A right R-module M is called a Utumi Module (U-module) if, whenever A and B are isomorphic submodules of M with A ∩ B = 0, there exist two summands K and T of M such that A is an essential submodule of K, B is an essential submodule of T and K ⊕ T is a direct summand of M . The class of U -modules is a simultaneous and strict generalization of three fundamental classes of modules; namely the quasi-continuous, the square-free and the automorphism-invariant modules.

# Seminars

TBA

Dynamical systems with an underlying network structure are a subject of great interest as they arise frequently in applications and exhibit many staggering phenomena some of which resemble those in equivariant dynamics. We introduce a theory developed by Rink and Sanders that connects a class of network dynamical systems - namely homogeneous coupled cell systems - to equivariant dynamical systems. The symmetries, however, are generalized in the sense that they do not necessarily form a group but more general structures such as monoids or semigroups.

TBA

We examine the relation between a heteroclinic network as a flow-invariant set and directed graphs of possible connections between nodes. In particular, we show that there are robust realizations of a large class of transitive directed graphs that are not complete (i.e. not all unstable manifolds of nodes are included) but almost complete (i.e. complete up to a set of zero measure in the unstable manifold) and equable (i.e. all sets of connections from a node have the same dimension).

Pseudovarieties of ordered semigroups have been introduced as a refined algebraic classifying tool for regular languages in the sense of Eilenberg's Correspondence Theorem. Forgetting the order, each such pseudovariety generates a pseudovariety of semigroups. A natural question is which pseudovarieties of semigroups arise in this way from non-selfdual pseudovarieties of ordered semigroups which, for the purpose of the talk, we call orderable.

Although one could naively expect that random Lorentz gases are easier to investigate than deterministic periodic ones, this seems not to be the case as essentially no results are available in the non periodic case. In this talk, I will present some general ideas towards studying random Lorentz gases and I will show how to apply them for a class of deterministic walks in random environments wit hone-dimensional uniformly expanding local dynamics. This is a joint work with Carlangelo Liverani.

The modular class is a generalization of the divergence of vector fields to other geometric structures such as Poisson manifolds and Lie algebroids. The modular class of a regular foliation involves a volume form on the conormal bundle and the associated Bott connection. It is a closed one form along the leaves. The vanishing of this modular class implies that there exists a volume form which is invariant along the leaves. In the singular case, the above definition can not be summoned since the conormal bundle may not be even well defined.

Links of Gorenstein toric isolated singularities are good toric contact

manifolds with zero first Chern class. In this talk I will present some

results relating contact and singularity invariants in this particular

toric context. Namely,

(i) I will explain why the contact mean Euler characteristic is equal

to the Euler characteristic of any crepant toric smooth resolution of

the singularity (joint work with Leonardo Macarini).

(ii) I will discuss applications of contact invariants of Lens spaces

At the beginning of this century, Fomin and Zelevinsky invented a new class of algebras called cluster algebras motivated by total positivity in algebraic groups and canonical bases in quantum groups. Since their introduction, cluster algebras have found application in a diverse variety of settings which include Poisson geometry, Teichmüller theory, tropical geometry, algebraic combinatorics and last not least the representation theory of quivers and finite dimensional algebras.