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.

Topological aspects of the human protein interaction network

It is currently widely accepted that the understanding of complex cell functions depends on an integrated network theoretical approach and not on an isolated view of the different molecular agents. Aim of this thesis was the examination of topological properties that mirror known biological aspects by depicting the human protein network with methods from graph- and network theory. The presented network is a partial human interactome of 9222 proteins and 36324 interactions, consisting of single interactions reliably extracted from peer-reviewed scientific publications. In general, one can focus on intra- or intermodular characteristics, where a functional module is defined as "a discrete entity whose function is separable from those of other modules". It is found that the presented human network is also scale-free and hierarchically organised, as shown for yeast networks before. The interactome also exhibits proteins with high betweenness and low connectivity which are biologically analyzed and interpreted here as shuttling proteins between organelles (e.g. ER to Golgi, internal ER protein translocation, peroxisomal import, nuclear pores import/export) for the first time. As an optimisation for finding proteins that connect modules, a new method is developed here based on proteins located between highly clustered regions, rather than regarding highly connected regions. As a proof of principle, the Mediator complex is found in first place, the prime example for a connector complex. Focusing on intramodular aspects, the measurement of k-clique communities discriminates overlapping modules very well. Twenty of the largest identified modules are analysed in detail and annotated to known biological structures (e.g. proteasome, the NFκB-, TGF-β complex). Additionally, two large and highly interconnected modules for signal transducer and transcription factor proteins are revealed, separated by known shuttling proteins. These proteins yield also the highest number of redundant shortcuts (by calculating the skeleton), exhibit the highest numbers of interactions and might constitute highly interconnected but spatially separated rich-clubs either for signal transduction or for transcription factors. This design principle allows manifold regulatory events for signal transduction and enables a high diversity of transcription events in the nucleus by a limited set of proteins. Altogether, biological aspects are mirrored by pure topological features, leading to a new view and to new methods that assist the annotation of proteins to biological functions, structures and subcellular localisations. As the human protein network is one of the most complex networks at all, these results will be fruitful for other fields of network theory and will help understanding complex network functions in general.

Symplectic Lagrangian fibrations

In this work we investigate the deformation theory of pairs of an irreducible symplectic manifold X together with a Lagrangian subvariety Y in X, where the focus is on singular Lagrangian subvarieties. Among other things, Voisin's results [Voi92] are generalized to the case of simple normal crossing subvarieties; partial results are also obtained for more complicated singularities.rnAs done in Voisin's article, we link the codimension of the subspace of the universal deformation space of X parametrizing those deformations where Y persists, to the rank of a certain map in cohomology. This enables us in some concrete cases to actually calculate or at least estimate the codimension of this particular subspace. In these cases the Lagrangian subvarieties in question occur as fibers or fiber components of a given Lagrangian fibration f : X --> B. We discuss examples and the question of how our results might help to understand some aspects of Lagrangian fibrations.

High precision swept volume approximation with conservative error bounds

In technical design processes in the automotive industry, digital prototypes rapidly gain importance, because they allow for a detection of design errors in early development stages. The technical design process includes the computation of swept volumes for maintainability analysis and clearance checks. The swept volume is very useful, for example, to identify problem areas where a safety distance might not be kept. With the explicit construction of the swept volume an engineer gets evidence on how the shape of components that come too close have to be modified.rnIn this thesis a concept for the approximation of the outer boundary of a swept volume is developed. For safety reasons, it is essential that the approximation is conservative, i.e., that the swept volume is completely enclosed by the approximation. On the other hand, one wishes to approximate the swept volume as precisely as possible. In this work, we will show, that the one-sided Hausdorff distance is the adequate measure for the error of the approximation, when the intended usage is clearance checks, continuous collision detection and maintainability analysis in CAD. We present two implementations that apply the concept and generate a manifold triangle mesh that approximates the outer boundary of a swept volume. Both algorithms are two-phased: a sweeping phase which generates a conservative voxelization of the swept volume, and the actual mesh generation which is based on restricted Delaunay refinement. This approach ensures a high precision of the approximation while respecting conservativeness.rnThe benchmarks for our test are amongst others real world scenarios that come from the automotive industry.rnFurther, we introduce a method to relate parts of an already computed swept volume boundary to those triangles of the generator, that come closest during the sweep. We use this to verify as well as to colorize meshes resulting from our implementations.

