Hyper-bent Boolean functions were introduced in 2001 by Youssef and Gong (and initially proposed by Golomb and Gong in 1999 as a component of S-boxes) to ensure the security of symmetric cryptosystems but no cryptographic attack has been identified till 2016.

Hyper-bent functions have properties still stronger than the well-known bent functions which were already studied by Dillon and

# Seminars

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.

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.

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.

This talk is about describing dynamics on primitive network objects and finding conditions that allow a good "reductive" description of network dynamics. We will give a number of examples when feedback is added. The examples range from surprising synchrony, an example of the "bullwhip" effect and a remarkable layered network mixing synchrony and chaotic dynamics. Some of this work is part of a joint project with Ana Dias and Manuela Aguiar.

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).

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.

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.

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