In this thesis we discuss amenability properties of the Banach algebra-valued continuous functions on a compact Hausdorff space X. Let A be a Banach algebra. The space of A-valued continuous functions on X, denoted by C(X,A), form a new Banach algebra. We show that C(X,A) has a bounded approximate diagonal (i.e. it is amenable) if and only if A has a bounded approximate diagonal. We also show that if A has a compactly central approximate diagonal then C(X,A) has a compact approximate diagonal. We note that, unlike C(X), in general C(X,A) is not a C*-algebra, and is no longer commutative if A is not so. Our method is inspired by a work of M. Abtahi and Y. Zhang. In addition to the above investigation, we directly construct a bounded approximate diagonal for C0(X), the Banach algebra of the closure of compactly supported continuous functions on a locally compact Hausdorff space X.

Approximation theory is a fundamental tool in order to study the representation theory of a ring R. Roughly speaking, it consists in determining suitable additive or abelian subcategories of the whole module category Mod-R with nice enough functorial properties. For example, torsion theory is a well suited incarnation of approximation theory. Of course, such an idea has been generalised to the additive setting itself, so that both Mod-R and other interesting categories related with R may be linked functorially. By the seminal work of Beilinson, Bernstein and Deligne (1982), the derived category of the ring turns out to admit useful torsion theories, called t-structures: they are pairs of full subcategories of D(R) whose intersection, called the heart, is always an abelian category. The so-called standard t-structure of D(R) has as its heart the module category Mod-R itself. Since then a lot of results devoted to the module theoretic characterisation of the hearts have been achieved, providing evidence of the usefulness of the t-structures in the representation theory of R. In 2020, following a research line promoted by many other authors, Saorin and Stovicek proved that the heart of any compactly generated t-structure is always a locally finitely presented Grothendieck categories (actually, this is true for any t-structure in a triangulated category with coproducts). Essentially, this means that the hearts of D(R) come equipped with a finiteness condition miming that one valid in Mod-R. In the present thesis we tackle the problem of characterising when the hearts of certain compactly generated t-structures of a commutative ring are even locally coherent. In this commutative context, after the works of Neeman and Alonso, Jeremias and Saorin, compactly generated t-structures turned out to be very interesting over a noetherian ring, for they are in bijection with the Thomason filtrations of the prime spectrum. In other words, they are classified by geometric objects, moreover their constituent subcategories have a precise cohomological description. However, if the ascending chain condition lacks, such classification is somehow partial, though provided by Hrbek. The crucial point is that the constituents of the t-structures have a different description w.r.t. that available in the noetherian setting, yet if one copies the latter for an arbitrary ring still obtains a t-structure, but it is not clear whether it must be compactly generated. Consequently, pursuing the study of the local coherence of the hearts given by a Thomason filtration, we ended by considering two t-structures. Our technique in order to face the lack of the ascending chain condition relies on a further approximation of the hearts by means of suitable torsion theories. The main results of the thesis are the following: we prove that for the so-called weakly bounded below Thomason filtrations the two t-structures have the same heart (therefore it is always locally finitely presented), and we show that they coincide if and only they are both compactly generated. Moreover, we achieve a complete characterisation of the local coherence for the hearts of the Thomason filtrations of finite length.

