This thesis presents an algorithm for producing 4–manifold triangulations with boundary an arbitrary orientable, closed, triangulated 3–manifold. The research is an extension of Costantino and Thurston’s work on determining upper bounds on the number of 4–dimensional simplices necessary to construct such a triangulation. Our first step in this bordism construction is the geometric partitioning of an initial 3–manifold M using smooth singularity theory. This partition provides handle attachment sites on the 4–manifold Mx[0,1] and the ensuing handle attachments eliminate one of the boundary components of Mx[0,1], yielding a 4-manifold with boundary exactly M. We first present the construction in the smooth case before extending the smooth singularity theory to triangulated 3–manifolds. / Graduate

Le résultat principal est une caractérisation homotopique des orbifolds de dimension 3 qui sont fibrés de Seifert : si O est un orbifold de dimension 3 fermé, orientable et petit dont le groupe fondamental admet un sous-groupe infini cyclique normal, alors O est de Seifert. Ce théorème généralise un résultat de Scott, Mess, Tukia, Gabai et Casson-Jungreis pour les variétés. Il repose sur une caractérisation des groupes de surfaces virtuels comme groupes quasi-isométriques à un plan riemannien complet. D'autres résultats sur les quasi-isométries entre groupes et surfaces sont obtenus.

<p>Complex transition metals oxyhalides containing a lone pair element, such as tellurium (IV), form an attractive research field because there is a high probability of finding new low-dimensionally arranged compounds and, particularly, a low-dimensionally arranged transition metals substructures, leading to interesting physical properties. Tellurium (IV) can drive the formation of many unusual structures because of its stereochemically active lone pair electrons, E. It commonly takes a coordination of three or four oxygen atoms to form either a TeO3E square pyramid or a TeO3+1E trigonal bipyramid. These lone pairs are very important players involved in lowering the dimensionality of crystal structures. Previous studies in transition metal tellurium (IV) oxohalide quarternary systems revealed a family of compounds, many of which exhibit interesting properties e.g. magnetic frustration. The unique point of this thesis is to employ alkaline earth elements (AE) to augment this ability of lone pair elements to lower the dimensionality of the transition metal arrangements. By this double usage of “chemical scissors” (a lone pair element used in conjunction with alkaline earth elements) we obtained new types of low-dimensionally arranged compounds.</p><p>This thesis is focused on the syntheses and characterization of a series of compounds in the pentanary (five components) system AE-TeIV-TM-O-X (AE=alkaline earth metal, TM=transition metal and X=halogen), in which nine new compounds were found. The crystal structures of each of these compounds were determined by the single crystal X-ray diffraction data.</p>

Complex transition metals oxyhalides containing a lone pair element, such as tellurium (IV), form an attractive research field because there is a high probability of finding new low-dimensionally arranged compounds and, particularly, a low-dimensionally arranged transition metals substructures, leading to interesting physical properties. Tellurium (IV) can drive the formation of many unusual structures because of its stereochemically active lone pair electrons, E. It commonly takes a coordination of three or four oxygen atoms to form either a TeO3E square pyramid or a TeO3+1E trigonal bipyramid. These lone pairs are very important players involved in lowering the dimensionality of crystal structures. Previous studies in transition metal tellurium (IV) oxohalide quarternary systems revealed a family of compounds, many of which exhibit interesting properties e.g. magnetic frustration. The unique point of this thesis is to employ alkaline earth elements (AE) to augment this ability of lone pair elements to lower the dimensionality of the transition metal arrangements. By this double usage of “chemical scissors” (a lone pair element used in conjunction with alkaline earth elements) we obtained new types of low-dimensionally arranged compounds. This thesis is focused on the syntheses and characterization of a series of compounds in the pentanary (five components) system AE-TeIV-TM-O-X (AE=alkaline earth metal, TM=transition metal and X=halogen), in which nine new compounds were found. The crystal structures of each of these compounds were determined by the single crystal X-ray diffraction data.

