Isothermic constrained Willmore tori in 3-space

We show that the homogeneous and the 2-lobe Delaunay tori in the 3-sphere provide the only isothermic constrained Willmore tori in 3-space with Willmore energy below 8 π. In particular, every constrained Willmore torus with Willmore energy below 8 π and non-rectangular conformal class is non-degenerated

Une fonction zêta motivique pour l'étude des singularités réelles

The main purpose of this thesis is to study real singularities using arguments from motivic integration as initiated by S. Koike and A. Parusiński and then continued by G. Fichou. In order to classify real singularities, T.-C. Kuo introduced the blow-analytic equivalence which is an equivalence relation on real analytic germs without moduli for isolated singularities. This notion is closely related to the notion of arc-analytic maps introduced by K. Kurdyka, thus it is natural to adapt arguments from motivic integration to the study of the relation. The difficulty lies in finding efficient ways to prove that two germs are equivalent and in constructing invariants that distinguish germs which are not in the same class. We focus on the blow-Nash equivalence, a more algebraic notion which was introduced by G. Fichou. The first part of this thesis consists in an inverse theorem for blow-Nash maps. Under certain assumptions, this ensures that the inverse of a homeomorphism which is blow-Nash is also blow-Nash. Such maps are involved in the definition of the blow-Nash equivalence. In the second part, we associate a power series to an analytic germ, called the zeta function of the germ. This construction generalizes the zeta functions of Koike-Parusiński and Fichou. Furthermore, it admits a convolution formula while being an invariant for the blow-Nash equivalence.Nous nous intéressons à l'étude des singularités réelles à l'aide d'arguments provenant de l'intégration motivique. Une telle démarche a été initiée par S. Koike et A. Parusiński puis poursuivie par G. Fichou. Afin de donner une classification des singularités réelles, T.-C. Kuo a défini la notion d'équivalence blow-analytique. Il s'agit d'une relation d'équivalence pour les germes analytiques réels n'admettant pas de module continu pour les singularités isolées. Cette notion est étroitement liée à la notion d'applications analytiques par arcs définie par K. Kurdyka. Il est donc naturel d'adapter des arguments provenant de l'intégration motivique pour l'étude de l'équivalence blow-analytique. La difficulté réside désormais dans le fait de trouver des méthodes permettant de montrer que deux germes sont équivalents et de construire des invariants permettant de distinguer deux germes qui ne sont pas dans la même classe. Nous travaillons avec une variante plus algébrique de cette notion, l'équivalence blow-Nash introduite par G. Fichou. La première partie de la thèse consiste en un théorème d'inversion donnant des conditions pour que l'inverse d'un homéomorphisme blow-Nash soit encore blow-Nash. L'intérêt d'un tel énoncé est que de telles applications apparaissent dans la définition de l'équivalence blow-Nash. La seconde partie est consacrée à l'étude d'une nouvelle fonction zêta motivique. Il s'agit d'associer à un germe analytique une série formelle. Cette fonction zêta motivique généralise les fonctions zêta de Koike-Parusiński et de Fichou et admet une formule de convolution. Il s'agit d'un invariant pour l'équivalence blow-Nash

