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Hermitian-Yang-Mills Metrics on Hilbert Bundles and in the Space of Kahler PotentialsKuang-Ru Wu (9132815) 05 August 2020 (has links)
<div>The two main results in this thesis have a common point: Hermitian--Yang--Mills (HYM) metrics. In the first result, we address a Dirichlet problem for the HYM equations in bundles of infinite rank over Riemann surfaces. The solvability has been known since the work of Donaldson \cite{Donaldson92} and Coifman--Semmes \cite{CoifmanSemmes93}, but only for bundles of finite rank. So the novelty of our first result is to show how to deal with infinite rank bundles. The key is an a priori estimate obtained from special feature of the HYM equation.</div><div> </div><div> In the second result, we take on the topic of the so-called ``geometric quantization." This is a vast subject. In one of its instances the aim is to approximate the space of K\"ahler potentials by a sequence of finite dimensional spaces. The approximation of a point or a geodesic in the space of K\"ahler potentials is well-known, and it has many applications in K\"ahler geometry. Our second result concerns the approximation of a Wess--Zumino--Witten type equation in the space of K\"ahler potentials via HYM equations, and it is an extension of the point/geodesic approximation. </div><div> </div>
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Topological Aspects of Dirac Fermions in Condensed Matter SystemsZirnstein, Heinrich-Gregor 23 April 2021 (has links)
Dirac fermions provide a prototypical description of topological insulators and their gapless boundary states, which are predicted by the bulk-boundary correspondence. Motivated by the unusual physical properties of these states, we study them in two different Hermitian quantum systems. In non-Hermitian systems, we investigate the failure of the bulk-boundary correspondence and show that non-Hermitian topological invariants impact a system’s bulk response.
First, we study electronic topological insulators in three dimensions with time-reversal symmetry. These can be characterized by a quantized magnetoelectric coefficient in the bulk, which, however, does not yield an experimentally observable response. We show that the signature response of a time-reversal-invariant topological insulator is a nonlinear magnetoelectric effect, which in the presence of a small electric field leads to the appearance of half-integer charges bound to a magnetic flux quantum.
Next, we consider topological superconducting nanowires. These feature Majorana zero modes at their ends, which combine nonlocally into a single electronic state. An electron tunneling through such a state will be transmitted phase-coherently from one end of the wire to the other. We compute the transmission phase for nanowires with broken time-reversal symmetry and confirm that it is independent of the wire length.
Turning to non-Hermitian systems, we consider planar optical microcavities with an anisotropic cavity material, which may feature topological degeneracies known as excep- tional points in their complex frequency spectrum. We present a quantitative method to extract an effective non-Hermitian Hamiltonian for the eigenmodes, and describe how a pair of exceptional points arises from a Dirac point due to the cavity loss.
Finally, we investigate generalized topological invariants that can be defined for non- Hermitian systems, but which have no counterpart (i.e. vanish) in Hermitian systems, for example the so-called non-Hermitian winding number in one dimension. Contrary to Hermitian systems, the bulk-boundary correspondence breaks down: Comparing Green functions for periodic and open boundary conditions, we find that in general there is no correspondence between topological invariants computed for periodic boundary con- ditions, and boundary eigenstates observed for open boundary conditions. Instead, we prove that the non-Hermitian winding number in one dimension signals a topological phase transition in the bulk: It implies spatial growth of the bulk Green function, which we define as the response of a gapped system to an external perturbation on timescales where the induced excitations have not propagated to the boundary yet. Since periodic systems cannot accommodate such spatial growth, they differ from open ones.