Note sul ruolo dell'infinito in matematica

Questo elaborato realizzato assieme alla creazione di un link nel sito "progettomatematic@" tratta dell'infinito in tre modi diversi: la storia, l'applicazione ai frattali e alla crittografia. Inizia con una breve storia dai greci all'antinomia di Russel; poi si parla dei frattali in natura, di misura e dimensione di Hausdorff, polvere di Cantor e fiocco di neve di Koch. Infine si trova un riassunto dei cifrari storici famosi, con particolare attenzione al cifrario di Vernam, alla teoria dell'entropia di Shannon e alla dimostrazione che otp ha sicurezza assoluta.

Misura di Hausdorff e frattali

In questa tesi sono presentate la misura e la dimensione di Hausdorff, gli strumenti matematici che permettono di descrivere e analizzare alcune delle più importanti proprietà degli insiemi frattali. Inoltre viene introdotto il carattere di autosimilarità, comune a questi insiemi, e vengono mostrati alcuni tra i più noti esempi di frattali, come l'insieme di Cantor, la curva di Koch, l'insieme di Mandelbrot e gli insiemi di Julia. Di quest'ultimi sono presenti immagini ottenute tramite un codice Matlab.

Analysis of the Kohn Laplacian on the Heisenberg Group and on Cauchy-Riemann Manifolds

The purpose of this study is to analyse the regularity of a differential operator, the Kohn Laplacian, in two settings: the Heisenberg group and the strongly pseudoconvex CR manifolds. The Heisenberg group is defined as a space of dimension 2n+1 with a product. It can be seen in two different ways: as a Lie group and as the boundary of the Siegel UpperHalf Space. On the Heisenberg group there exists the tangential CR complex. From this we define its adjoint and the Kohn-Laplacian. Then we obtain estimates for the Kohn-Laplacian and find its solvability and hypoellipticity. For stating L^p and Holder estimates, we talk about homogeneous distributions. In the second part we start working with a manifold M of real dimension 2n+1. We say that M is a CR manifold if some properties are satisfied. More, we say that a CR manifold M is strongly pseudoconvex if the Levi form defined on M is positive defined. Since we will show that the Heisenberg group is a model for the strongly pseudo-convex CR manifolds, we look for an osculating Heisenberg structure in a neighborhood of a point in M, and we want this structure to change smoothly from a point to another. For that, we define Normal Coordinates and we study their properties. We also examinate different Normal Coordinates in the case of a real hypersurface with an induced CR structure. Finally, we define again the CR complex, its adjoint and the Laplacian operator on M. We study these new operators showing subelliptic estimates. For that, we don't need M to be pseudo-complex but we ask less, that is, the Z(q) and the Y(q) conditions. This provides local regularity theorems for Laplacian and show its hypoellipticity on M.

Monodromy calculations for some differential equations

In vielen Teilgebieten der Mathematik ist es w"{u}nschenswert, die Monodromiegruppe einer homogenen linearen Differenzialgleichung zu verstehen. Es sind nur wenige analytische Methoden zur Berechnung dieser Gruppe bekannt, daher entwickeln wir im ersten Teil dieser Arbeit eine numerische Methode zur Approximation ihrer Erzeuger.rnIm zweiten Abschnitt fassen wir die Grundlagen der Theorie der Uniformisierung Riemannscher Fl"achen und die der arithmetischen Fuchsschen Gruppen zusammen. Auss erdem erkl"aren wir, wie unsere numerische Methode bei der Bestimmung von uniformisierenden Differenzialgleichungen dienlich sein kann. F"ur arithmetische Fuchssche Gruppen mit zwei Erzeugern erhalten wir lokale Daten und freie Parameter von Lam'{e} Gleichungen, welche die zugeh"origen Riemannschen Fl"achen uniformisieren. rnIm dritten Teil geben wir einen kurzen Abriss zur homologischen Spiegelsymmetrie und f"uhren die $widehat{Gamma}$-Klasse ein. Wir erkl"aren wie diese genutzt werden kann, um eine Hodge-theoretische Version der Spiegelsymmetrie f"ur torische Varit"aten zu beweisen. Daraus gewinnen wir Vermutungen "uber die Monodromiegruppe $M$ von Picard-Fuchs Gleichungen von gewissen Familien $f:mathcal{X}rightarrow bbp^1$ von $n$-dimensionalen Calabi-Yau Variet"aten. Diese besagen erstens, dass bez"uglich einer nat"urlichen Basis die Monodromiematrizen in $M$ Eintr"age aus dem K"orper $bbq(zeta(2j+1)/(2 pi i)^{2j+1},j=1,ldots,lfloor (n-1)/2 rfloor)$ haben. Und zweitens, dass sich topologische Invarianten des Spiegelpartners einer generischen Faser von $f:mathcal{X}rightarrow bbp^1$ aus einem speziellen Element von $M$ rekonstruieren lassen. Schliess lich benutzen wir die im ersten Teil entwickelten Methoden zur Verifizierung dieser Vermutungen, vornehmlich in Hinblick auf Dimension drei. Dar"uber hinaus erstellen wir eine Liste von Kandidaten topologischer Invarianten von vermutlich existierenden dreidimensionalen Calabi-Yau Variet"aten mit $h^{1,1}=1$.

