Lie algebras of maximal class (or filiform Lie algebras) are the Lie-theoretic analogue of pro-p-groups of maximal class. In particular, they are 2-generated. If one further assumes that the algebras are graded over the positive integers, then over a field of characteristic p it has been shown that a classification is possible provided one generator has degree 1 and the other has either degree 1 or 2. In this thesis I give a classification of graded Lie algebras of maximal class with generators of degree 1 and p, respectively.

We consider smooth complex projective varieties X which are rationally connected by rational curves of degree d with respect to a fixed ample line bundle L on X, and we focus our attention on conic connected manifolds (d=2) and on rationally cubic connected manifolds (d=3). Conic connected manifolds were studied by Ionescu and Russo; they considered conic connected manifolds embedded in projective space (i.e. L is very ample) and they proved a classification theorem for these manifolds. We show that their classification result holds true assuming just the ampleness of L. Moreover we give a different proof of a theorem due to Kachi and Sato; this result characterizes a special subclass of conic connected manifolds. As already said before, we study also rationally cubic connected manifolds. We prove that if rationally cubic connected manifolds are covered by “lines†, i.e. by curves of degree 1 with respect to L, then the Picard number of X is equal to or less than 3; moreover we show that if the Picard number is equal to 3 then there is a covering family of “lines†whose numerical class spans a negative extremal ray of the Kleiman-Mori cone of X. Unfortunately, for rationally cubic connected manifolds which don't admit a covering family of “lines†there isn't an upper bound on the Picard number. However we prove that if we consider rationally cubic connected manifolds which are not covered by “lines†but are Fano then up to a few exceptions in dimension 2 also the Picard number of these manifolds is equal to or less than 3. In particular, supposing that the dimension of X is greater than 2, we show that either the Picard number is equal to or less than 2 or X is the blow up of projective space along two disjoint subvarieties that are linear subspaces or quadrics.

Given a linear code, it is important both to identify fast decoding algorithms and to estimate the rst terms of its weight distribution. Ecient decoding algorithms allow the exploitation of the code in practical situations, while the knowledge of the number of small-weight codewords allows to estimate its decoding performance. For ane-variety codes and its subclass formed by Hermitian codes, both problems are as yet unsolved. We investigate both and provide some solutions for special cases of interest. The rst problem is faced with use of the theory of Gröbner bases for zero-dimensional ideals. The second problem deals in particular with small-weight codewords of high-rate Hermitian codes. We determine them by studying some geometrical properties of the Hermitian curve, specically the intersection number of the curve with lines and parabolas.

Channel coding is the branch of Information Theory which studies the noise that can occur in data transmitted through a channel. Algebraic Coding Theory is the part of Channel Coding which studies the possibility to detect and correct errors using algebraic and geometric techniques. Nowadays, the best performing linear codes are known to be mostly algebraic geometry codes, also named Goppa codes, which arise from an algebraic curve over a finite field, by the pioneering construction due to V. D. Goppa. The best choices for curves on which Goppa's construction and its variants may provide codes with good parameters are those with many rational points, especially maximal curves attaining the Hasse-Weil upper bound for the number of rational points compared with the genus of the curve. Unfortunately, maximal curves are difficult to find. The best known examples of maximal curves are the Hermitian curve, the Ree curve, the Suzuki curve, the GK curve and the GGS curve. In the present thesis, we construct and investigate algebraic geometry codes (shortly AG codes), their parameters and automorphism groups.

Public key cryptography allows two or more users to communicate in a secure way on an insecure channel, using two different keys: a public key, which has the function to encrypt the messages, and a private key, employed in the decryption of the ciphertext. Because of the importance of these keys,their generation is a sensible issue and it is often based on an underlying mathematical problem, which is considered hard to be solved. Among these difficult problems, the Integer Factorization Problem (IFP) is one of the most famous: given a composite integer number, recovering its factors is commonly believed to be hard (worst-case complexity). In this thesis, after a brief explaination of the developments on integer factorization and a description of the General Number Field Sieve (GNFS), we will present different strategies to face this well-known problem of Number Theory. First, we will employ elementary remarks on modular arithmetic to produce a formula that describes the remainders of a given integer, starting from just three monotonic remainders and we will link this result to the behaviour of a second-degree interpolating polynomial. Secondly, we will show an attempt to improve GNFS, considering two linearly disjoint quadratic fields and study the relation between first-degree prime ideals. Finally, we will characterize the elements used in GNFS through some systems having integer solutions, that can be found using Groebner Bases.

In this dissertation I face three main arguments. 1) Classification of Fano-Mori contractions. 2) Chern numbers on smooth threefolds. 3) Pluricanonical systems.