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Spectral factorization of matricesGaoseb, Frans Otto 06 1900 (has links)
Abstract in English / The research will analyze and compare the current research on the spectral
factorization of non-singular and singular matrices. We show that a nonsingular non-scalar matrix A can be written as a product A = BC where the eigenvalues of B and C are arbitrarily prescribed subject to the condition that the product of the eigenvalues of B and C must be equal to the determinant of A. Further, B and C can be simultaneously triangularised as a lower and upper triangular matrix respectively. Singular matrices will
be factorized in terms of nilpotent matrices and otherwise over an arbitrary
or complex field in order to present an integrated and detailed report on the
current state of research in this area. Applications related to unipotent, positive-definite, commutator, involutory and Hermitian factorization are studied for non-singular matrices, while applications related to positive-semidefinite matrices are investigated for singular matrices. We will consider the theorems found in Sourour [24] and Laffey [17] to show
that a non-singular non-scalar matrix can be factorized spectrally. The same
two articles will be used to show applications to unipotent, positive-definite
and commutator factorization. Applications related to Hermitian factorization will be considered in [26]. Laffey [18] shows that a non-singular matrix
A with det A = ±1 is a product of four involutions with certain conditions
on the arbitrary field. To aid with this conclusion a thorough study is made
of Hoffman [13], who shows that an invertible linear transformation T of a
finite dimensional vector space over a field is a product of two involutions
if and only if T is similar to T−1. Sourour shows in [24] that if A is an
n × n matrix over an arbitrary field containing at least n + 2 elements and
if det A = ±1, then A is the product of at most four involutions.
We will review the work of Wu [29] and show that a singular matrix A of
order n ≥ 2 over the complex field can be expressed as a product of two
nilpotent matrices, where the rank of each of the factors is the same as A,
except when A is a 2 × 2 nilpotent matrix of rank one.
Nilpotent factorization of singular matrices over an arbitrary field will also
be investigated. Laffey [17] shows that the result of Wu, which he established
over the complex field, is also valid over an arbitrary field by making use
of a special matrix factorization involving similarity to an LU factorization.
His proof is based on an application of Fitting's Lemma to express, up to
similarity, a singular matrix as a direct sum of a non-singular and nilpotent matrix, and then to write the non-singular component as a product of a lower and upper triangular matrix using a matrix factorization theorem of Sourour [24]. The main theorem by Sourour and Tang [26] will be investigated to highlight the necessary and sufficient conditions for a singular matrix to be written as a product of two matrices with prescribed eigenvalues. This result is used to prove applications related to positive-semidefinite matrices for singular matrices. / National Research Foundation of South Africa / Mathematical Sciences / M Sc. (Mathematics)
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E_1 ring structures in Motivic Hermitian K-theoryLópez-Ávila, Alejo 02 March 2018 (has links)
This Ph.D. thesis deals with E1-ring structures on the Hermitian K-theory in the motivic setting, more precisely, the existence of such structures on the motivic spectrum representing the hermitianK-theory is proven. The presence of such structure is established through two different approaches. In both cases, we consider the category of algebraic vector bundles over a scheme, with the usual requirements to do motivic homotopy theory. This category has two natural symmetric monoidal structures given by the direct sum and the tensor product, together with a duality coming from the functor represented by the structural sheaf. The first symmetric monoidal structure is the one that we are going to group complete along this text, and we will see that the second one, the tensor product, is preserved giving rise to an E1-ring structure in the resulting spectrum. In the first case, a classic infinite loop space machine applies to the hermitian category of the category of algebraic vector bundles over a scheme. The second approach abords the construction using a new hermitian infinite loop space machine which uses the language of infinity categories. Both assemblies applied to our original category have like output a presheaf of E1-ring spectra. To get an spectrum representing the hermitian K-theory in the motivic context we need a motivic spectrum, i.e, a P1-spectrum. We use a delooping construction at the end of the text to obtain a presheaf of E1-ring P1-spectra from the two presheaves of E1-ring spectra indicated above.
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Mixed Witt rings of algebras with involutionGarrel, Nicolas 04 April 2024 (has links)
Although there is no natural internal product for hermitian forms over an algebra with involution of the first kind, we describe how tomultiply two ε-hermitian forms to obtain a quadratic form over the base field. This allows to define a commutative graded ring structure by taking together bilinear forms and ε-hermitian forms, which we call the mixedWitt ring of an algebra with involution. We also describe a less powerful version of this construction for unitary involutions, which still defines a ring, but with a grading over Z instead of the Klein group. We first describe a general framework for defining graded rings out of monoidal functors from monoidal categories with strong symmetry properties to categories of modules. We then give a description of such a strongly symmetric category Brₕ(K, ι) which encodes the usual hermitian Morita theory of algebras with involutions over a field K. We can therefore apply the general framework to Brₕ(K, ι) and theWitt group functors to define our mixed Witt rings, and derive their basic properties, including explicit formulas for products of diagonal forms in terms of involution trace forms, explicit computations for the case of quaternion algebras, and reciprocity formulas relative to scalar extensions. We intend to describe in future articles further properties of those rings, such as a λ-ring structure, and relations with theMilnor conjecture and the theory of signatures of hermitian forms.