L-2-Kohomologie von Calabi-Yau-Familien über Kurven

Ist $f: X \to S$ eine glatte Familie von Calabi-Yau-Mannigfaltigkeiten der Dimension $m$ über einer quasiprojektiven Kurve, so trägt nach einem Resultat von Zucker die erste $L^2$-Kohomologiegruppe $H^1_{(2)}(S, R^m f_* \mathbb{C}_X)$ eine reine Hodgestruktur vom Gewicht $m+1$. In dieser Arbeit berechnen wir die Hodgezahlen solcher Hodgestrukturen für $m= 1, 2, 3$ und verallgemeinern dabei Formeln aus einem Artikel von del Angel, Müller-Stach, van Straten und Zuo auf den Fall, in dem die lokalen Monodromiematrizen bei Unendlich nicht unipotent, sondern echt quasi-unipotent sind. Wir verwenden dazu den $L^2$-Higgs-Komplex nach Jost, Yang und Zuo. Für Familien von Kurven führt dies auf eine bereits bekannte Formel von Cox und Zucker. Schließlich wenden wir die Ergebnisse im Fall $m=3$ auf 14 Familien von Calabi-Yau-Mannigfaltigkeiten an, die eine Rolle in der Spiegelsymmetrie spielen, sowie auf eine von Rohde konstruierte Familie ohne Punkte mit maximal unipotenter Monodromie.

L'assioma di scelta : Storia e influenza sulla matematica del novecento.

L’assioma di scelta ha una preistoria, che riguarda l’uso inconsapevole e i primi barlumi di consapevolezza che si trattasse di un nuovo principio di ragionamento. Lo scopo della prima parte di questa tesi è quello di ricostruire questo percorso di usi più o meno impliciti e più o meno necessari che rivelarono la consapevolezza non solo del fatto che fosse indispensabile introdurre un nuovo principio, ma anche che il modo di “fare matematica” stava cambiando. Nei capitoli 2 e 3, si parla dei moltissimi matematici che, senza rendersene conto, utilizzarono l’assioma di scelta nei loro lavori; tra questi anche Cantor che appellandosi alla banalità delle dimostrazioni, evitava spesso di chiarire le situazioni in cui era richiesta questa particolare assunzione. Il capitolo 2 è dedicato ad un caso notevole e rilevante dell’uso inconsapevole dell’Assioma, di cui per la prima volta si accorse R. Bettazzi nel 1892: l’equivalenza delle due nozioni di finito, quella di Dedekind e quella “naturale”. La prima parte di questa tesi si conclude con la dimostrazione di Zermelo del teorema del buon ordinamento e con un’analisi della sua assiomatizzazione della teoria degli insiemi. La seconda parte si apre con il capitolo 5 in cui si parla dell’intenso dibattito sulla dimostrazione di Zermelo e sulla possibilità o meno di accettare il suo Assioma, che coinvolse i matematici di tutta Europa. In quel contesto l’assioma di scelta trovò per lo più oppositori che si appellavano ad alcune sue conseguenze apparentemente paradossali. Queste conseguenze, insieme alle molte importanti, sono analizzate nel capitolo 6. Nell’ultimo capitolo vengono riportate alcune tra le molte equivalenze dell’assioma di scelta con altri enunciati importanti come quello della tricotomia dei cardinali. Ci si sofferma poi sulle conseguenze dell’Assioma e sulla sua influenza sulla matematica del Novecento, quindi sulle formulazioni alternative o su quelle più deboli come l’assioma delle scelte dipendenti e quello delle scelte numerabili. Si conclude con gli importanti risultati, dovuti a Godel e a Cohen sull’indipendenza e sulla consistenza dell’assioma di scelta nell’ambito della teoria degli insiemi di Zermelo-Fraenkel.

