Spherical, Euclidean, and hyperbolic simplices can be characterized by the dihedral angles on their codimension-two faces. These characterizations analyze the Gram matrix, a matrix with entries given by cosines of dihedral angles. Hyperideal hyperbolic simplices are non-compact generalizations of hyperbolic simplices wherein the vertices lie outside hyperbolic space. We extend recent characterization results to include fully general hyperideal simplices. Our analysis utilizes the Gram matrix, however we use inversive distances instead of dihedral angles to accommodate fully general hyperideal simplices.For two-dimensional triangulations, an angle structure is an assignment of three face angles to each triangle. An angle structure permits a globally consistent scaling provided the faces can be simultaneously scaled so that any two contiguous faces assign the same length to their common edge. We show that a class of symmetric Euclidean angle structures permits globally consistent scalings. We develop a notion of virtual scaling to accommodate spherical and hyperbolic triangles of differing curvatures and show that a class of symmetric spherical and hyperbolic angle structures permit globally consistent virtual scalings.The double tetrahedron is a triangulation of the three-sphere obtained by gluing two congruent tetrahedra along their boundaries. The pentachoron is a triangulation of the three-sphere obtained from the boundary of the 4-simplex. As piecewise flat manifolds, the geometries of the double tetrahedron and pentachoron are determined by edge lengths that gives rise to a notion of a metric. We study notions of Einstein metrics on the double tetrahedron and pentachoron. Our analysis utilizes Regge's Einstein-Hilbert functional, a piecewise flat analogue of the Einstein-Hilbert (or total scalar curvature) functional on Riemannian manifolds.A notion of conformal structure on a two dimensional piecewise flat manifold is given by a set of edge constants wherein edge lengths are calculated from the edge constants and vertex based parameters. A conformal variation is a smooth one parameter family of the vertex parameters. The analysis of conformal variations often involves the study of degenerating triangles, where a face angle approaches zero. We show for a conformal variation that remains weighted Delaunay, if the conformal parameters are bounded then no triangle degenerations can occur.

Three problems are investigated in the context of quantum collective dynamics. First, we examine the optomechanics of a Bose-Einstein condensate trapped in an optical ring cavity and coupled to counter-propagating light fields. Virtual dipole transitions cause the light to recoil elastically from the condensate and to excite its atoms into momentum side modes. These momentum side modes produce collective density oscillations. We contrast the situation to a condensate trapped in a Fabry-Perot cavity, where only symmetric ("cosine") side modes are excited. In the ring cavity case, antisymmetric ("sine") modes can be excited also. We explore the mean field limit and find that even when the counter-propagating light fields are symmetrically pumped, there are parameter regions where spontaneous symmetry breaking occurs and the sine mode becomes occupied. In addition, quantum fluctuations are taken into account and shown to be particularly significant for parameter values near bifurcations of the mean field dynamics. The next system studied is a hybrid composed of a high quality micromechanical membrane coupled magnetically to a spinor condensate. This coupling entangles the membrane and the condensate and can produce position superposition states of the membrane. Successive spin measurements of the condensate can put the membrane into an increasingly complicated state. It is possible in principle to produce nonclassical states of the membrane. We also examine a model of weaker, nonprojective measurements of the condensate's spin using phase contrast imaging. We find an upper limit on how quickly such measurements can be made without severely disrupting the unitary dynamics. The third situation analyzed is the string breaking mechanism in ultrahigh energy collisions. When quark-antiquark pairs are produced in a collision, they are believed to be linked by a tube of chromoelectric field flux, the color string. The energy of the string grows linearly with quark separation. This energy is converted into real particles by the Schwinger mechanism. Screening of the color fields by new particles breaks the string. By quantizing excitations of the string using the conjugate coordinates of field strength and string cross-section, we recover the observed exponential spectrum of outgoing particles.

