We define and study the fundamental pro-finite 2-groupoid of varieties X defined over a field k. This is a higher algebraic invariant of a scheme X, analogous to the higher fundamental path 2-groupoids as defined for topological spaces. This invariant is related to previously defined invariants, for example the absolute Galois group of a field, and Grothendieck’s étale fundamental group. The special case of Brauer-Severi varieties is considered, in which case a “sections conjecture” type theorem is proved. It is shown that a Brauer-Severi variety X has a rational point if and only if its étale fundamental 2-groupoid has a special sort of section.

It can be deduced from the Burnside Basis Theorem that if G is a finite p-group with d(G)=r then given any generating set A for G there exists a subset of A of size r that generates G. We have denoted this property B. A group is said to have the basis property if all subgroups have property B. This thesis is a study into the nature of these two properties. Note all groups are finite unless stated otherwise. We begin this thesis by providing examples of groups with and without property B and several results on the structure of groups with property B, showing that under certain conditions property B is inherited by quotients. This culminates with a result which shows that groups with property B that can be expressed as direct products are exactly those arising from the Burnside Basis Theorem. We also seek to create a class of groups which have property B. We provide a method for constructing groups with property B and trivial Frattini subgroup using finite fields. We then classify all groups G where the quotient of G by the Frattini subgroup is isomorphic to this construction. We finally note that groups arising from this construction do not in general have the basis property. Finally we look at groups with the basis property. We prove that groups with the basis property are soluble and consist only of elements of prime-power order. We then exploit the classification of all such groups by Higman to provide a complete classification of groups with the basis property.

Nous étudions la clôture algébrique et définissable dans les groupes libres. Les résultats principaux peuvent être résumés comme suit. Nous montrons un résultat de constructibilité des groupes hyperboliques sans torsion au-dessus de la clôture algébrique d'un sous-ensemble engendrant un groupe non abélien. Nous avons cherché à comprendre la place qu'occupe la clôture algébrique acl_G(A) dans certaines décompositions de G. Nous avons étudié la possibilité de la généralisation de la méthode de Bestvina-Paulin dans d'autres directions, en considérant les groupes de type fini qui agissent d'une manière acylindrique (au sens de Bowditch) sur les graphes hyperboliques. Enfin, nous avons étudié les relations qui existent entre les différentes notions de clôture algébrique et entre la clôture algébrique et la clôture définissable.

Many applications lead to signals with nonnegative function values. Understanding the structure of the spaces of nonnegative signals is therefore of interest in many different areas. Hence, constructing effective representation spaces with suitable metrics and natural transformations is an important research topic. In this thesis, we present our investigations of the structure of spaces of nonnegative signals and illustrate the results with applications in the fields of multispectral color science and content-based image retrieval. The infinite-dimensional Hilbert space of nonnegative signals is conical and convex. These two properties are preserved under linear projections onto lower dimensional spaces. The conical nature of these coordinate vector spaces suggests the use of hyperbolic geometry. The special case of three-dimensional hyperbolic geometry leads to the application of the SU(1,1) or SO 2,1) groups. We introduce a new framework to investigate nonnegative signals. We use PCA-based coordinates and apply group theoretical tools to investigate sequences of signal coordinate vectors. We describe these sequences with oneparameter subgroups of SU(1,1) and show how to compute the one-parameter subgroup of SU(1,1) from a given set of nonnegative signals. In our experiments we investigate the following signal sequences: (i) blackbody radiation spectra; (ii) sequences of daylight/twilight spectra measured in Norrk¨oping, Sweden and in Granada, Spain; (iii) spectra generated by the SMARTS2 simulation program; and (iv) sequences of image histograms. The results show that important properties of these sequences can be modeled in this framework. We illustrate the usefulness with examples where we derive illumination invariants and introduce an efficient visualization implementation. Content-Based Image Retrieval (CBIR) is another topic of the thesis. In such retrieval systems, images are first characterized by descriptor vectors. Retrieval is then based on these content-based descriptors. Selection of contentbased descriptors and defining suitable metrics are the core of any CBIR system. We introduce new descriptors derived by using group theoretical tools. We exploit the symmetry structure of the space of image patches and use the group theoretical methods to derive low-level image filters in a very general framework. The derived filters are simple and can be used for multispectral images and images defined on different sampling grids. These group theoretical filters are then used to derive content-based descriptors, which will be used in a real implementation of a CBIR.