Plasmonic waves in solid-state are caused by collective oscillation of mobile charges inside or at the surface of conductors. In particular, surface plasmonic waves propagating at the skin of metals have recently attracted interest, as they reduce the wavelength of electromagnetic waves coupled to them by up to ~10 times, allowing one to create miniaturized wave devices at optical frequencies. In contrast, plasmonic waves on two-dimensional (2D) conductors appear at much lower infrared and THz-GHz frequencies, near or in the electronics regime, and can achieve far stronger wavelength reduction factor reaching well above 100. In this thesis, we study the unique machinery of 2D plasmonic waves behind this ultra-subwavelength confinement and explore how it can be used to create various interesting devices. To this end, we first develop a physically intuitive theoretical formulation of 2D plasmonic waves, whose two main components---the Coulomb restoration force and inertia of the collectively oscillating charges---are combined into a transmission-line-like model. We then use this formulation to create various ultra-subwavelength 2D plasmonic devices. For the 2D conductor, we first choose GaAs/AlGaAs heterostructure---a 2D electron gas consisting of massive (m*>0) electrons---demonstrating plasmonic bandgap crystals, interferometers, and negatively refracting metamaterials. We then examine a 2D plasmonic device based on graphene, a 2D electron gas consisting of effectively massless (m*=0) electrons. We theoretically show and experimentally demonstrate that the massless electrons in graphene can surprisingly exhibit a collective mass when subjected to a collective excitation, providing the inertia that is essential for the propagation of 2D plasmonic waves. Lastly, we theoretically investigate the thermal current fluctuation behaviors in massive and massless electron gases. While seemingly unrelated on first sight, we show that the thermal current fluctuation is actually intimately linked to the collective mass of the massive or massless electron gas. Thus, we show that the thermal current fluctuation behaviors can also be described by the same theoretical framework introduced earlier, suggesting a possibility to design new concept devices and experiments based on this linkage. / Engineering and Applied Sciences

Berge introduced knots that are primitive/primitive with respect to the standard genus 2 Heegaard surface, F, for the 3-sphere; surgery on such knots at the surface slope yields a lens space. Later Dean described a similar class of knots that are primitive/Seifert with respect to F; surgery on these knots at the surface slope yields a Seifert fibered space. The examples Dean worked with are among the twisted torus knots. In Chapter 3, we show that a given knot can have distinct primitive/Seifert representatives with the same surface slope. In Chapter 4, we show that a knot can also have a primitive/primitive and a primitive/Seifert representative that share the same surface slope. In Section 5.2, we show that these two results are part of the same phenomenon, the proof of which arises from the proof that a specific class of twisted torus knots are fibered, demonstrated in Section 5.1. / text

