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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.

Geometria analítica a Batxillerat: un enfocament didàctic contextualitzat i amb eines TIC desenvolupa, implementa i analitza un enfocament didàctic innovador per a l'ensenyament de la geometria analítica al primer curs de Batxillerat. Els fonaments teòrics dels treball prenen com a referents els textos de diversos autors agrupats en tres blocs temàtics: contextualització, matematització i modelització (els autors de referència són principalment: Biembengut i Hein; Chamoso i Rawson; English; Filloy; Freudenthal; Niss; Niss, Lesh i Lee; Niss i Blum; Peralta; Treffers; Van den Heuvel-Panhuizen; Van Reeuwijk). El treball es posiciona en el sentit que la contextualització, la matematització i la modelització són diferents facetes d'una única realitat didàctica en la qual els alumnes realitzen activitats contextualitzades (en l'entorn del programari interactiu GeoGebra), les quals els indueixen a matematitzar i a construir models matemàtics senzills. També hi és present un complet recorregut per la història de la geometria analítica i la seva didàctica motivat per la consideració que la perspectiva històrica és imprescindible: la història de les matemàtiques, i en concret la de la geometria analítica, evidencia que els continguts i la seva presentació emergeixen a partir de les necessitats humanes en contextos concrets, i que l'estructuració formal es produeix amb posterioritat. El plantejament didàctic del treball consisteix en una seqüència que s'inicia amb activitats contextualitzades que indueixen la matematització en els alumnes, en l'entorn del programari GeoGebra, i que es completa amb una posterior formalització dels continguts. És un plantejament "de baix a dalt", ja que, a diferència de la metodologia tradicional, no comença amb la presentació formal i perfectament estructurada dels continguts per passar a continuació als exercicis d'aplicació, sinó que la formalització arriba després que la matematització induïda per les activitats contextualitzades hagi preparat el terreny per a la fixació formal dels continguts. La implementació queda completament integrada (no superposada) en el currículum i en la programació didàctica del centre educatiu de secundària on es duu a terme. Essent globalment innovador, l'estudi supera la dicotomia innovador - tradicional i la dicotomia constructivisme - empirisme perquè, encara que és innovador i potencia el procés de descobriment personal de l'alumne en una primera fase, conté també en una segona fase elements vàlids i útils de la metodologia tradicional. L'anàlisi de la matematització que realitzen els alumnes té una importància absolutament central en la investigació. L'estudi analitza amb instruments quantitatius i també qualitatius el resultats grupals i els resultats individuals dels alumnes en el procés de matematització, i realitza una classificació dels alumnes en diferents categories. També analitza les valoracions subjectives dels alumnes sobre la realització de les activitats amb GeoGebra, tant de la perspectiva grupal com des de la perspectiva individual. Aquestes anàlisis permeten constatar que, en comparació amb una metodologia tradicional, millora el rendiment en la matematització, alhora que augmenta l'autoconsciència, la motivació i la implicació dels alumnes en el procés d'aprenentatge. A partir de l'anàlisi detallat de la matematització i les valoracions subjectives de l'alumnat, el treball realitza una sèrie de consideracions de les quals emergeix una síntesi interpretativa. És la sistematització del plantejament didàctic, un cop analitzat i interpretat, sota la denominació de "plataforma de matematització". Això comporta que a partir dels resultats del treball d'implementació el plantejament esdevé també una proposta didàctica que pot ser extensible a altres unitats didàctiques i fins i tot a altres nivells educatius. / Geometria anal'tica a Batxillerat: un enfocament did ctic contextualitzat i amb eines TIC (Analytic geometry in Batxillerat: a contextualized learning approach with ICT tools) develops, implements and analyzes an innovative approach for the teaching of analytic geometry in the first year of post-compulsory education in Catalonia (Spain). The theoretical references come from texts by various authors and are grouped into three areas: contextualization, mathematization and mathematical modeling. The main authors of reference are: Biembengut and Hein; Chamoso and Rawson; English; Filloy; Freudenthal; Niss; Niss, Lesh and Lee; Niss and Blum; Peralta; Treffers; Van den Heuvel-Panhuizen; Van Reeuwijk. Under the terms of the study, contextualization, mathematization and modeling are different aspects of a unique educational reality in which students carry out contextualized activities. All this takes place in the realm of the interactive software GeoGebra. The activities induce mathematization and construction of simple mathematical models. There is also a complete journey through the history of analytic geometry and its teaching. The history of mathematics, and particularly the history of analytic geometry, shows that contents and presentations emerge from human needs in specific contexts, and that formal structures occur later. The didactic approach consists of a sequence that begins with contextualized activities that induce mathematization, and is completed with a further formalization of the contents. Unlike the traditional methodology, the sequence does not start with formal presentations and perfectly structured contents. The completion comes after the mathematization induced by contextualized activities has paved the way for the formal establishment of the contents. The implementation is fully integrated (not overlapped) in the curriculum and the educational programs for secondary education in Catalonia. As a global innovation, the study overcomes the innovative vs. traditional dichotomy and the constructivism vs. empiricism dichotomy because, although it is innovative and it encourages the discovery process to students at an early stage, in a second phase it also contains valid and helpful elements from the traditional methodology. The analysis of the mathematization made by the students has a central role in this research. The study uses quantitative and qualitative instruments to analyze group results and individual results of the students in the process of mathematization and performs a classification of students into different categories. It also examines the students' subjective assessments on the implementation of activities with GeoGebra. These analysis show that, compared to the traditional methodology, there is an improved performance in mathematization, as well as an increased self awareness, motivation and involvement of students in the learning process. From the detailed analysis of mathematization and the subjective ratings of the students, the work makes a number of considerations. An interpretive synthesis emerges, which is the systematization of the teaching approach, once analyzed and interpreted, under the name "mathematization platform". This means that through the results of the implementation the work also becomes a didactic approach that can be extended to other teaching units and even to other educational levels.