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The global log canonical threshold, algebraic counterpart to Tian's alpha-invariant, plays an important role when studying the geometry of Fano varieties. In particular, Tian showed that Fano manifolds with big alpha-invariant can be equipped with a Kahler-Einstein metric. In recent years Donaldson drafted a programme to precisely determine when a smooth Fano variety X admits a Kahler-Einstein metric. It was conjectured that the existence of such a metric is equivalent to X being K-stable, an algebraic-geometric property. A crucial step in Donaldson's programme consists on finding a Kahler-Einstein metric with edge singularities of small angle along a smooth anticanonical boundary. Jeffres, Mazzeo and Rubinstein showed that a dynamic version of the alpha-invariant could be used to find such metrics. The global log canonical threshold measures how anticanonical pairs fail to be log canonical. In this thesis we compute the global log canonical threshold of del Pezzo surfaces in various settings. First we extend Cheltsov's computation of the global log canonical threshold of complex del Pezzo surfaces to non-singular del Pezzo surfaces over a ground field which is algebraically closed and has arbitrary characteristic. Then we study which anticanonical pairs fail to be log canonical. In particular, we give a very explicit classifiation of very singular anticanonical pairs for del Pezzo surfaces of degree smaller or equal than 3. We conjecture under which circumstances such a classifcation is plausible for an arbitrary Fano variety and derive several consequences. As an application, we compute the dynamic alpha-invariant on smooth del Pezzo surfaces of small degree, where the boundary is any smooth elliptic curve C. Our main result is a computation of the dynamic alpha-invariant on all smooth del Pezzo surfaces with boundary any smooth elliptic curve C. The values of the alpha-invariant depend on the choice of C. We apply our computation to find Kahler-Einstein metrics with edge singularities of angle β along C.

Toric geometry studies manifolds M2n acted on effectively by a torus of half their dimension, Tn. Joyce shows that for such a 4-manifold sufficient conditions for a conformal class of metrics on the free part of the action to be self-dual can be given by a pair of linear ODEs and gives criteria for a metric in this class to extend to the degenerate orbits. Joyce and Calderbank-Pedersen use this result to find representatives which are scalar flat K¨ahler and self-dual Einstein respectively. We review some results concerning the topology of toric manifolds and the construction of Joyce metrics. We then extend this construction to give explicit complete scalar-flat K¨ahler and self-dual Einstein metrics on manifolds of infinite topological type, and to find a new family of Joyce metrics on open submanifolds of toric spaces. We then give two applications of these extensions — first, to give a large family of scalar flat K¨ahler perturbations of the Ooguri-Vafa metric, and second to search for a toric scalar flat K¨ahler metric on a neighbourhood of the origin in C2 whose restriction to an annulus on the degenerate hyperboloid {(z1, z2)|z1z2 = 0} is the cusp metric.

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In this thesis we have basically studied two aspects of BoseEinstein condensation (BEC) in trapped dilute gases: (i) superfluidity with the possible onset of quantum turbulence (QT), and (ii) nonuniformity, which suggests the definition of new variables in order to build a global thermodynamic description. Both analyses were performed in a 87Rb BEC magnetically trapped in a QuadrupoleIoffe configuration (QUIC) trap. Concerning the first item, vortices and QT were generated by applying an oscillatory excitation formed by a quadrupole magnetic field superimposed onto the QUIC trapping potential. Scanning both the excitation amplitude and its duration allowed us to observe different regimes, particularly one with regular, welldefined vortices and another, where the onset of QT is believed to have occurred. The transition between this two regimes were explained by considering the finitesize characteristic of trapped gases. Additionally, data analyses on three vortex configurations suggested the presence of both vortices and antivortices (opposite circulation sign), and the vortex nucleation mechanism was proposed to be related to a relative motion between the condensate and thermal components, namely a counterflow. As for the second item, the BEC transition in our experiment was characterized in terms of new global thermodynamic variables. A phase diagram was constructed and compared to the superfluid helium phase transition. Finally, we provide preliminary results on the calculation of a global heat capacity, and briefly discuss the advantages of this new approach over the local density approximation alternative, particularly on BEC clouds in the presence of vortices and QT. / Nesta tese, nós estudamos dois aspectos da condensação de BoseEinstein (CBE) em gases diluídos aprisionados: (i) superfluidez e a possível ocorrência de turbulência quântica (TQ); e (ii) nãouniformidade, o que sugere um tratamento termodinâmico diferente pela definição de novas variáveis globais. Ambos os estudos foram realizados em amostras condensadas de átomos de 87Rb aprisionados magneticamente numa armadilha do tipo QUIC. Em relação ao primeiro item, a geração de vórtices e TQ ocorreu pela aplicação de uma excitação oscilatória gerada pela adição de um campo quadrupolar ao potencial confinante do QUIC. Como dependência da amplitude e duração da excitação, diferentes regimes foram observados. Particularmente, num dos regimes, apenas vórtices bem definidos foram observados e em outro, imagens consistentes com a ocorrência de TQ foram obtidas. A transição entre estes dois regimes foi explicada em termos do tamanho finito característico de gases aprisionados. Além disto, através da análise de dados mostrando configurações com três vórtices, pudemos inferir a presença de vórtices e antivórtices (circulação oposta). Para explicar o mecanismo de nucleação de vórtices, analisamos, como possível causa, um movimento relativo entre as componentes térmicas e condensadas das amostras, conhecido como contrafluxo. Já em relação ao segundo item, a transição de fase da CBE foi descrita em termos de novas variáveis termodinâmicas globais. Um diagrama de fase foi construído ressaltando as semelhanças com a transição observada no hélio superfluido. Por fim, apresentamos resultados preliminares sobre o cálculo de uma capacidade térmica global e discutimos as vantagens desta nova abordagem em relação à alternativa usual baseada na aproximação de densidade local. Estas vantagens são particularmente relevantes no caso de nuvens condensadas que apresentam vórtices e TQ.