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Kähler and almost-Kähler geometric flows / Flots géométriques kähleriens et presque-kähleriensPook, Julian 21 March 2014 (has links)
Les objects d'étude principaux de la thèse "Flots géométriques kähleriens et presque-kähleriens" sont des généralisations du flot de Calabi et du flot hermitienne de Yang--Mills. <p> Le flot de Calabi $partial_t omega = -i delbar del S(omega) =- i delbar del Lambda_omega <p> ho(omega) $ tente de déformer une forme initiale kählerienne vers une forme kählerienne $omega_c$ de courbure scalaire constante caractérisée par $S(omega_c) = Lambda_{omega_c} <p> ho(omega_c) = underline{S}$ dans la même classe de cohomologie. La généralisation étudiée est le flot de Calabi twisté qui remplace la forme de Kähler--Ricci $ho$ par $ho + alpha(t)$, où le emph{twist} $alpha(t)$ est une famille de $2$-formes qui converge vers $alpha_infty$. Le but de ce flot est de trouver des métriques kähleriennes $omega_{tc}$ de courbure scalaire twistées constantes caractérisées par $Lambda_{omega_{tc}} (ho(omega_{tc}) +alpha_infty) = underline{S} + underline{alpha}_infty$. L'existence et la convergence de ce flot sont établies sur des surfaces de Riemann à condition que le twist soit défini négatif et reste dans une classe de cohomologie fixe. <p>Si $E$ est un fibré véctoriel holomorphe sur une varieté kählerienne $(X,omega)$, une métrique de Hermite--Einstein $h_{he}$ est caractérisée par la condition $Lambda_omega i F_{he} = lambda id_E$. Le flot hermitien de Yang--Mills donné par $h^{-1}partial_t h =- [Lambda_omega iF_{h} - lambda id_E]$ tente de déformer une métrique hermitienne initiale vers une métrique Hermite--Einstein. La version classique du flot fixe la forme kählerienne $omega$. Le cas où $omega$ varie dans sa classe de cohomologie et converge vers $omega_infty$ est considéré dans la thèse. Il est démontré que le flot existe pour tout $t$ sur des surfaces de Riemann et converge vers une métrique Hermite--Einstein (par rapport à $omega_infty$) si le fibré $E$ est stable. <p> Les généralisations du flot de Calabi et du flot hermitien de Yang--Mills ne sont pas arbitraires, mais apparaissent naturellement comme une approximation du flot de Calabi sur des fibrés adiabatiques. Si $Z,X$ sont des variétés complexes compactes, $pi colon Z \ / Doctorat en Sciences / info:eu-repo/semantics/nonPublished
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The differential geometry of the fibres of an almost contract metric submersionTshikunguila, Tshikuna-Matamba 10 1900 (has links)
Almost contact metric submersions constitute a class of Riemannian submersions whose
total space is an almost contact metric manifold. Regarding the base space, two types
are studied. Submersions of type I are those whose base space is an almost contact
metric manifold while, when the base space is an almost Hermitian manifold, then the
submersion is said to be of type II.
After recalling the known notions and fundamental properties to be used in the
sequel, relationships between the structure of the fibres with that of the total space
are established. When the fibres are almost Hermitian manifolds, which occur in the
case of a type I submersions, we determine the classes of submersions whose fibres
are Kählerian, almost Kählerian, nearly Kählerian, quasi Kählerian, locally conformal
(almost) Kählerian, Gi-manifolds and so on. This can be viewed as a classification of
submersions of type I based upon the structure of the fibres.