High-dimensional data such as multi-modal neuroimaging data and large-scale networks carry excessive amount of information, and can be used to test various scientific hypotheses or discover important patterns in complicated systems. While considerable efforts have been made to analyze high-dimensional data, existing approaches often rely on simple summaries which could miss important information, and many challenges on modeling complex structures in data remain unaddressed. In this proposal, we focus on analyzing structured high-dimensional data, including functional data with important local regions and network data with community structures. The first part of this dissertation concerns the detection of ``important'' regions in functional data. We propose a novel Bayesian approach that enables region selection in the functional data regression framework. The selection of regions is achieved through encouraging sparse estimation of the regression coefficient, where nonzero regions correspond to regions that are selected. To achieve sparse estimation, we adopt compactly supported and potentially over-complete basis to capture local features of the regression coefficient function, and assume a spike-slab prior to the coefficients of the bases functions. To encourage continuous shrinkage of nearby regions, we assume an Ising hyper-prior which takes into account the neighboring structure of the bases functions. This neighboring structure is represented by an undirected graph. We perform posterior sampling through Markov chain Monte Carlo algorithms. The practical performance of the proposed approach is demonstrated through simulations as well as near-infrared and sonar data. The second part of this dissertation focuses on constructing diversified portfolios using stock return data in the Center for Research in Security Prices (CRSP) database maintained by the University of Chicago. Diversification is a risk management strategy that involves mixing a variety of financial assets in a portfolio. This strategy helps reduce the overall risk of the investment and improve performance of the portfolio. To construct portfolios that effectively diversify risks, we first construct a co-movement network using the correlations between stock returns over a training time period. Correlation characterizes the synchrony among stock returns thus helps us understand whether two or multiple stocks have common risk attributes. Based on the co-movement network, we apply multiple network community detection algorithms to detect groups of stocks with common co-movement patterns. Stocks within the same community tend to be highly correlated, while stocks across different communities tend to be less correlated. A portfolio is then constructed by selecting stocks from different communities. The average return of the constructed portfolio over a testing time period is finally compared with the SandP 500 market index. Our constructed portfolios demonstrate outstanding performance during a non-crisis period (2004-2006) and good performance during a financial crisis period (2008-2010). / PHD / High dimensional data, which are composed by data points with a tremendous number of features (a.k.a. attributes, independent variables, explanatory variables), brings challenges to statistical analysis due to their “high-dimensionality” and complicated structure. In this dissertation work, I consider two types of high-dimension data. The first type is functional data in which each observation is a function. The second type is network data whose internal structure can be described as a network. I aim to detect “important” regions in functional data by using a novel statistical model, and I treat stock market data as network data to construct quality portfolios efficiently

Dans le Chapitre 1 nous étudions les groupes localement compacts lacunaires hyperboliques. Nous caractérisons les groupes ayant un cône asymptotique qui est un arbre réel et dont l'action naturelle est focale. Nous étudions également la structure des groupes lacunaires hyperboliques, et montrons que dans le cas unimodulaire les sous-groupes ne satisfont pas de loi. Nous appliquons au Chapitre 2 les résultats précédents pour résoudre le problème de l'existence de points de coupure dans un cône asymptotique dans le cas des groupes de Lie connexes. Dans le Chapitre 3 nous montrons que le groupe de Neretin est compactement présenté et donnons une borne supérieure sur sa fonction de Dehn. Nous étudions également les propriétés métriques du groupe de Neretin, et prouvons que certains sous-groupes remarquables sont quasi-isométriquement plongés. Nous étudions dans le Chapitre 4 une famille de groupes agissant sur un arbre, et dont l'action locale est prescrite par un groupe de permutations. Nous montrons entre autres que ces groupes ont la propriété (PW), et exhibons des groupes simples au sein de cette famille. Dans le Chapitre 5 nous introduisons l'éventail des relations d'un groupe de type fini, qui est l'ensemble des longueurs des relations non engendrées par des relations plus courtes. Nous établissons un lien entre la simple connexité d'un cône asymptotique et l'éventail des relations du groupe, et donnons une grande classe de groupes dont l'éventail des relations est aussi grand que possible. / In Chapter 1 we investigate the class of locally compact lacunary hyperbolic groups. We characterize locally compact groups having one asymptotic cone that is a real tree and whose natural isometric action is focal. We also study the structure of lacunary hyperbolic groups, and prove that in the unimodular case subgroups cannot satisfy a law. We apply the previous results in Chapter 2 to solve the problem of the existence of cut-points in asymptotic cones for connected Lie groups. In Chapter 3 we prove that Neretin's group is compactly presented and give an upper bound on its Dehn function. We also study metric properties of Neretin's group, and prove that some remarkable subgroups are quasi-isometrically embedded. In Chapter 4 we study a family of groups acting on a tree, and whose local action is prescribed by some permutation group. We prove among other things that these groups have property (PW), and exhibit some simple groups in this family. In Chapter 5 we introduce the relation range of a finitely generated group, which is the set of lengths of relations that are not generated by relations of smaller length. We establish a link between simple connectedness of asymptotic cones and the relation range of the group, and give a large class of groups having a relation range as large as possible.