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Persistent Topology is an innovative way of matching topology and geometry, and it proves to be an effective mathematical tool in shape analysis. In order to express its full potential for applications, it has to interface with the typical environment of Computer Science: It must be possible to deal with a finite sampling of the object of interest, and with combinatorial representations of it. Following that idea, the main result claims that it is possible to construct a relation between the persistent Betti numbers (PBNs; also called rank invariant) of a compact, Riemannian submanifold X of R^m and the ones of an approximation U of X itself, where U is generated by a ball covering centered in the points of the sampling. Moreover we can state a further result in which, this time, we relate X with a finite simplicial complex S generated, thanks to a particular construction, by the sampling points. To be more precise, strict inequalities hold only in "blind strips'', i.e narrow areas around the discontinuity sets of the PBNs of U (or S). Out of the blind strips, the values of the PBNs of the original object, of the ball covering of it, and of the simplicial complex coincide, respectively.

The aim of this dissertation is to improve the knowledge of knots and links in lens spaces. If the lens space L(p,q) is defined as a 3-ball with suitable boundary identifications, then a link in L(p,q) can be represented by a disk diagram, i.e. a regular projection of the link on a disk. In this contest, we obtain a complete finite set of Reidemeister-type moves establishing equivalence, up to ambient isotopy. Moreover, the connections of this new diagram with both grid and band diagrams for links in lens spaces are shown. A Wirtinger-type presentation for the group of the link and a diagrammatic method giving the first homology group are described. A class of twisted Alexander polynomials for links in lens spaces is computed, showing its correlation with Reidemeister torsion. One of the most important geometric invariants of links in lens spaces is the lift in 3-sphere of a link L in L(p,q), that is the counterimage of L under the universal covering of L(p,q). Starting from the disk diagram of the link, we obtain a diagram of the lift in the 3-sphere. Using this construction it is possible to find different knots and links in L(p,q) having equivalent lifts, hence we cannot distinguish different links in lens spaces only from their lift. The two final chapters investigate whether several existing invariants for links in lens spaces are essential, i.e. whether they may assume different values on links with equivalent lift. Namely, we consider the fundamental quandle, the group of the link, the twisted Alexander polynomials, the Kauffman Bracket Skein Module and an HOMFLY-PT-type invariant.

Let C be a compact Riemann surface. Let us consider a finite group acting on CxC, having some elements that exchange the factors, and assume that the subgroup of those elements that do not exchange the factors acts freely. We call the quotient a Semi-isogenous Mixed Surface. In this work we investigate these surfaces and we explain how their geometry is encoded in the group. Based on this, we present an algorithm to classify the Semi-isogenous Mixed Surfaces with given geometric genus, irregularity and self-intersection of the canonical class. In particular we give the classification of Semi-isogenous Mixed Surfaces with K^2>0 and holomorphic Euler-Poincaré characteristic equal to 1, where new examples of minimal surfaces of general type appear. Minimality of Semi-isogenous Mixed Surfaces is discussed using two different approaches. The first one involves the study of the bicanonical system of such surfaces: we prove that we can relate the dimension of its first cohomology group to the rank of a linear map that involves only curves. The second approach exploits Hodge index theorem to bound the number of exceptional curves that live on a Semi-isogenous Mixed Surface.

In this thesis I've explored the theory of quaternionic slice regular functions. More precisely I've studied some rigidity properties, some differential issues and an application in complex differential geometry. This application concerns the constructions and classifications of orthogonal complex structures on open domains of the four dimensional euclidean space.

In this thesis we define and study the mixed quasi-étale surfaces. In particularwe classify all the mixed quasi-étale surfaces whose minimal resolution of the singularities is a regular surface with p_g=0 and K^2>0. It is a well known fact that each Riemann surface with p_g=0 is isomorphic to P^1. At the end of XIX century M. Noether conjectured that an analogous statement holds for the surfaces: in modern words, he conjectured that every smooth projective surface with p_g=q=0 be rational. The first counterexample to this conjecture is due to F. Enriques (1869). He constructed the so called Enriques surfaces. The Enriques-Kodaira classification divides compact complex surfaces in four main classes according to their Kodaira dimension k: -oo, 0, 1, 2. A surface is said to be of general type if k=2. Nowadays this class is much less understood than the other three. The Enriques surfaces have k=0. The first examples of surfaces of general type with p_g=0 have been constructed in the 30's by L. Campedelli e L. Godeaux. The idea of Godeaux to construct surfaces was to consider the quotient of simpler surfaces by the free action of a finite group. In this spirit, Beauville proposed a simple construction of surfaces of general type, considering the quotient of a product of two curves C_1 and C_2 by the free action of a finite group G. Moreover he gave an explicit example with p_g=q=0 considering the quotient of two Fermat curves of degree 5 in P^2. There is no hope at the moment to achieve a classification of the whole class of the surfaces of general type. Since for a surface in this class the Euler characteristic of the structure sheaf \chi is strictly positive, one could hope that a classification of the boundary case \chi=1 is more affordable. Some progresses in this direction have been done in the last years through the work of many authors, but this (a priori small) case has proved to be very challenging, and we are still very far from a classification of it. At the same time, this class of surfaces, and in particular the subclass of the surfaces with p_g=0 contains some of the most interesting surfaces of general type. If S is a surface of general type with \chi=1, which means p_g=q, then p_g = q < 5, and if p_g=q=4, then S is birational to the product of curves of genus 2. The surfaces with p_g = q = 3 are completely classified. The cases p_g = q < 3 are still far from being classified. Generalizing the Beauville example, we can consider the quotient (C_1 x C_2)/G, where the C_i are Riemann surfaces of genus at least two, and G is a finite group. There are two cases: the mixed case where the action of G exchanges the two factors (and then C_1 = C_2); and the unmixed case where G acts diagonally. Many authors studied the surfaces birational to a quotient of a product of two curves, mainly in the case of surfaces of general type with \chi=1. In all these works the authors work either in the unmixed case or in the mixed case under the assumption that the group acts freely. The main purpose of this thesis is to extend the results and the strategies of the above mentioned papers in the non free mixed case. Let C be a Riemann surface of genus at least 2, let G be a finite group that acts on C x C with a mixed action, i.e. there exists an element in G that exchanges the two factors. Let G^0 be the index two subgroup of the elements that do not exchange the factors. We say that X=(C x C)/G is a mixed quasi-étale surface if the quotient map C x C -> (C x C)/G has finite branch locus. We present an algorithm to construct regular surfaces as the minimal resolution of the singularities of mixed quasi-étale surfaces. We give a complete classification of the regular surfaces with p_g=0 and K^2>0 that arise in this way. Moreover we show a way to compute the fundamental group of these surfaces and we apply it to the surfaces we construct. Some of our construction are more interesting than others. We have constructed two numerical Campedelli surfaces (K^2 = 2) with topological fundamental group Z/4Z. Two of our constructions realize surfaces whose topological type was not present in the literature before. We also have three examples of Q-homology projective planes, and two of them realize new examples of Q-homology projective planes.

ABSTRACT: Roughly speaking, the moduli space of higher spin curves parametrizes equivalence classes of pairs (C, L) where C is a smooth genus g algebraic curve and L is a line bundle on it whose r-th tensor power is isomorphic to the canonical bundle of the curve. The aim of the talk is to discuss important geometrical properties of these spaces under different points of view: one possible compactification together with the description of the rational Picard group, their birational geometry in some low genus cases and their relation with some special locus inside the classical moduli spaces of curves.