by Tsoi, Man. / Thesis submitted in: July 1999. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (leaves 56-57). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 2 --- Preliminary --- p.12 / Chapter 2.1 --- Lie Group and Lie Algebra --- p.12 / Chapter 2.2 --- Harmonic Maps --- p.15 / Chapter 2.3 --- Some Factorization theorems --- p.17 / Chapter 3 --- A Survey on Unlenbeck's Results --- p.22 / Chapter 3.1 --- Preliminary --- p.24 / Chapter 3.2 --- Extended Solutions --- p.26 / Chapter 3.3 --- The Variational Formulas for the Extended Solutions --- p.30 / Chapter 3.4 --- "The Representation of A(S2, G) on holomorphic maps C* → G" --- p.33 / Chapter 3.5 --- An Action of G) on extended solutions and Backlund Transformations --- p.39 / Chapter 3.6 --- The Additional S1 Action --- p.42 / Chapter 3.7 --- Harmonic Maps into Grassmannians --- p.43 / Chapter 4 --- Harmonic Maps into Compact Lie Groups --- p.47 / Chapter 4.1 --- Symmetry group of the harmonic map equation --- p.48 / Chapter 4.2 --- A New Formulation --- p.49 / Chapter 4.3 --- "Harmonic Maps into Grassmannian, Another Point of View" --- p.53 / Bibliography

Cheng Man Chuen. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2006. / Includes bibliographical references (leaves 78-80). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Different Notions of Negative Curvatures for Kahler Manifolds --- p.3 / Chapter 2.1 --- Definitions --- p.3 / Chapter 2.2 --- Adequate negativity of curvatures of classical domains of type I --- p.13 / Chapter 2.3 --- Adequate negativity of curvatures of classical domains of type IV --- p.23 / Chapter 2.4 --- Structure of complex semi-simple Lie algebra and its relation with Hermitian symmetric spaces --- p.28 / Chapter 2.5 --- Adequate negativity of curvatures of classical domains of type II and III --- p.34 / Chapter 2.6 --- Adequate negativity of curvatures of the two excep- tional bounded symmetric domains --- p.45 / Chapter 3 --- Complex-analyticity of Harmonic Maps between Compact Kahler Manifolds --- p.50 / Chapter 3.1 --- Existence of harmonic maps --- p.50 / Chapter 3.2 --- A Bochner type identity --- p.51 / Chapter 3.3 --- Complex-analyticity of harmonic maps --- p.58 / Chapter 3.4 --- Strong rigidity theorems --- p.62 / Chapter 3.5 --- Some further results from the Bochner technique --- p.64 / Chapter 4 --- Generalization to the Non-compact case --- p.67 / Chapter 4.1 --- A strong rigidity theorem for non-compact Kahler manifolds --- p.67 / Chapter 4.2 --- An existence theorem of harmonic map for Rieman- nian manifolds of finite volumes --- p.69 / Chapter 4.3 --- Bochner formula in the non-compact case --- p.71 / Bibliography --- p.78

by Wu Fung Leung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1996. / Includes bibliographical references (leaves 92-95). / Acknowledgements --- p.i / Notations --- p.ii / Introduction --- p.1 / Chapter 1 --- Preliminaries --- p.8 / Chapter 1.1 --- Formulations of Harmonic Maps --- p.8 / Chapter 1.2 --- Function Spaces --- p.11 / Chapter 1.3 --- Penalized Equations --- p.13 / Chapter 2 --- Main Lemmas --- p.15 / Chapter 2.1 --- Short Time Existence --- p.16 / Chapter 2.2 --- Energy Inequalities --- p.18 / Chapter 2.3 --- The Monotonicity Inequalities --- p.23 / Chapter 2.4 --- e - Regularity Theorem --- p.30 / Chapter 3 --- The Compact Case --- p.39 / Chapter 3.1 --- Existence and Regularity for dim M = 2 --- p.39 / Chapter 3.2 --- Existence and Regularity for dim M ≥ 2 --- p.49 / Chapter 3.3 --- Blow-up Results --- p.61 / Chapter 3.4 --- Existence of Harmonic maps --- p.69 / Chapter 4 --- The Noncompact Case --- p.74 / Chapter 4.1 --- Heat-flows from Rm --- p.75 / Chapter 4.2 --- Basic Lemmas --- p.77 / Chapter 4.3 --- Nonpositive Curvature Target Manifolds --- p.83 / Chapter 4.4 --- Dirichlet Problem at Infinity --- p.88 / Bibliography --- p.92

We seek to discover combinatorial explanations of known identities involving harmonic numbers. Harmonic numbers do not readily lend themselves to combinatorial interpretation, since they are sums of fractions, and combinatorial arguments involve counting whole objects. It turns out that we can transform these harmonic identities into new identities involving Stirling numbers, which are much more apt to combinatorial interpretation. We have proved four of these identities, the first two being special cases of the third.

