Dans cette thèse, nous nous concentrons sur l’étude des algèbres de Hopf de type I, à savoir de type sélection/quotient. Nous présentons une structure d’algèbre de Hopf sur l’espace vectoriel engendré par les mots tassés avec du coproduit sélection/quotient. C’est un algèbre libre sur ses mots irreductible. Nous montrons que la série de Hilbert de cette algèbre de Hopf. Nous donnons une nouvelle preuve de l’universalité du polynôme de Tutte pour les matroïdes.Cette preuve utilise des caractères appropriés de l’algèbre de Hopf des matroïdes introduite par Schmitt (1994). Nous montrons que ces caractères sont des solutions des équations différentielles du même type que les équations différentielles utilisées pour décrire le flux du groupe de renormalisation en théorie quantique de champs. Cette approche nous permet aussi de démontrer,d’une manière différente, une formule de convolution du polynôme de Tutte des matroïdes,formule publiée par Kook, Reiner et Stanton (1999) et par Etienne et Las Vergnas (1998). Dans la dernière partie, nous définissons une algèbre de Hopf non-commutative de graphes. Lanon-commutativité du produit est obtenue grâce à des étiquettes entières distinctes associées aux arrêtes du graphe. Cette idée est inspirée de certaines techniques analytiques utilisées en renormalisation en théories quantiques des champs. Nous définissons ensuite une structure d’algèbre de Hopf, avec un coproduit basé sur une règle de type sélection/quotient, et nous démontrons la coassociativité de ce coproduit. Nous analysons finalement la structure de quadri-cogèbre et les structures codendriformes associées. / In this thesis, we focus on the study of Hopf algebras of type I, namely the selection/quotient.We study the new Hopf algebra structure on the vector space spanned by packed words. Weshow that this algebra is free on its irreducible packed words. We also compute the Hilbertseries of this Hopf algebra.We provide a new way to obtain the universality of the Tutte polynomial for matroids. Thisproof uses appropriate characters of Hopf algebra of matroids, algebra introduced by Schmitt(1994). We show that these Hopf algebra characters are solutions of some differential equationswhich are of the same type as the differential equations used to describe the renormalizationgroup flow in quantum field theory. This approach allows us to also prove, in a different way, amatroid Tutte polynomial convolution formula published by Kook, Reiner and Stanton (1999)and by Etienne and Las Vergnas (1998).We define a non-commutative Hopf algebra of graphs. The non-commutativity of the productis obtained thanks to some discrete labels associated to the graph edges. This idea is inspiredfrom certain analytic techniques used in quantum field theory renormalization. We then definea Hopf algebra structure, with a coproduct based on a selection/quotient rule, and prove thecoassociativity of this coproduct. We analyze the associated quadri-coalgebra and codendrifromstructures.

In this thesis we generalize the problem of phase retrieval of vector to that of multi-vector. The identification of the multi-vector is done up to some special classes of isometries in the space. We give some upper and lower estimates on the minimal number of multi-linear operators needed for the retrieval. The results are preliminary and far from sharp.

We develop a theory of unbounded derived categories of quasi-coherent sheaves on algebraic stacks. In particular, we show that these categories are compactly generated by perfect complexes for stacks that either have finite stabilizers or are local quotient stacks. We also extend Toën and Antieau–Gepner’s results on derived Azumaya algebras and compact generation of sheaves on linear categories from derived schemes to derived Deligne–Mumford stacks. These are all consequences of our main theorem: compact generation of a presheaf of triangulated categories on an algebraic stack is local for the quasi-finite flat topology.

Let G be a simple simply connected algebraic group defined over an algebraically closed field K and V an irreducible module defined over K on which G acts. Let E denote the set of vectors in V which are eigenvectors for some non-central semisimple element of G and some eigenvalue in K*. We prove, with a short list of possible exceptions, that the dimension of Ē is strictly less than the dimension of V provided dim V > dim G + 2 and that there is equality otherwise. In particular, by considering only the eigenvalue 1, it follows that the closure of the union of fixed point spaces of non-central semisimple elements has dimension strictly less than the dimension of V provided dim V > dim G + 2, with a short list of possible exceptions. In the majority of cases we consider modules for which dim V > dim G + 2 where we perform an analysis of weights. In many of these cases we prove that, for any non-central semisimple element and any eigenvalue, the codimension of the eigenspace exceeds dim G. In more difficult cases, when dim V is only slightly larger than dim G + 2, we subdivide the analysis according to the type of the centraliser of the semisimple element. Here we prove for each type a slightly weaker inequality which still suffices to establish the main result. Finally, for the relatively few modules satisfying dim V ≤ dim G + 2, an immediate observation yields the result for dim V < dim B where B is a Borel subgroup of G, while in other cases we argue directly.

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This project's purpose was to understand the workings of a new theorem introduced in a professional paper on the cutwidth of meshes and then use this knowledge to apply it to the search for the cyclic cutwidth of the n-cube.

This project centers around finding specific examples of quintic polynomials that were and were not solvable. This helped to devise a method for finding examples of solvable and unsolvable quintics.