The objects of study in this thesis are automorphism groups of free and right-angled Artin groups. Right-angled Artin groups are defined by a presentation where the only relations are commutators of the generating elements. When there are no relations the right-angled-Artin group is a free group and if we take all possible relations we have a free abelian group. We show that if no finite index subgroup of a group $G$ contains a normal subgroup that maps onto $mathbb{Z}$, then every homomorphism from $G$ to the outer automorphism group of a free group has finite image. The above criterion is satisfied by SL$_m(mathbb{Z})$ for $m geq 3$ and, more generally, all irreducible lattices in higher-rank, semisimple Lie groups with finite centre. Given a right-angled Artin group $A_Gamma$ we find an integer $n$, which may be easily read off from the presentation of $A_G$, such that if $m geq 3$ then SL$_m(mathbb{Z})$ is a subgroup of the outer automorphism group of $A_Gamma$ if and only if $m leq n$. More generally, we find criteria to prevent a group from having a homomorphism to the outer automorphism group of $A_Gamma$ with infinite image, and apply this to a large number of irreducible lattices as above. We study the subgroup $IA(A_Gamma)$ of $Aut(A_Gamma)$ that acts trivially on the abelianisation of $A_Gamma$. We show that $IA(A_Gamma)$ is residually torsion-free nilpotent and describe its abelianisation. This is complemented by a survey of previous results concerning the lower central series of $A_Gamma$. One of the commonly used generating sets of $Aut(F_n)$ is the set of Whitehead automorphisms. We describe a geometric method for decomposing an element of $Aut(F_n)$ as a product of Whitehead automorphisms via Stallings' folds. We finish with a brief discussion of the action of $Out(F_n)$ on Culler and Vogtmann's Outer Space. In particular we describe translation lengths of elements with regards to the `non-symmetric Lipschitz metric' on Outer Space.

Pour l’apprentissage automatisé de données régulières comme des images ou des signaux sonores, les réseaux convolutifs profonds s’imposent comme le modèle de deep learning le plus performant. En revanche, lorsque les jeux de données sont irréguliers (par example : réseaux de capteurs, de citations, IRMs), ces réseaux ne peuvent pas être utilisés. Dans cette thèse, nous développons une théorie algébrique permettant de définir des convolutions sur des domaines irréguliers, à l’aide d’actions de groupe (ou, plus généralement, de groupoïde) agissant sur les sommets d’un graphe, et possédant des propriétés liées aux arrêtes. A l’aide de ces convolutions, nous proposons des extensions des réseaux convolutifs à des structures de graphes. Nos recherches nous conduisent à proposer une formulation générique de la propagation entre deux couches de neurones que nous appelons la contraction neurale. De cette formule, nous dérivons plusieurs nouveaux modèles de réseaux de neurones, applicables sur des domaines irréguliers, et qui font preuve de résultats au même niveau que l’état de l’art voire meilleurs pour certains. / Convolutional neural networks have proven to be the deep learning model that performs best on regularly structured datasets like images or sounds. However, they cannot be applied on datasets with an irregular structure (e.g. sensor networks, citation networks, MRIs). In this thesis, we develop an algebraic theory of convolutions on irregular domains. We construct a family of convolutions that are based on group actions (or, more generally, groupoid actions) that acts on the vertex domain and that have properties that depend on the edges. With the help of these convolutions, we propose extensions of convolutional neural netowrks to graph domains. Our researches lead us to propose a generic formulation of the propagation between layers, that we call the neural contraction. From this formulation, we derive many novel neural network models that can be applied on irregular domains. Through benchmarks and experiments, we show that they attain state-of-the-art performances, and beat them in some cases.

Deux groupes sont dits élémentairement équivalents s'ils satisfont les mêmes énoncés du premier ordre dans le langage des groupes. Aux environs de l'année 1945, Tarski posa la question suivante, connue désormais comme le problème de Tarski : les groupes libres non abéliens sont-ils élémentairement équivalents ? Une réponse positive à cette fameuse question fut apportée plus d'un demi-siècle plus tard par Sela, et en parallèle par Kharlampovich et Myasnikov, comme le point d'orgue de deux volumineuses séries de travaux. Dans la foulée, Sela généralisa aux groupes hyperboliques sans torsion, dont les groupes libres sont des représentants emblématiques, les méthodes de nature géométrique qu'il avait précédemment introduites à l'occasion de son travail sur le problème de Tarski. Les résultats rassemblés ici s'inscrivent dans cette lignée, en s'en démarquant toutefois dans la mesure où ils traitent des théories du premier ordre des groupes hyperboliques en présence de torsion. Dans un premier chapitre, on démontre, entre autres, que tout groupe de type fini qui est élémentairement équivalent à un groupe hyperbolique est lui-même hyperbolique. On démontre ensuite que les groupes virtuellement libres sont presque homogènes, ce qui signifie que deux éléments qui sont indiscernables du point de vue de la logique du premier ordre sont dans la même orbite sous l'action du groupes des automorphismes du groupe ambiant, à une indétermination finie près. Enfin, on donne une classification complète des groupes virtuellement libres de type fini du point de l'équivalence élémentaire à deux quantificateurs. / Two groups are said to be elementarily equivalent if they satisfy the same first-order sentences in the language of groups, that is the same mathematical statements whose variables are only interpreted as elements of a group. Around 1945, Tarski asked the following question : are non-abelian free groups elementarily equivalent? An affirmative answer to this famous Tarski's problem was given in 2006 by Sela and independently by Kharlampovich and Myasnikov, as the culmination of two voluminous series of papers. Then, Sela gave a classification of all finitely generated groups that are elementarily equivalent to a given torsion-free hyperbolic group. The results contained in the present thesis fall into this context and deal with first-order theories of hyperbolic groups with torsion. In the first chapter, we prove that any finitely generated group that is elementarily equivalent to a hyperbolic group is itself a hyperbolic group. Then, we prove that virtually free groups are almost homogeneous, meaning that elements are almost determined up to automorphism by their type, i.e. the first-order formulas they satisfy. In the last chapter, we give a complete classification of finitely generated virtually free groups up to elementary equivalence with two quantifiers.