Let $S$ be a $2n$-dimensional complex manifold equipped with a line bundle with a real-analytic complex connection such that its curvature is of type $(1,1)$, and with a real analytic h-projective structure such that its h-projective curvature is of type $(1,1)$. For $n=1$ we assume that $S$ is equipped with a real-analytic M\"obius structure. Using the structure on $S$, we construct a twistor space of a quaternionic $4n$-manifold $M$. We show that $M$ can be identified locally with a neighbourhood of the zero section of the twisted (by a unitary line bundle) tangent bundle of $S$ and that $M$ admits a quaternionic $S^1$ action given by unit scalar multiplication in the fibres. We show that $S$ is a totally complex submanifold of $M$ and that a choice of a connection $D$ in the h-projective class on $S$ gives extensions of a complex structure from $S$ to $M$. For any such extension, using $D$, we construct a hyperplane distribution on $Z$ which corresponds to the unique quaternionic connection on $M$ preserving the extended complex structure. We show that, in a special case, the construction gives the Feix--Kaledin construction of hypercomplex manifolds, which includes the construction of hyperk\"ahler metrics on cotangent bundles. We also give an example in which the construction gives the quaternion-K\"ahler manifold $\mathbb{HP}^n$ which is not hyperk\"ahler. We show that the same construction and results can be obtained for $n=1$. By convention, in this case, $M$ is a self-dual conformal $4$-manifold and from Jones--Tod correspondence we know that the quotient $B$ of $M$ by an $S^1$ action is an asymptotically hyperbolic Einstein--Weyl manifold. Using a result of LeBrun \cite{Le}, we prove that $B$ is an asymptotically hyperbolic Einstein--Weyl manifold. We also give a natural construction of a minitwistor space $T$ of an asymptotically hyperbolic Einstein--Weyl manifold directly from $S$, such that $T$ is the Jones--Tod quotient of $Z$. As a consequence, we deduce that the Einstein--Weyl manifold constructed using $T$ is equipped with a distinguished Gauduchon gauge.

The Schrödinger equations have had a profound impact on a wide range of fields of modern science, including quantum mechanics, superfluidity, geometrical optics, Bose-Einstein condensates, and the analysis of dispersive phenomena in the theory of PDE. The main purpose of this thesis is to explore two Schrödinger-type equations appearing in the so-called Bohmian formulation of quantum mechanics and in the study of exciton-polariton condensates. For the first topic, the linear Schrödinger equation is the starting point in the formulation of a phase-space model proposed in [1] for the Bohmian interpretation of quantum mechanics. We analyze this model, a nonlinear Vlasov-type equation, as a Hamiltonian system defined on an appropriate Poisson manifold built on Wasserstein spaces, the aim being to establish its existence theory. For this purpose, we employ results from the theory of PDE, optimal transportation, differential geometry and algebraic topology. The second topic of the thesis is the study of a nonlinear Schrödinger equation, called the complex Gross-Pitaevskii equation, appearing in the context of Bose-Einstein condensation of exciton-polaritons. This model can be roughly described as a driven-damped Gross-Pitaevskii equation which shares some similarities with the complex Ginzburg-Landau equation. The difficulties in the analysis of this equation stem from the fact that, unlike the complex Ginzburg-Landau equation, the complex Gross-Pitaevskii equation does not include a viscous dissipation term. Our approach to this equation will be in the framework of numerical computations, using two main tools: collocation methods and numerical continuation for the stationary solutions and a time-splitting spectral method for the dynamics. After performing a linear stability analysis on the computed stationary solutions, we are led to postulate the existence of radially symmetric stationary ground state solutions only for certain values of the parameters in the equation; these parameters represent the “strength” of the driving and damping terms. Moreover, numerical continuation allows us to show, for fixed parameters, the ground and some of the excited state solutions of this equation. Finally, for the values of the parameters that do not produce a stable radially symmetric solution, our dynamical computations show the emergence of rotating vortex lattices.