Dearomatizzazione organocatalitica regio- ed enantioselettiva di sali di piridinio attivati

In this work, we present the first regio- and enantioselective organocatalytic nucleophilic dearomatization of activated N-alkyl pyridinium salts. In particular, N-benzyl pyridinium bromides bearing electron-withdrawing substituents at the C3 position of the pyridine ring were chosen as substrates. These compounds were easily obtained through an alkylation reaction between benzyl bromides and the corresponding 3-substituted pyridines. Then, a wide range of nucleophiles and organocatalysts was tested, providing the best results when indole, a thiourea derived from quinidine and 1-benzyl-3-nitropyridinum bromide were employed as the nucleophile, the catalyst and the pyridinium salt, respectively. Subsequently, the reaction conditions were optimised evaluating different bases, solvents, N-benzylic protecting groups, molar concentrations and temperatures. With the optimized condition in hand, the scope of the reaction with different substituted indoles was explored, affording the corresponding 1,4-dihydropyridines in good yields, regio- and enantio-selectivities. In addition, several experiments were carried out in order to understand the mechanism of the reaction, showing an unusual pathway involving a covalently bound intermediate formed by addition of the catalyst to the pyridine unit.

Lewis acid catalysed direct glycosylation of N-acetyl-D-glucosamine

Incorporation of the relevant monosaccharide N-Acetyl-D-glucosamine (GlcNAc) into synthetic oligosaccharides by chemical glycosylation is still a very challenging object of studies, since direct reactions are low yielding. This issue is generally ascribed to its low solubility in common solvents and to the formation of a poorly reactive oxazoline intermediate, which is typically bypassed by introducing extra synthetic steps to avoid the presence of the NHAc moiety during glycosylation. Recently, a new direct Lewis acids-catalysed GlcNAc-ylation protocol has been disclosed, with acylated donors appearing to hold potential for high yielding glycosylation reactions. This master project focused indeed on a novel synthesis of promising 1-acyl GlcNAc donors, in order to test them in direct Lewis acid catalysed glycosylation without the need of N-protecting groups. Screening of various Lewis acids and reaction conditions with these acylated donors has been carried out, in presence of reactive primary alcohols as well as more challenging carbohydrate acceptor alcohols. These experiments demonstrated that the fine tuning of the leaving group combined with a suitable metal triflate could lead to a successful reaction outcome in the direct glycosylation. Successful methodology of this kind would provide rapid access to naturally occurring N-glycan motifs, such as the highly relevant human milk oligosaccharides (HMOs).

Quantum groups and vertex models

Nella tesi vengono presentate alcune relazioni fra gruppi quantici e modelli reticolari. In particolare si associa un modello vertex a una rappresentazione di un'algebra inviluppante quantizzata affine e si mostra che, specializzando il parametro quantistico ad una radice dell'unità, si manifestano speciali simmetrie.

Differential forms on Carnot groups

The main goal of this thesis is to understand and link together some of the early works by Michel Rumin and Pierre Julg. The work is centered around the so-called Rumin complex, which is a construction in subRiemannian geometry. A Carnot manifold is a manifold endowed with a horizontal distribution. If further a metric is given, one gets a subRiemannian manifold. Such data arise in different contexts, such as: - formulation of the second principle of thermodynamics; - optimal control; - propagation of singularities for sums of squares of vector fields; - real hypersurfaces in complex manifolds; - ideal boundaries of rank one symmetric spaces; - asymptotic geometry of nilpotent groups; - modelization of human vision. Differential forms on a Carnot manifold have weights, which produces a filtered complex. In view of applications to nilpotent groups, Rumin has defined a substitute for the de Rham complex, adapted to this filtration. The presence of a filtered complex also suggests the use of the formal machinery of spectral sequences in the study of cohomology. The goal was indeed to understand the link between Rumin's operator and the differentials which appear in the various spectral sequences we have worked with: - the weight spectral sequence; - a special spectral sequence introduced by Julg and called by him Forman's spectral sequence; - Forman's spectral sequence (which turns out to be unrelated to the previous one). We will see that in general Rumin's operator depends on choices. However, in some special cases, it does not because it has an alternative interpretation as a differential in a natural spectral sequence. After defining Carnot groups and analysing their main properties, we will introduce the concept of weights of forms which will produce a splitting on the exterior differential operator d. We shall see how the Rumin complex arises from this splitting and proceed to carry out the complete computations in some key examples. From the third chapter onwards we will focus on Julg's paper, describing his new filtration and its relationship with the weight spectral sequence. We will study the connection between the spectral sequences and Rumin's complex in the n-dimensional Heisenberg group and the 7-dimensional quaternionic Heisenberg group and then generalize the result to Carnot groups using the weight filtration. Finally, we shall explain why Julg required the independence of choices in some special Rumin operators, introducing the Szego map and describing its main properties.

Higher order Sobolev Spaces and polyharmonic boundary value problems in Carnot Groups

The main task of this work is to present a concise survey on the theory of certain function spaces in the contexts of Hörmander vector fields and Carnot Groups, and to discuss briefly an application to some polyharmonic boundary value problems on Carnot Groups of step 2.

Acknowledging linguistic diversity in a multicultural society: the issue of indigenous languages in Colombia.

Nowadays, modern society is gradually becoming multicultural. However, only in the last few years awareness on its importance has been raised. In the case of Colombia, multiculturalism has existed since the pre-Columbian period and today there are more than 80 ethnic groups and 65 indigenous languages in the country. The aim of this work is to illustrate the status of indigenous languages in Colombia and to enlighten about the importance of recognizing, protecting and strengthening the use of these native languages. Subsequent to this, it will be point out that linguistic diversity should be considered a resource and not a barrier to achieve unity in diversity. Finally, ethno-education will be presented as an adequate educational program that may guarantee an equal linguistic representation in the country.