Historically, the study of modules over finite dimensional algebras has started with the study of the ones with finite dimension. This is sufficient when dealing with a finite dimensional algebra of finite representation type, where there are only finitely many indecomposable modules of finite length. Indecomposable modules of infinite length occur when dealing with algebras of infinite representation type and the study of pure-injective modules over a finite dimensional algebra is crucial for the problem of describing infinite dimensional modules. In this talk, we consider a specific class of finite dimensional algebras of infinite representation type, called "tubular algebras". Pure-injective modules over tubular algebra have been partially classified by Angeleri Hügel and Kussin, in 2016, and we want to give a contribution to the classification of the ones of "irrational slope". In this talk, first, via a derived equivalence, we move to a more geometrical framework, ie. we work in the category of quasi-coherent sheaves over a tubular curve, and we approach our classification problem from the point of view of tilting/cotilting theory. More precisely, we consider specific torsion pairs cogenerated by infinite dimensional cotilting sheaves and we study the Happel-Reiten-Smalø heart of the corresponding t-structure in the derived category. These hearts are locally coherent Grothendieck categories and, in these categories, the pure-injective sheaves over the tubular curve become injective objects. In order to study injective objects in a Grothendieck category is fundamental the classification of the simple objects. In the seminar, we use some techniques coming from continued fractions and universal extensions to provide a method to construct an infinite dimensional sheaf of a prescribed irrational slope that becomes simple in the Grothendieck category given as the heart of a precise t-structure.

Galois spaces, that is affine and projective spaces of dimension N ≥ 2 defined over a finite (Galois) field F_q, are well known to be rich of nice geometric, combinatorial and group theoretic properties that have also found wide and relevant applications in several branches of Combinatorics, as well as in more practical areas, notably Coding Theory and Cryptography. The systematic study of Galois spaces was initiated in the late 1950’s by the pioneering work of B. Segre [59]. The trilogy [34, 36, 42] covers the general theory of Galois spaces including the study of objects which are linked to linear codes. Typical such objects are plane arcs and their generalizations - especially caps and arcs in higher dimensions - whose code theoretic counterparts are distinguished types of error-correcting and covering linear codes. Their investigation has received a great stimulus from Coding Theory, especially in the last decades; see the survey papers [40, 41]. An important issue in this context is to ask for explicit constructions of small complete arcs and small complete caps. A cap in a Galois space is a set of points no three of which are collinear. A cap is complete if its secants (lines through two points of the set) cover the whole space. An arc in a Galois space of dimension N is a set of points no N+1 of which lying on the same hyperplane. In analogy with caps, an arc which is maximal with respect to set-theoretical inclusion is said to be complete. Also, arcs coincide with caps in Galois planes. From these geometrical objects, there arise linear codes which turn out to have very good covering properties, provided that the size of the set is small with respect to the dimension N and the order q of the ambient space. For the size t(AG(N,q)) of the smallest complete caps in a Galois affine space AG(N,q) of dimension N over F_q, the trivial lower bound is √2q^{N−1/2}. General constructions of complete caps whose size is close to this lower bound are only known for q even and N odd, see [16, 19, 29, 52]. When N is even, complete caps of size of the same order of magnitude as cq^{N/2}, with c a constant independent of q, are known to exist for both the odd and the even order case, see [16, 18, 28, 29, 31] (see also Section 2.2 and the references therein). Whereas, few constructions of small complete arcs in Galois spaces of dimension N>2 are known. In [65, 66, 67], small complete arcs having many points in common with the normal rational curve are investigated (see Section 4.2.3 for comparisons with our results). In this thesis, new infinite families of complete arcs and caps in higher dimensional spaces are constructed from algebraic curves defined over a finite field. In most cases, no smallest complete caps/arcs were previously known in the literature. Although caps and arcs are rather combinatorial objects, constructions and proofs sometimes heavily rely on concepts and results from Algebraic Geometry in positive characteristic.

The aim of this thesis is to begin the study of graded Lie algebras of maximal class over a field of odd characteristic, which are generated by two elements of different weights.

Biofilm communities host complex biogeochemical processes and play a role in the formation of many carbonate rocks by influencing both carbonate precipitation and dissolution. In this study, the biogeochemistry of microbial mats from a hypersaline pond and biofilm from a coral reef are described using SEM, microelectrode profiling, Biolog, fatty acid methyl ester (FAME) and carbon nitrogen analysis. Results show that the microbial mats are distinctly layered, having an oxic upper portion and an H2S-rich lower portion. The most significant conclusions are that the mats have exceptionally high TOC values and display significant differences in microbial communities present, both between layers and between cores. Additionally, organic matter is abundant in microbial mat and biofilm samples, but evidence of precipitation is surprisingly lacking.