial. However, many of the classes of languages that are of interest to study are not regular, and so, those theories no longer apply. Boolean spaces with biactions of monoids were recently proposed as suitable objects to handle classes of (not necessarily regular) languages.

In this introduction for geometers and topologists, we explain the role that central extensions of L-infinity algebras, a concept we will define, plays in physics. This connection first appeared with the work of physicists D’Auria and Fré in 1982, but is beautifully captured by the “brane bouquet” of Fiorenza, Sati and Schreiber which shows how physical objects such as “strings”, “D-branes” and “M- branes” can be classified by taking successive central extensions of an especially simple L-infinity algebra called the “supertranslation algebra”.

We investigate the dynamics near a heteroclinic cycle between a hyperbolic equilibrium E and a hyperbolic periodic solution P such that both connections are of codimension one. Such a cycle can be seen as the center of a two-parameter bifurcation scenario and, depending on properties of the transition maps, we find different types of (chaotic) shift dynamics near the cycle. Through our study we further explore the bifurcation diagrams previously outlined by others.

In this work an extension of the spectral Lanczos' tau method to systems of nonlinear integro-differential equations is proposed. This extension includes (i) linearization coefficients of orthogonal polynomials products issued from nonlinear terms and (ii) recursive relations to implement matrix inversion whenever a polynomial change of basis is required and (iii) orthogonal polynomial evaluations directly on the orthogonal basis.

We will begin by presenting some history of the Krull-Schmidt-Remak Theorem. From groups, we will pass to modules over a ring R, introducing some direct-sum decompositions that follow a special pattern. We will consider invariants that also appear in factorisation of polynomials. Then we will go back from (right) R-modules to groups. Here the category that appears in a natural way is that of G-groups, which substitutes the category of right R-modules. In this category, Remak's result has a natural interpretation.

Mumford introduced in the 1960ies an algebraic approach to the construction of (almost) canonical bases of sections of ample line bundles on abelian varieties that permitted him to construct quasi-projective moduli spaces. His construction was later re-interpreted by Welters as a flat projective connection before being generalized by Hitchin to the non-abelian setting.

One of the key problems that has been considered on a pseudovariety V of finite monoids consists in deciding whether a given finite system of equations with rational constraints has a solution in every monoid from V. An instance of this problem is the separation problem, which has received considerable attention lately due to its role in investigations on the Straubing-Thérien hierarchy of star-free languages.

Let *S* be a smooth projective surface over **C** and* B *a smooth projective curve. A fibration *f* : *S* → *B* is a surjective morphism such that the general fibre is a smooth connected curve.

This talk will focus on some properties of fibrations with general fibre of genus ≥ 2, discussing in particular the existence and number of singular fibres on a fibration.

In 2008, Gehrke, Grigorieff, and Pin proposed Stone duality as a means for studying Boolean algebras of (non-necessarily regular) languages. They showed how that tool could be understood as a generalization of some key concepts found in the study of varieties of regular languages, such as the syntactic monoid of a language. In this talk, I will present the main points of this approach.