In this thesis we study the geography of irregular complex projective (or compact Kähler) varieties, paying special attention to the existence of fibrations. The thesis is divided into two parts. In the first one we consider irregular varieties of arbitrary dimension, looking for bounds for the Hodge numbers in the absence of fibrations. In first place, by truncating the BGG complex of the variety (an object recently introduced by Lazarsfeld and Popa), we get lower bounds on the partial Euler characteristics. In order to improve these first results, we define the higher-rank derivative complexes (generalizing the derivative complex introduced by Green and Lazarsfeld). We study their exactness by means of the Eagon-Northcott complexes, and we obtain some inequalities between the Hodge numbers of varieties admitting some kind of subspaces of 1-forms (¿non-degenerate subspaces¿). In the case of subvarieties of Abelian varieties, the existence of non-degenerate subspaces of any dimension allows us to obtain better inequalities than in the general case. In the case of h^(2,0), a different method gives a much better result. In fact, the bound is much stronger, and the only hypothesis needed is the non-existence of higher irrational pencils (a priori, less restrictive than the existence of non-degenerate subspaces). To close, using the Grassmannian BGG complex (a generalization of the BGG complex that aggregates all the higher-rank derivative complexes) and computing the Chern classes of its last cokernel, we recover the same bound for h^(2,0) using the general results mentioned in the previous paragraph. In the second part, the scope is restricted to surfaces fibred over a curve. We look for upper bounds for the relative irregularity in terms of properties of the general fibre, in the spirit of the inequality obtained by Xiao for non-isotrivial fibrations over a rational curve. Xiao conjectured the same inequality to hold for fibrations over any base, but Pirola found a counterexample. After that, a corrected conjecture was proposed. The result obtained in this thesis is a bound depending on the genus and the Clifford index of a general fibre, which coincides with the corrected conjecture in the case of maximal Clifford index. We have used several techniques in our proof. On the one hand, the ¿adjoint images¿ play a crucial role. The adjoint images were introduced by Collino and Pirola to study infinitesimal deformations of smooth curves, and generalized later by Pirola and Zucconi to higher-dimensional varieties. In this thesis we construct the ¿global adjoint map¿, which allows to find subspaces with vanishing adjoint image assuming that the kernel of the infinitesimal deformation has dimension (at least) half the genus of the cruve. More generally, the global adjoint map can also be defined for infinitesimal deformations of irregular varieties of any dimension, and allows to find numerical conditions that guarantee the existence of subspaces with vanishing adjoint. On the other hand, we have extended to arbitrary (one-dimensional) families of curves some well-stablished concepts of infinitesimal deformations, related with the bicanonical embedding of the curve. As the global adjoint map, some of these constructions can also be extended to families of irregular varieties of arbitrary dimension. Finally, all these previous constructions lead to a structural result for fibrations supported on a relatively rigid divisor. With this result we can treat some cases of the conjecture of Xiao. The remaining cases are solved using an inequality for the rank of an infinitesimal deformation in terms of a supporting divisor (its degree and the dimension of its complete linear series). This inequality, which we reprove, is originally due to Ginensky.

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.