Low-dimensional signal structures naturally arise in a large set of applications in various fields such as medical imaging, machine learning, signal, and array processing. A ubiquitous low-dimensional structure in signals and images is sparsity, and a new sampling theory; namely, compressive sensing, proves that the sparse signals and images can be reconstructed from incomplete measurements. The signal recovery is achieved using efficient algorithms such as \ell_1-minimization. Recently, the research focus has spun-off to encompass other interesting low-dimensional signal structures such as group-sparsity and low-rank structure. This thesis considers low-rank matrix recovery (LRMR) from various structured-random measurement ensembles. These results are then employed for the in depth investigation of the classical blind-deconvolution problem from a new perspective, and for the development of a framework for the efficient sampling of correlated signals (the signals lying in a subspace). In the first part, we study the blind deconvolution; separation of two unknown signals by observing their convolution. We recast the deconvolution of discrete signals w and x as a rank-1 matrix wx* recovery problem from a structured random measurement ensemble. The convex relaxation of the problem leads to a tractable semidefinite program. We show, using some of the mathematical tools developed recently for LRMR, that if we assume the signals convolved with one another live in known subspaces, then this semidefinite relaxation is provably effective. In the second part, we design various efficient sampling architectures for signals acquired using large arrays. The sampling architectures exploit the correlation in the signals to acquire them at a sub-Nyquist rate. The sampling devices are designed using analog components with clear implementation potential. For each of the sampling scheme, we show that the signal reconstruction can be framed as an LRMR problem from a structured-random measurement ensemble. The signals can be reconstructed using the familiar nuclear-norm minimization. The sampling theorems derived for each of the sampling architecture show that the LRMR framework produces the Shannon-Nyquist performance for the sub-Nyquist acquisition of correlated signals. In the final part, we study low-rank matrix factorizations using randomized linear algebra. This specific method allows us to use a least-squares program for the reconstruction of the unknown low-rank matrix from the samples of its row and column space. Based on the principles of this method, we then design sampling architectures that not only acquire correlated signals efficiently but also require a simple least-squares program for the signal reconstruction. A theoretical analysis of all of the LRMR problems above is presented in this thesis, which provides the sufficient measurements required for the successful reconstruction of the unknown low-rank matrix, and the upper bound on the recovery error in both noiseless and noisy cases. For each of the LRMR problem, we also provide a discussion of a computationally feasible algorithm, which includes a least-squares-based algorithm, and some of the fastest algorithms for solving nuclear-norm minimization.

An efficient numerical method for solving Schr&ouml / dinger&#039 / s and Poisson&#039 / s equations using a basis set of cubic B-splines is investigated. The method is applied to find both the wave functions and the corresponding eigenenergies of low-dimensional semiconductor structures. The computational efficiency of the method is explicitly shown by the multiresolution analysis, non-uniform grid construction and imposed boundary conditions by applying it to well-known single electron potentials. The method compares well with the results of analytical solutions and of the finite difference method.

The observation of graphene’s extraordinary electrical properties has stirred great interest in two dimensional (2D) materials. The rapid pace of discovery for low dimensional materials with exciting properties continue with graphene allotropes, multiple polymorphs of borophene, germanene, and many others. The future of 2D materials goes beyond synthesis and characterization of free standing materials and on to the construction of heterostructures or sophisticated multilayer devices. Knowledge about the resulting local structure and composition of such systems will be key to understanding and optimizing their performance characteristics. 2D materials do not have a repeating crystal structure which can be easily characterized using bulk methods and therefore a localized high resolution method is needed. Electron microscopy is well suited for characterizing 2D materials as a repeating coherent structure is not necessary to produce a measureable signal as may be the case for diffraction methods. A unique opportunity for fine local scale measurements in low dimensional systems exists with a specific class of materials known as ferecrystals, the rotationally disordered relative of misfit layer compounds. Ferecrystals provide an excellent test system to observe effects at heterostructure interfaces as the whole film is composed of interdigitated two dimensional layers. Therefore bulk methods can be used to corroborate local scale measurements. From the qualitative interpretation of high resolution scanning transmission electron microscope (STEM) images to the quantitative application of STEM energy dispersive X-ray spectroscopy (EDX), this thesis uses numerous methods electron microscopy. The culmination of this work is seen at the end of the thesis where atomically resolved STEM-EDX hyperspectral maps could be used to measure element specific atomic distances and the atomically resolved fractional occupancies of a low dimensional alloy. These local scale measurements are corroborated by additional experimental data. The input of multiple techniques leads to improved certainty in local scale measurements and the applicability of these methods to non-ferecrystal low dimensional systems.