Topological effects in particle physics phenomenology

This thesis is devoted to the study of topological effects in quantum field theories, with a particular focus on phenomenological applications. We begin by deriving a general classification of topological terms appearing in a non-linear sigma model based on maps from an arbitrary worldvolume manifold to a homogeneous space $G/H$ (where $G$ is an arbitrary Lie group and $H \subset G$). Such models are ubiquitous in phenomenology; in three or more dimensions they cover all cases in which only some subgroup $H$ of a dynamical symmetry group $G$ is linearly realized in vacuo. The classification is based on the observation that, for topological terms, the maps from the worldvolume to $G/H$ may be replaced by singular homology cycles on $G/H$. We find that such terms come in one of two types, which we refer to as `Aharonov-Bohm' (AB) and `Wess-Zumino' (WZ) terms. We derive a new condition for their $G$-invariance, which we call the `Manton condition', which is necessary and sufficient when the Lie group $G$ is connected. Armed with this classification of topological terms, we then apply it to Composite Higgs models based on a variety of coset spaces $G/H$ and discuss their phenomenology. For example, we point out the existence of an AB term in the minimal Composite Higgs model based on $SO(5)/SO(4)$, whose phenomenological effects arise only at the non-perturbative level, and lead to $P$ and $CP$ violation in the Higgs sector. Consideration of the Manton condition leads us to discover a rather subtle anomaly in a non-minimal model based on $SO(5)\times U(1)/SO(4)$ (a model which does, however, feature an AB term not previously noticed in the literature). A particularly rich topological structure, with six distinct terms of various types, is uncovered for the model based on $SO(6)/SO(4)$, which features two Higgs doublets and one singlet. Perhaps most importantly for phenomenology, measuring the coefficients of WZ terms that appear in any of these Composite Higgs models can allow one to probe the gauge group of the underlying microscopic theory. As a further application of our results, we analyse quantum mechanics models featuring such topological terms. In this context, a topological term couples the particle to a background magnetic field. The usual methods for formulating and solving the quantum mechanics of a particle moving in a magnetic field respect neither locality nor any global symmetries which happen to be present. We show how both locality and symmetry can be made manifest, by passing to an otherwise redundant description on a principal bundle over the original configuration space, and by promoting the original symmetry group to a central extension thereof. We then demonstrate how harmonic analysis on the extended symmetry group can be used to solve the Schr{\"o}dinger equation. To conclude our study of topological terms in sigma models, we show that the classification we have proposed may be rigorously justified (and generalised) using differential cohomology theory. In doing so, we introduce the notion of the `$G$-invariant differential characters' of a manifold $M$. Within this language, the Manton condition follows from the homotopy formula for differential characters, and we show that it remains necessary and sufficient under weaker conditions than connectedness of $G$. We prove that the abelian group of $G$-invariant differential characters sits inside various exact sequences and commutative diagrams, which thus provide us with some powerful algebraic tools for classifying topological terms in quantum field theories. In the remainder of the thesis we depart from the topic of sigma models and turn to gauge theories. We analyse anomalies (which may be understood as arising from topological effects) in both the Standard Model (SM) and theories Beyond the Standard Model (BSM). This analysis consists of two parts, in which we consider `local' and `global' anomalies in a gauge symmetry $G$; the former depend only on the Lie algebra of $G$, while the latter are sensitive also to its global structure, {\em i.e.} its topology. We first chart the space of anomaly-free extensions of the SM by a flavour-dependent $U(1)$ gauge symmetry, using arithmetic techniques from Diophantine analysis to cancel all possible local anomalies. We then develop some of these anomaly-free theories into phenomenological models featuring a heavy $Z^\prime$ gauge boson, that can account for a collection of recent measurements involving $b\rightarrow s\mu\mu$ transitions which are discrepant with SM predictions. We discuss how these models might also explain coarse features of the fermion mass problem, such as the heaviness of the third family. We then turn to global anomalies, which we analyse using the Dai-Freed theorem. Our principal tool here is to compute the bordism groups of the classifying spaces of various Lie groups, preserving particular spin structures, using the Atiyah-Hirzebruch spectral sequence. We show that there are no global anomalies (beyond the Witten anomaly associated with the electroweak factor) in four different `versions' of the SM, in which the gauge group is taken to be $G_{\text{SM}}/\Gamma$, with $G_{\text{SM}}=SU(3)\times SU(2) \times U(1)$ and $\Gamma \in \{0,\mathbb{Z}_2,\mathbb{Z}_3,\mathbb{Z}_6\}$. We also show that there are no new global anomalies in $U(1)^m$ extensions of the SM, which feature multiple $Z^\prime$ bosons, or in the Pati-Salam model.Vice-Chancellor's Award (Cambridge Trust

EXTENDING CONVOLUTION THROUGH SPATIALLY ADAPTIVE ALIGNMENT

Convolution underlies a variety of applications in computer vision and graphics, including efficient filtering, analysis, simulation, and neural networks. However, convolution has an inherent limitation: when convolving a signal with a filter, the filter orientation remains fixed as it travels over the domain, and convolution loses effectiveness in the presence of deformations that change alignment of the signal relative to the local frame. This problem metastasizes when attempting to generalize convolution to domains without a canonical orientation, such as the surfaces of 3D shapes, making it impossible to locally align signals and filters in a consistent manner. This thesis presents a unified framework for transformation-equivariant convolutions on arbitrary homogeneous spaces and 2D Riemannian manifolds called extended convolution. This approach is based on the the following observation: to achieve equivariance to an arbitrary class of transformations, we only need to consider how the positions of points as seen in the frames of their neighbors deform. By defining an equivariant frame operator at each point with which we align the filter, we correct for the change in the relative positions induced by the transformations. This construction places no constraints on the filters, making extended convolution highly descriptive. Furthermore, the framework can handle arbitrary transformation groups, including higher-dimensional non-compact groups that act non-linearly on the domain. Critically, extended convolution naturally generalizes to arbitrary 2D Riemannian manifolds as it does not need a canonical coordinate system to be applied. The power and utility of extended convolution is demonstrated in several applications. A unified framework for image and surface feature descriptors called Extended Convolution Histogram of Orientations (ECHO) is proposed, based on the optimal filters maximizing the response of the extended convolution at a given point. Using the generalization of extended convolution to surface vector fields, state-of-the-art surface convolutional neural networks (CNNs) are constructed. Last, we move beyond rotations and isometries and use extended convolution to design spherical CNNs equivariant to Mobius transformations, representing a first step toward conformally-equivariant surface networks