Consider a region Ω in the plane and a point z0 in Ω. If a particle which travels randomly, by Brownian motion, is released from z0, then it will eventually cross the boundary of Ω somewhere. We define the harmonic measure distribution function, or h-function hΩ, in the following way. For each r > 0, let hΩ(r) be the probability that the point on the boundary where the particle first exits the region is at a distance at most r from z0. We would like to know, given a function f, whether there is some region Ω such that f is the h-function of that region. We investigate this question using convergence properties of domains and of h-functions. We show that any well behaved sequence of regions must have a convergent subsequence. This, together with previous results, implies that any function f that can be written as the limit of the h-functions hΩn of a sufficiently well behaved sequence{Ωn}ofregionsis the h-function of some region. We also make some progress towards finding sequences {Ωn} of regions whose h-functions converge to some predetermined function f, and which are sufficiently well behaved for our results to apply. Thus, we make some progress towards showing that certain functions f are in fact the h- function of some region.

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The limits of current micro-scale technology is approaching rapidly. As the technology is going toward nano-scale devices, physical phenomena involved are fundamentally different from micro-scale ones [1], [2]. Principles in classical physics are no longer powerful enough to explicate the phenomena involved in nano-scale devices. At this stage, quantum mechanic sheds some light on those topics which cannot be described by classical physics. The primary focus of this research work is the development of an analysis technique for understanding the behavior of strongly perturbed harmonic oscillators. Developing ``auxiliary'' boundary value problems we solve monomially perturbed harmonic oscillators. Thereby, we assume monomial terms of arbitrary degree and any finite coefficient desired. The corresponding eigenvalues and eigenvectors can be utilized to solve more complex anharmonic oscillators with non polynomial anharmonicity or numerically defined anharmonicity. A large number of numerical calculations demonstrate the robustness and feasibility of our technique. Particular attention has been paid to the details as have implemented the underlying formula. We have developed iterative expressions for the involved integrals and the introduced ``Universal Functions.'' The latter are applications and adaptations of a concept which was developed in 1990's to accelerate computations in the Boundary Element Method.

Thesis (Ph. D.)--University of Texas at Austin, 2006. / Vita. Includes bibliographical references.

Given a surface of genus g with fundamental group π, a representation of π into PSL(2,R) is a homomorphism that assigns to each generator of π an element of P SL(2, R). The group P SL(2, R) acts on Hom(π, P SL(2, R)) by conjugation. Define therepresentationspaceRg tobethequotientspaceHom(π,PSL(2,R))\PSL(2,R). Associated to each representation ρ is a number e(ρ) called its Euler class. Goldman showed that the space Rg has components that can be indexed by Euler classes of rep- resentations, and that there is one component for each integer e satisfying |e| ≤ 2g−2. The two maximal components correspond to Teichmu ̈ller space, the space of isotopy classes of hyperbolic structures on a surface. Teichmu ̈ller space is known to be homeomorphic to a ball of dimension 6g − 6. The other components of Rg are not as well understood. The theory of harmonic maps between non-positively curved manifolds has been used to study Teichmu ̈ller space. Given a harmonic map between hyperbolic surfaces, there is an associated quadratic differential on the domain surface called the Hopf differential. Wolf, following Sampson, proved that via the Hopf differential, harmonic maps parametrize Teichmu ̈ller space. This thesis extends his work to the case of branched hyperbolic structures, which correspond to certain elements in non- maximal components of representation space. More precisely, a branched hyperbolic structure is a pair (M, σ|dz|2) where M is a compact surface of genus g and σ|dz|2 is a hyperbolic metric with integral order cone singularities at a finite number of points expressed in terms of a conformal parameter. Fix a base surface (M, σ|dz|2). For each target surface (M, ρ|dw|2) with the same number and orders of cone points as (M,σ|dz|2), there is a unique harmonic map w : (M,σ|dz|2) → (M,ρ|dw|2) homotopic to the identity that fixes the cone points of M pointwise. Thus we may define another map from the space of branched hyperbolic structures with the same number and orders of cone points to the space of meromorphic quadratic differentials on the base surface M. This map, Φ, takes the harmonic map w associated with a metric ρ|dw|2 to the Hopf differential of w. This thesis shows that the map Φ is injective.