Geometric constraint solving is a central topic in many fields such as parametric solid modeling, computer-aided design or chemical molecular docking. A geometric constraint problem consists of a set geometric objects on which a set of constraints is defined. Solving the geometric constraint problem means finding a placement for the geometric elements with respect to each other such that the set of constraints holds. Clearly, the primary goal of geometric constraint solving is to define rigid shapes. However an interesting problem arises when we ask whether allowing parameter constraint values to change with time makes sense. The answer is in the positive. Assuming a continuous change in the variant parameters, the result of the geometric constraint solving with variant parameters would result in the generation of families of different shapes built on top of the same geometric elements but governed by a fixed set of constraints. Considering the problem where several parameters change simultaneously would be a great accomplishment. However the potential combinatorial complexity make us to consider problems with just one variant parameter. Elaborating on work from other authors, we develop a new algorithm based on a new tool we have called h-graphs that properly solves the geometric constraint solving problem with one variant parameter. We offer a complete proof for the soundness of the approach which was missing in the original work. Dynamic geometry is a computer-based technology developed to teach geometry at secondary school, which provides the users with tools to define geometric constructions along with interaction tools such as drag-and-drop. The goal of the system is to show in the user's screen how the geometry changes in real time as the user interacts with the system. It is argued that this kind of interaction fosters students interest in experimenting and checking their ideas. The most important drawback of dynamic geometry is that it is the user who must know how the geometric problem is actually solved. Based on the fact that current user-computer interaction technology basically allows the user to drag just one geometric element at a time, we have developed a new dynamic geometry approach based on two ideas: 1) the underlying problem is just a geometric constraint problem with one variant parameter, which can be different for each drag-and-drop operation, and, 2) the burden of solving the geometric problem is left to the geometric constraint solver. Two classic and interesting problems in many computational models are the reachability and the tracing problems. Reachability consists in deciding whether a certain state of the system can be reached from a given initial state following a set of allowed transformations. This problem is paramount in many fields such as robotics, path finding, path planing, Petri Nets, etc. When translated to dynamic geometry two specific problems arise: 1) when intersecting geometric elements were at least one of them has degree two or higher, the solution is not unique and, 2) for given values assigned to constraint parameters, it may well be the case that the geometric problem is not realizable. For example computing the intersection of two parallel lines. Within our geometric constraint-based dynamic geometry system we have developed an specific approach that solves both the reachability and the tracing problems by properly applying tools from dynamic systems theory. Finally we consider Henneberg graphs, Laman graphs and tree-decomposable graphs which are fundamental tools in geometric constraint solving and its applications. We study which relationships can be established between them and show the conditions under which Henneberg constructions preserve graph tree-decomposability. Then we develop an algorithm to automatically generate tree-decomposable Laman graphs of a given order using Henneberg construction steps.

El principal objectiu d'aquesta tesi és l'estudi de l'evolució mitjançant el flux de Ricci de superfícies amb singularitats de tipus cònic. Un segon objectiu, sorgit de les tècniques que utilitzem, és l'estudi de famílies de solitons del flux de Ricci en dimensió 2 i 3. El flux de Ricci és una equació d'evolució per a varietats Riemannianes, introduïda per R. Hamilton el 1982. És des dels avenços assolits per G. Perelman amb aquesta tècnica el 2002 quan el flux de Ricci s'ha establert com a una disciplina pròpia, aixecant un gran interès per la comunitat. Aquesta tesi conté quatre resultats originals. El primer resultat és una classificació exhaustiva dels solitons en superfícies llises i còniques. Amb aquesta classificació completem els precedents trobats per Hamilton, Chow i Wu entre d'altres, i obtenim descripcions explícites de tots els solitons en dimensió 2. El segon resultat és una Geometrització de les superfícies còniques mitjançant el flux de Ricci. Aquest resultat, que utilitza el primer resultat ja esmentat, estén la teoria de Hamilton al cas singular. Aquest és el resultat més extens, per al qual fem servir i desenvolupem tècniques tant d'anàlisi i EDPs com de geometria de comparació . El tercer resultat és l'existència d'un flux de Ricci que elimina les singularitats còniques . Això exposa clarament la no unicitat de solucions al flux, en analogia als fluxos de Ricci amb cusps de P. Topping . El quart resultat és la construcció d'un nou solitó gradient expansiu en dimensió 3. De la mateixa manera que amb els solitons cònics, donem una construcció explícita utilitzant tècniques de retrats de fase. Demostrem també que és l'únic solitó amb la seva topologia i la seva cota inferior de la curvatura, i que és un cas crític entre tots els solitons expansius en dimensió 3 amb curvatura acotada inferiorment. A més, mostrem que l'evolució de la seva curvatura escalar no és monòtona. / El principal objetivo de esta tesis es el estudio de la evolución mediante el flujo de Ricci de superficies con singularidades de tipo cónico. Un segundo objetivo, surgido de las