String inflationary models with non-monotonic slow-roll and detectable tensor modes

The first chapter of this work has the aim to provide a brief overview of the history of our Universe, in the context of string theory and considering inflation as its possible application to cosmological problems. We then discuss type IIB string compactifications, introducing the study of the inflaton, a scalar field candidated to describe the inflation theory. The Large Volume Scenario (LVS) is studied in the second chapter paying particular attention to the stabilisation of the Kähler moduli which are four-dimensional gravitationally coupled scalar fields which parameterise the size of the extra dimensions. Moduli stabilisation is the process through which these particles acquire a mass and can become promising inflaton candidates. The third chapter is devoted to the study of Fibre Inflation which is an interesting inflationary model derived within the context of LVS compactifications. The fourth chapter tries to extend the zone of slow-roll of the scalar potential by taking larger values of the field φ. Everything is done with the purpose of studying in detail deviations of the cosmological observables, which can better reproduce current experimental data. Finally, we present a slight modification of Fibre Inflation based on a different compactification manifold. This new model produces larger tensor modes with a spectral index in good agreement with the date released in February 2015 by the Planck satellite.

La teoria dei numeri transfiniti nei suoi aspetti matematici e filosofici

La presente tesi si occupa, da un punto di vista matematico e filosofico, dello studio dei numeri transfiniti introdotti da Georg Cantor. Vengono introdotti i concetti di numero cardinale ed ordinale, la loro aritmetica ed i principali risultati riguardo al concetto di insieme numerabile. Si discutono le nozioni di infinito potenziale ed attuale e quella di esistenza secondo la concezione di Cantor. Viene infine presentata l'induzione transfinita, una generalizzazione al caso transfinito del principio di induzione matematica.

Misure di Hausdorff e applicazioni

Obiettivo della tesi è fornire nozioni di teoria della misura tramite cui è possibile l'analisi e la descrizione degli insiemi frattali. A tal fine vengono definite la Misura e la Dimensione di Hausdorff, strumenti matematici che permettono di "misurare" tali oggetti particolari, per i quali la classica Misura di Lebesgue non risulta sufficientemente precisa. Viene introdotto, inoltre, il carattere di autosimilarità, comune a molti di questi insiemi, e sono forniti alcuni tra i più noti esempi di frattali, come l'insieme di Cantor, l'insieme di Mandelbrot e il triangolo di Sierpinski. Infine, viene verificata l'ipotesi dell'esistenza di componenti di natura frattale in serie storiche di indici borsistici e di titoli finanziari (Ipotesi dei Mercati Frattali, Peters, 1990).

Fractal sets and their applications in medicine

La geometria euclidea risulta spesso inadeguata a descrivere le forme della natura. I Frattali, oggetti interrotti e irregolari, come indica il nome stesso, sono più adatti a rappresentare la forma frastagliata delle linee costiere o altri elementi naturali. Lo strumento necessario per studiare rigorosamente i frattali sono i teoremi riguardanti la misura di Hausdorff, con i quali possono definirsi gli s-sets, dove s è la dimensione di Hausdorff. Se s non è intero, l'insieme in gioco può riconoscersi come frattale e non presenta tangenti e densità in quasi nessun punto. I frattali più classici, come gli insiemi di Cantor, Koch e Sierpinski, presentano anche la proprietà di auto-similarità e la dimensione di similitudine viene a coincidere con quella di Hausdorff. Una tecnica basata sulla dimensione frattale, detta box-counting, interviene in applicazioni bio-mediche e risulta utile per studiare le placche senili di varie specie di mammiferi tra cui l'uomo o anche per distinguere un melanoma maligno da una diversa lesione della cute.

Studio retrospettico sul ruolo della strumentazione nei fallimenti meccanici a seguito di Trattamento Chirurgico nelle Scoliosi dell'adulto

The spinal column performs important functions in the body, including the support of the entire weight of the human body, the ability to orientate the head in space, bending, flexing and rotating the body. Diseases affecting the spine are manifold: the most frequent is scoliosis, which often affects the female population. It is often treated surgically with a very high percentage of failures. The aim of the thesis is to study the role of instrumentation in mechanical failures encountered 12 months after surgery in the treatment of scoliosis. For the purposes of the study, we analyzed specific biomechanical parameters. The pelvic angles determine the position of the pelvis, while the imbalance parameters the structure of the body. We infer other parameters by analyzing the characteristics of the implanted instrumentation. Initially, the anatomy is described of the spine and vertebrae, the equipment used and the possible failures that may occur after surgery. Subsequently, the materials and methods used for the analysis of the above-mentioned parameters for the 61 patients are reported. All data are obtained by the observation of pre and post-operative x-rays with a special program, by reading reports from operators and by medical records. In the fourth chapter, we report the results: the overall failure rate is 60.9%; the types of failures that occurred are rupture of bars and rupture of bars simultaneously to PJK. The most influential parameters on results of the progress of the surgery are the type of material used and the BMI. It is estimated a high percentage of failures in patients treated with implants of cobalt chromium alloys (90.0%). According to the results obtained, it is possible to understand the aspects that in the future should be studied, in order to find a solution to the most frequent surgical failures.