Comprendre la supraconductivité des pnictures et séléniures de fer nécessite de bien connaître leurs propriétés électroniques et magnétiques. Dans ce cadre, j'ai aidé à réaliser et j'ai automatisé un dispositif cryogénique capable d'étudier ces propriétés, à basse température (jusqu'à ~1 K) et sous application d'un champ magnétique statique (jusqu'à 2 T) ou oscillant. Les techniques implémentées sont la spectroscopie de pointe et la microscopie à sonde de Hall, et le dispositif est conçu de sorte qu'il soit possible de basculer de l'une à l'autre sans manipuler l'échantillon.J'ai ensuite utilisé ce dispositif pour étudier par spectroscopie de pointe les gaps supraconducteurs du pnicture Ba(Fe,Ni)2As2, puis rechercher la signature du couplage de ses électrons de conduction à des modes bosoniques. Ce faisant, il a été observé dans la conductance différentielle de jonctions métal-supraconducteur un signal oscillant dont la période varie en température comme le gap supraconducteur de l'échantillon. Ce signal dépend de la résistance de contact de la jonction d'une façon qui prouve clairement qu'il est issu d'effets hors équilibre.J'ai modélisé ce signal en étudiant théoriquement la physique de jonctions métal-métal-supraconducteur, dont la seconde région métallique serait formée par transition locale du supraconducteur vers l'état normal. Le modèle que j'ai ainsi construit permet de prédire la conductance différentielle d'une telle jonction, moyennant une connaissance préalable de la loi L(V) reliant la taille de la seconde région métallique à la tension aux bornes de la jonction. J'ai ensuite proposé plusieurs modèles pour cette loi.Après comparaison avec l'expérience, il semble possible que la région métallique se forme par dépassement local de la densité de courant critique Jd du supraconducteur associée à la brisure de paires de Cooper, ou "courant de depairing". Mais il serait aussi vraisemblable que l'injection d'électrons perturbe localement la distribution électronique f(E) de l'échantillon, au point de déstabiliser l'état supraconducteur. Cette dernière interprétation suppose une forte dépendance en température du couplage électron-boson à basse énergie.Enfin, je présente en annexe les fonctionnalités de microscopie magnétique du dispositif réalisé, ainsi que les premiers résultats scientifiques qu'elles ont permis d'obtenir : la mise en évidence du rôle joué par le fluage quantique dans la relaxation des vortex piégés au sein de Fe(Se,Te). / Reaching a good understanding of the superconductivity of iron pnictides and selenides requires an accurate knowledge of their electronic and magnetic properties. To this end, I helped building and I automated a cryogenic device that is suitable for the study of these properties, at low temperatures (down to ~1 K) and under the application of a magnetic field, either static (up to 2 T) or oscillating. The device implements the experimental techniques of point contact spectroscopy and scanning hall probe microscopy, and it allows switching between them without requiring sample manipulations.I subsequently used this device to study the superconducting gaps of Ba(Fe,Ni)2As2 by point contact spectroscopy, before I began looking for signatures of the coupling of conduction electrons to bosonic modes. However, in this process, the differential conductance of metal-superconductor junctions turned out to exhibit oscillating features, whose period evolves in temperature like the superconducting gap of the sample. This signal also depends on a junction's contact resistance in such a manner that it appears unmistakably out-of-equilibrium in nature.I derived a model of this signal by undertaking a theoretical study of metal-metal-superconductor junctions. In these junctions, the second metallic region would emerge from a local transition of the superconductor to the normal state. The resulting model is able to predict the differential conductance of such a junction, given prior knowledge of the L(V) law linking the size of the second metallic region to the voltage being applied across the junction. I subsequently derived several models for this law.Comparing these models to experimental data, it appears that the observed phenomenology could emerge from a local increase of current density above the "depairing current" Jd associated to Cooper pair breaking in the superconductor. Alternatively, electron injection could also locally alter the electron energy distribution f(E) of the sample to the point of destabilizing the superconducting state. This last explanation requires a strong temperature dependence of the electron-boson coupling at low energies.Finally, in an appendix, I describe the magnetic microscopy capabilities of the experimental device. Those capabilities enabled us to understand the role played by quantum creep in the relaxation of trapped vortices within Fe(Se,Te).