Reconstruction from Spatio-Spectrally Coded Multispectral Light Fields

In dieser Arbeit werden spektral kodierte multispektrale Lichtfelder untersucht, wie sie von einer Lichtfeldkamera mit einem spektral kodierten Mikrolinsenarray aufgenommen werden. Für die Rekonstruktion der kodierten Lichtfelder werden zwei Methoden entwickelt, eine basierend auf den Prinzipien des Compressed Sensing sowie eine Deep Learning Methode. Anhand neuartiger synthetischer und realer Datensätze werden die vorgeschlagenen Rekonstruktionsansätze im Detail evaluiert

Reconstruction from Spatio-Spectrally Coded Multispectral Light Fields

In this work, spatio-spectrally coded multispectral light fields, as taken by a light field camera with a spectrally coded microlens array, are investigated. For the reconstruction of the coded light fields, two methods, one based on the principles of compressed sensing and one deep learning approach, are developed. Using novel synthetic as well as a real-world datasets, the proposed reconstruction approaches are evaluated in detail

Reconstruction from Spatio-Spectrally Coded Multispectral Light Fields

In dieser Arbeit werden spektral codierte multispektrale Lichtfelder, wie sie von einer Lichtfeldkamera mit einem spektral codierten Mikrolinsenarray aufgenommen werden, untersucht. Für die Rekonstruktion der codierten Lichtfelder werden zwei Methoden entwickelt und im Detail ausgewertet. Zunächst wird eine vollständige Rekonstruktion des spektralen Lichtfelds entwickelt, die auf den Prinzipien des Compressed Sensing basiert. Um die spektralen Lichtfelder spärlich darzustellen, werden 5D-DCT-Basen sowie ein Ansatz zum Lernen eines Dictionary untersucht. Der konventionelle vektorisierte Dictionary-Lernansatz wird auf eine tensorielle Notation verallgemeinert, um das Lichtfeld-Dictionary tensoriell zu faktorisieren. Aufgrund der reduzierten Anzahl von zu lernenden Parametern ermöglicht dieser Ansatz größere effektive Atomgrößen. Zweitens wird eine auf Deep Learning basierende Rekonstruktion der spektralen Zentralansicht und der zugehörigen Disparitätskarte aus dem codierten Lichtfeld entwickelt. Dabei wird die gewünschte Information direkt aus den codierten Messungen geschätzt. Es werden verschiedene Strategien des entsprechenden Multi-Task-Trainings verglichen. Um die Qualität der Rekonstruktion weiter zu verbessern, wird eine neuartige Methode zur Einbeziehung von Hilfslossfunktionen auf der Grundlage ihrer jeweiligen normalisierten Gradientenähnlichkeit entwickelt und gezeigt, dass sie bisherige adaptive Methoden übertrifft. Um die verschiedenen Rekonstruktionsansätze zu trainieren und zu bewerten, werden zwei Datensätze erstellt. Zunächst wird ein großer synthetischer spektraler Lichtfelddatensatz mit verfügbarer Disparität Ground Truth unter Verwendung eines Raytracers erstellt. Dieser Datensatz, der etwa 100k spektrale Lichtfelder mit dazugehöriger Disparität enthält, wird in einen Trainings-, Validierungs- und Testdatensatz aufgeteilt. Um die Qualität weiter zu bewerten, werden sieben handgefertigte Szenen, so genannte Datensatz-Challenges, erstellt. Schließlich wird ein realer spektraler Lichtfelddatensatz mit einer speziell angefertigten spektralen Lichtfeldreferenzkamera aufgenommen. Die radiometrische und geometrische Kalibrierung der Kamera wird im Detail besprochen. Anhand der neuen Datensätze werden die vorgeschlagenen Rekonstruktionsansätze im Detail bewertet. Es werden verschiedene Codierungsmasken untersucht -- zufällige, reguläre, sowie Ende-zu-Ende optimierte Codierungsmasken, die mit einer neuartigen differenzierbaren fraktalen Generierung erzeugt werden. Darüber hinaus werden weitere Untersuchungen durchgeführt, zum Beispiel bezüglich der Abhängigkeit von Rauschen, der Winkelauflösung oder Tiefe. Insgesamt sind die Ergebnisse überzeugend und zeigen eine hohe Rekonstruktionsqualität. Die Deep-Learning-basierte Rekonstruktion, insbesondere wenn sie mit adaptiven Multitasking- und Hilfslossstrategien trainiert wird, übertrifft die Compressed-Sensing-basierte Rekonstruktion mit anschließender Disparitätsschätzung nach dem Stand der Technik

Reconstruction from Spatio-Spectrally Coded Multispectral Light Fields

In this work, spatio-spectrally coded multispectral light fields, as taken by a light field camera with a spectrally coded microlens array, are investigated. For the reconstruction of the coded light fields, two methods, one based on the principles of compressed sensing and one deep learning approach, are developed. Using novel synthetic as well as a real-world datasets, the proposed reconstruction approaches are evaluated in detail