Within undergraduate mathematics education, there are few validated instruments designed for large-scale usage. The Group Concept Inventory (GCI) was created as an instrument to evaluate student conceptions related to introductory group theory topics. The inventory was created in three phases: domain analysis, question creation, and field-testing. The domain analysis phase included using an expert consensus protocol to arrive at the topics to be assessed, analyzing curriculum, and reviewing literature. From this analysis, items were created, evaluated, and field-tested. First, 383 students answered open-ended versions of the question set. The questions were converted to multiple-choice format from these responses and disseminated to an additional 476 students over two rounds. Through follow-up interviews intended for validation, and test analysis processes, the questions were refined to best target conceptions and strengthen validity measures. The GCI consists of seventeen questions, each targeting a different concept in introductory group theory. The results from this study are broken into three papers. The first paper reports on the methodology for creating the GCI with the goal of providing a model for building valid concept inventories. The second paper provides replication results and critiques of previous studies by leveraging three GCI questions (on cyclic groups, subgroups, and isomorphism) that have been adapted from prior studies. The final paper introduces the GCI for use by instructors and mathematics departments with emphasis on how it can be leveraged to investigate their students' understanding of group theory concepts. Through careful creation and extensive field-testing, the GCI has been shown to be a meaningful instrument with powerful ability to explore student understanding around group theory concepts at the large-scale.

<p>Let $\mathbf{G}$ be a connected reductive algebraic group defined over an algebraic closure of the finite field of prime order $p>0$, which we assume to be good for $\mathbf{G}$. We denote by $F : \mathbf{G} \to \mathbf{G}$ a Frobenius endomorphism of $\mathbf{G}$ and by $G$ the corresponding $\mathbb{F}_q$-rational structure. If $\operatorname{Irr}(G)$ denotes the set of ordinary irreducible characters of $G$ then by work of Lusztig and Geck we have a well defined map $\Phi_{\mathbf{G}} : \operatorname{Irr}(G) \to \{F\text{-stable unipotent conjugacy classes of }\mathbf{G}\}$ where $\Phi_{\mathbf{G}}(\chi)$ is the unipotent support of $\chi$.</p> <p>Lusztig has given a classification of the irreducible characters of $G$ and obtained their degrees. In particular he has shown that for each $\chi \in \operatorname{Irr}(G)$ there exists an integer $n_{\chi}$ such that $n_{\chi}\cdot\chi(1)$ is a monic polynomial in $q$. Given a unipotent class $\mathcal{O}$ of $\mathbf{G}$ with representative $u \in \mathbf{G}$ we may define $A_{\mathbf{G}}(u)$ to be the finite quotient group $C_{\mathbf{G}}(u)/C_{\mathbf{G}}(u)^{\circ}$. If the centre $Z(\mathbf{G})$ is connected and $\mathbf{G}/Z(\mathbf{G})$ is simple then Lusztig and H\'zard have independently shown that for each $F$-stable unipotent class $\mathcal$ of $\mathbf$ there exists $\chi \in \operatorname(G)$ such that $\Phi_(\chi)=\mathcal$ and $n_ = |A_(u)|$, (in particular the map $\Phi_$ is surjective).</p> <p>The main result of this thesis extends this result to the case where $\mathbf$ is any simple algebraic group, (hence removing the assumption that $Z(\mathbf)$ is connected). In particular if $\mathbf$ is simple we show that for each $F$-stable unipotent class $\mathcal$ of $\mathbf$ there exists $\chi \in \operatorname(G)$ such that $\Phi_(\chi) = \mathcal$ and $n_ = |A_(u)^F|$ where $u \in \mathcal^F$ is a well-chosen representative. We then apply this result to prove, (for most simple groups), a conjecture of Kawanaka's on generalised Gelfand--Graev representations (GGGRs). Namely that the GGGRs of $G$ form a $\mathbf{Z}$-basis for the $\mathbf{Z}$-module of all unipotently supported class functions of $G$. Finally we obtain an expression for a certain fourth root of unity associated to GGGRs in the case where $\mathbf{G}$ is a symplectic or special orthogonal group.</p>

In any group G, we may extend the definition of the conjugacy class of an element to the conjugacy class of a k-tuple, for a positive integer k. When k = 2, we are forming the conjugacy classes of ordered pairs, when k = 3, we are forming the conjugacy classes of ordered triples, etc. In this report we explore a generalized question which Professor B. Doug Park has posed (for k = 2). For an arbitrary k, is it true that: (G has finitely many k-conjugacy classes) implies (G is finite)? Supposing to the contrary that there exists an infinite group G which has finitely many k-conjugacy classes for all k = 1, 2, 3, ..., we present some preliminary analysis of the properties that G must have. We then investigate known classes of groups having some of these properties: universal locally finite groups, existentially closed groups, and Engel groups.