Concerning the fibres of a type II submersions, which are almost contact metric
manifolds, we discuss how they inherit the structure of the total space.
Considering the curvature property on the total space, we determine its corresponding
on the fibres in the case of a type I submersions. For instance, the cosymplectic
curvature property on the total space corresponds to the Kähler identity on the fibres.
Similar results are obtained for Sasakian and Kenmotsu curvature properties.
After producing the classes of submersions with minimal, superminimal or umbilical
fibres, their impacts on the total or the base space are established. The minimality of
the fibres facilitates the transference of the structure from the total to the base space.
Similarly, the superminimality of the fibres facilitates the transference of the structure
from the base to the total space. Also, it is shown to be a way to study the integrability
of the horizontal distribution.
Totally contact umbilicity of the fibres leads to the asymptotic directions on the total
space.
Submersions of contact CR-submanifolds of quasi-K-cosymplectic and
quasi-Kenmotsu manifolds are studied. Certain distributions of the under consideration
submersions induce the CR-product on the total space. / Mathematical Sciences / D. Phil. (Mathematics)
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Mathematical Foundations of Quantum Mechanics / Kvantfysikens Matematiska GrunderIsraelsson, Anders January 2013 (has links)
No description available.
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Výpočet vlastních čísel a vlastních vektorů hermitovské matice / Computation of the eigenvalues and eigenvectors of Hermitian matrixŠtrympl, Martin January 2016 (has links)
This project deals with computation of eigenvalues and eigenvectors of Hermitian positive-semidefinite complex square matrix of order 4. The target is an implementation of computation in language VHDL to field-programmable gate array of type Xilinx Zynq-7000. This master project deals with algorithms used for computation of eigenvalues and eigenvectors of positive-semidefinite symmetric real square and positive-semidefinite complex Hermitian matrix and the analysis of algorithms by AnalyzeAlgorithm program assembled for this purpose. The closing part of this project describes implementation of the computation into field-programmable gate array with use of IP core Xilinx® Floating-Point \linebreak Operator and SVAOptimalizer, SVAInterpreter and SVAToDSPCompiler programs.
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The differential geometry of the fibres of an almost contract metric submersionTshikunguila, Tshikuna-Matamba 10 1900 (has links)
Almost contact metric submersions constitute a class of Riemannian submersions whose
total space is an almost contact metric manifold. Regarding the base space, two types
are studied. Submersions of type I are those whose base space is an almost contact
metric manifold while, when the base space is an almost Hermitian manifold, then the
submersion is said to be of type II.
After recalling the known notions and fundamental properties to be used in the
sequel, relationships between the structure of the fibres with that of the total space
are established. When the fibres are almost Hermitian manifolds, which occur in the
case of a type I submersions, we determine the classes of submersions whose fibres
are Kählerian, almost Kählerian, nearly Kählerian, quasi Kählerian, locally conformal
(almost) Kählerian, Gi-manifolds and so on. This can be viewed as a classification of
submersions of type I based upon the structure of the fibres.
Concerning the fibres of a type II submersions, which are almost contact metric
manifolds, we discuss how they inherit the structure of the total space.
Considering the curvature property on the total space, we determine its corresponding
on the fibres in the case of a type I submersions. For instance, the cosymplectic
curvature property on the total space corresponds to the Kähler identity on the fibres.
Similar results are obtained for Sasakian and Kenmotsu curvature properties.
After producing the classes of submersions with minimal, superminimal or umbilical
fibres, their impacts on the total or the base space are established. The minimality of
the fibres facilitates the transference of the structure from the total to the base space.
Similarly, the superminimality of the fibres facilitates the transference of the structure
from the base to the total space. Also, it is shown to be a way to study the integrability
of the horizontal distribution.
Totally contact umbilicity of the fibres leads to the asymptotic directions on the total
space.
Submersions of contact CR-submanifolds of quasi-K-cosymplectic and
quasi-Kenmotsu manifolds are studied. Certain distributions of the under consideration
submersions induce the CR-product on the total space. / Mathematical Sciences / D. Phil. (Mathematics)
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