Elkaïoum M. Moutuou

• PhD
• Concordia University

The brain's synaptic network, characterized by parallel connections and feedback loops, drives information flow between neurons through a large system with infinitely many degrees of freedom. This system is best modeled by the graph $C^*$-algebra of the underlying directed graph, the Toeplitz-Cuntz-Krieger algebra, which captures the diversity of p...

A fundamental paradigm in neuroscience is that cognitive functions -- such as perception, learning, memory, and locomotion -- are shaped by the brain's structural organization. However, the theoretical principles explaining how this physical architecture governs its function remain elusive. Here, we propose an algebraic quantum mechanics (AQM) fram...

Multilayer networks have permeated all areas of science as an abstraction for interdependent heterogeneous complex systems. However, describing such systems through a purely graph-theoretic formalism presupposes that the interactions that define the underlying infrastructures are only pairwise-based, a strong assumption likely leading to oversimpli...

Multilayer networks have permeated all the sciences as a powerful mathematical abstraction for interdependent heterogenous systems such as multimodal brain connectomes, transportation, ecological systems, and scientific collaboration. But describing such systems through a purely graph-theoretic formalism presupposes that the interactions that defin...

Multilayer networks have permeated all the sciences as a powerful mathematical abstraction for interdependent heterogenous complex systems such as multimodal brain connectomes, transportation, ecological systems, and scientific collaboration. But describing such systems through a purely graph-theoretic formalism presupposes that the interactions th...

We provide concrete models for generalized morphisms and Morita equivalences of topological 2-groupoids by introducing the notions of crossings and crossed extensions of groupoid crossed modules. A systematic study of these objects is elaborated and an explicit description of how they do yield a groupoid and geometric picture of weak 2-groupoid mor...

We define a new class of racks, called finitely stable racks, which, to some extent, share various flavours with Abelian groups. Characterization of finitely stable Alexander quandles is established. Further, we study twisted rack dynamical systems, construct their cross-products, and introduce representation theory of racks and quandles. We prove...

We define a new class of racks, called finitely stable racks, which, to some extent, share various flavors with Abelian groups. Characterization of finitely stable Alexander quandles is established. Further, we study twisted rack dynamical systems, construct their cross-products, and introduce representation theory of racks and quandles. We prove s...

We give a foundational account on topological racks and quandles. Specifically, we define the notions of ideals, kernels, units, and inner automorphism group in the context of topological racks. Further, we investigate topological rack modules and principal rack bundles. Central extensions of topological racks are then introduced providing a first...

We define a group $\wRBr(\cG)$ containing, in a sense, the graded complex and orthogonal Brauer groups of a locally compact groupoid $\cG$ equipped with an involution. When the involution is trivial, we show that the new group naturally provides a generalization of Donovan-Karoubi's graded orthogonal Brauer group $GBrO$. More generally, it is shown...

We develop equivariant KK-theory for locally compact groupoid actions by Morita equivalences on real and complex graded C*-algebras. Functoriality with respect to generalised morphisms and Bott periodicity are discussed. We introduce Stiefel-Whitney classes for real or complex equivariant vector bundles over locally compact groupoids to establish t...

In his 1966's paper "Ktheory and Reality", Atiyah introduced a variant of Ktheory of complex vector bundles called KRtheory, which, in some sense, is a mixture of complex Ktheory KU, real Ktheory (also called orthogonal Ktheory) KO, and Anderson's selfconjugate Ktheory KSc. The main purpose of this thesis is to generalize that theory to the noncomm...

We extend the definitions and main properties of graded extensions to the category of locally compact groupoids endowed with involutions. We introduce Real \v{C}ech cohomology, which is an equivariant-like cohomology theory suitable for the context of groupoids with involutions. The Picard group of such a groupoid is discussed and is given a cohomo...

B-fields over a groupoid with involution are defined as Real graded Dixmier-Douady bundles. We use these to introduce the Real graded Brauer group which constitutes the set of twistings for Atiyah's KR-functor in the category of locally compact groupoids with involutions. We interpret this group in terms of groupoid extensions and elements of some...

This paper is aimed at investigating links between Fell bundles over Morita equivalent groupoids and their corresponding reduced C*-algebras. Mainly, we review the notion of Fell pairs over a Morita equivalence of groupoids, and give the analogue of the Renault's Equivalence Theorem for the reduced C*-algebras of equivalent Fell systems. Eventually...

Atiyah et Singer donnent dans [1] une formule topologique de l’indice d’un op ́erateur pseudodiff ́erentiel elliptique d ́efini sur une vari ́et ́e compacte orient ́ee (sans bord). Plus pr ́ecis ́ement, pour un op ́erateur elliptique D : C∞(X, V ) −→ C∞(X,W), ou` V et W sont deux fibr ́es vectoriels sur X, l’indice de D est donn ́e par ind(D) = {Td...