técnicas que utilizamos, es el estudio de familias de solitones del flujo de Ricci en dimensión 2 y 3. El flujo de Ricci es una ecuación de evolución para variedades Riemannianas, introducida por R. Hamilton en 1982. Es desde los logros alcanzados por G. Perelman con esta técnica en 2002 cuando el flujo de Ricci se ha establecido en una disciplina propia, despertando un gran interés en la comunidad. Esta tesis contiene cuatro resultados originales. El primer resultado es una clasificación exhaustiva de los solitones en superficies lisas y cónicas. Con esta clasificación completamos los precedentes hallados por Hamilton, Chow y Wu entre otros, y obtenemos descripciones explícitas de todos los solitones en dimensión 2. El segundo resultado es una Geometrización de las superficies cónicas mediante el flujo de Ricci. Este resultado, que utiliza el primer resultado ya mencionado, extiende la teoría de Hamilton al caso singular. Este es el resultado más extenso, para el que usamos y desarrollamos técnicas tanto de análisis y EDPs como de geometría de comparación. El tercer resultado es la existencia de un flujo de Ricci que elimina las singularidades cónicas. Esto expone claramente la no unicidad de soluciones al flujo, en analogía a los flujos de Ricci con cúspides de P. Topping. El cuarto resultado es la construcción de un nuevo solitón gradiente expansivo en dimensión 3. Del mismo modo que con los solitones cónicos, damos una construcción explícita utilizando técnicas de retratos de fase. Demostramos también que es el único solitón con su topología y su cota inferior de la curvatura, y que es un caso crítico entre todos los solitones expansivos en dimensión 3 con curvatura acotada inferiormente. Además, mostramos que la evolución de su curvatura escalar no es monótona. / The main objective of this thesis is the study of the evolution under the Ricci flow of surfaces with singularities of cone type. A second objective, emerged from the techniques we use, is the study of families of Ricci flow solitons in dimension 2 and 3. The Ricci flow is an evolution equation for Riemannian manifolds, introduced by R. Hamilton in 1982. It is from the achievements made by G. Perelman with this technique in 2002 when the Ricci flow has been established in a discipline itself, generating a great interest in the community. This thesis contains four original results. First result is a complete classification of solitons in smooth and cone surfaces. This cllassification completes the preceding results found by Hamilton, Chow and Wu and others, and we obtain explicit descriptions of all solitons in dimension 2. Second result is a Geometrization of cone surfaces by Ricci flow. This result, which uses the aforementioned first result, extends the theory of Hamilton to the singular case. This is the most comprehensive result in the thesis, for which we use and develop analysis and PDE techniques, as well as comparison geometry techniques. Third result is the existence of a Ricci flow that removes cone singularities. This clearly exposes the non-uniqueness of solutions to the flow , in analogy to the Ricci flow with cusps of P. Topping. The fourth result is the construction of a new expanding gradient Ricci soliton in dimension 3. Just as we do with solitons on cone surfaces, we give an explicit construction using techniques of phase portraits. We also prove that this is the only soliton with its topology and its lower bound of the curvature, and besides this is a critical case amongst all expanding solitons in dimension 3 with curvature bounded below. Furthermore, we show that the evolution of its scalar curvature is not monotone.

Orientador: Luciana de Fátima Martins / Banca: Fábio Scalco Dias / Banca: João Carlos Ferreira da Costa / Resumo: Seja S uma superfície regular em R3 sem pontos parabólicos. O conjunto focal de S é o lugar geométrico dos centros das esferas que possuem contato degenerado com S em cada ponto. Tal contato é medido pelas singularidades da família de funcões distância ao quadrado D associada à S. O conjunto focal é uma superfície, porém não necessariamente regular, e pode também ser visto como o conjunto bifurcação da família D. A técnica de associar uma variedade singular X(S) a uma subvariedade suave S do espaço euclidiano e descobrir alguns aspectos da geometria de S a partir daqueles de X(S) está na essência das aplicações da Teoria das Singularidades á Geometria Diferencial. Neste trabalho, estudamos os modelos, a menos de difeomorfismos, para o conjunto focal de superfícies imersas em R3 genéricas, reunimos os principais resultados sobre a geometria da supefície focal encontrados na literatura e os apresentamos de forma mais explicativa e com uma linguagem moderna. Além disso, mostramos que a superfície focal pode ser parametrizada por uma frente de onda e utilizamos resultados conhecidos para tais aplicações no estudo da geometria da superfície focal / Abstract: Let S be a immersed surface in R3 without parabolic points. The focal set of S is the locus of the centres of spheres that have a degenerate contact with S in each point. This contact is measured by singularities of the family of distance squared function D associated with S. The focal set is a surface, but is not necessarily regular, and it can also be seen as the bifurcation set of the family D. The approach of associating a singular variety X(S) to a smooth submanifold S in an Euclidian space and recover some aspects of the geometry of S from that of X(S) is at the essence of applications of singularity theory to the Differential Geometry. In this work, we study models, unless diffeomorphism, of focal set of the immersed generics surfaces in R3. We have also gathered some results about the geometry of the focal set of the literature and we present them in a more explanatory way and in a modern notation. Furthermore, we show that the focal surface can be parametrized by a wave front and use the known results of such applications in the study of the focal set / Mestre