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The problem of embedding a manifold in Euclidean space is considered. Manifolds are introduced in Chapter I along with other basic definitions and examples. Chapter II contains a proof of the Regular Value Theorem along with the "Easy" Whitney Embedding Theorem. In Chapter III, vector bundles are introduced and some of their properties are discussed. Chapter IV introduces the Stiefel-Whitney classes and the four properties that characterize them. Finally, in Chapter V, the Stiefel-Whitney classes are used to produce a lower bound on the dimension of Euclidean space that is needed to embed real projective space.

Tsoi, Hung Ming. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 88-90). / Abstracts in English and Chinese. / Preface --- p.7 / Chapter 1 --- Kahler Manifold --- p.10 / Chapter 1.1 --- Hermitian Manifold --- p.12 / Chapter 1.2 --- Kahler Manifold --- p.13 / Chapter 1.2.1 --- "Positive (l,l)-form" --- p.15 / Chapter 2 --- Vector Bundle --- p.16 / Chapter 2.1 --- Holomorphic Vector Bundle and Connection --- p.17 / Chapter 2.2 --- Hermitian Connection and Chern Connection --- p.18 / Chapter 2.2.1 --- Existence of Chern connection on a holomorphic vector bundle --- p.19 / Chapter 2.3 --- Curvature --- p.21 / Chapter 2.4 --- Positivity of Vector Bundles --- p.23 / Chapter 2.5 --- Chern Classes and Holomorphic Line Bundle --- p.24 / Chapter 2.5.1 --- Chern class in axiomatic approach --- p.25 / Chapter 2.5.2 --- Chern class in algebraic topology --- p.26 / Chapter 2.5.3 --- Chern class in terms of curvature --- p.27 / Chapter 2.5.4 --- In the case of hermitian line bundle --- p.28 / Chapter 3 --- Analytic Technique on Kahler Manifold --- p.30 / Chapter 3.1 --- Dolbeault Cohomology --- p.30 / Chapter 3.2 --- Commutator Relations on Kahler Manifold --- p.31 / Chapter 3.2.1 --- Commutator relation on a line bundle --- p.32 / Chapter 3.3 --- Hodge Theory --- p.33 / Chapter 3.4 --- Bochner Technique --- p.35 / Chapter 3.4.1 --- Bochner-Kodaira-Nakano identity --- p.36 / Chapter 4 --- Kodaira Vanishing Theorem and L2 estimate of d --- p.38 / Chapter 4.1 --- Kodaira Vanishing Theorem --- p.39 / Chapter 4.2 --- Extension of Kodaira Vanishing Theorem by L2 Method --- p.44 / Chapter 4.2.1 --- Plurisubharmonic functions and weakly pseudoconvex Kahler manifold --- p.47 / Chapter 5 --- Multiplier Ideal Sheaf --- p.55 / Chapter 5.1 --- Algebraic Properties of Multiplier Ideal Sheaf --- p.56 / Chapter 5.2 --- Some Calculations of Multiplier Ideal Sheaf --- p.59 / Chapter 6 --- Nadel Vanishing Theorem --- p.62 / Chapter 6.1 --- Nadel Vanishing Theorem by L2 Estimate of d --- p.62 / Chapter 6.2 --- The Original Setting of Nadel --- p.64 / Chapter 6.2.1 --- S-bounded and S-null sequence --- p.65 / Chapter 6.2.2 --- Multiplier ideal sheaf by Nadel --- p.67 / Chapter 6.3 --- Nadel Vanishing Theorem by Computation of Cech Cohomology --- p.69 / Chapter 6.3.1 --- L2 estimate of d --- p.69 / Chapter 6.3.2 --- Koszul cochain --- p.70 / Chapter 6.3.3 --- The cohomology vanishing theorem --- p.73 / Chapter 7 --- Kawamata-Viehweg Vanishing Theorem --- p.77 / Chapter 7.1 --- Numerically Effective Line Bundle --- p.77 / Chapter 7.2 --- Kawamata-Viehweg Vanishing Theorem --- p.85 / Bibliography --- p.88

Thesis (Ph. D.)--Ohio State University, 2006. / Title from first page of PDF file. Includes bibliographical references (p. 69-71).

The instanton solutions to the Yang-Mills equations have a vast range of practical applications in field theories including gravitation and electro-magnetism. Solutions to Maxwell's equations, for example, are abelian gauge instantons on Minkowski space. Since these discoveries, a generalised theory of instantons has been emerging for manifolds with special holonomy. Beginning with connections and curvature on complex vector bundles, this thesis provides some of the essential background for studying moduli spaces of instantons. Manifolds with exceptional holonomy are special types of seven and eight dimensional manifolds whose holonomy group is contained in G2 and Spin(7), respectively. Focusing on the G2 case, instantons on G2 manifolds are defined to be solutions to an analogue of the four dimensional anti-self-dual equations. These connections are known as Donaldson-Thomas connections and a couple of examples are noted.

The instanton solutions to the Yang-Mills equations have a vast range of practical applications in field theories including gravitation and electro-magnetism. Solutions to Maxwell's equations, for example, are abelian gauge instantons on Minkowski space. Since these discoveries, a generalised theory of instantons has been emerging for manifolds with special holonomy. Beginning with connections and curvature on complex vector bundles, this thesis provides some of the essential background for studying moduli spaces of instantons. Manifolds with exceptional holonomy are special types of seven and eight dimensional manifolds whose holonomy group is contained in G2 and Spin(7), respectively. Focusing on the G2 case, instantons on G2 manifolds are defined to be solutions to an analogue of the four dimensional anti-self-dual equations. These connections are known as Donaldson-Thomas connections and a couple of examples are noted.

The Hilbert scheme of \(n\) points in the projective plane parameterizes degree \(n\) zero-dimensional subschemes of the projective plane. We examine the dual cones of effective divisors and moving curves on the Hilbert scheme. By studying interpolation, restriction, and stability properties of certain vector bundles on the plane we fully determine these cones for just over three fourths of all values of \(n\). A general Steiner bundle on \(\mathbb{P}^N\) is a vector bundle \(E\) admitting a resolution of the form \(0 \rightarrow \mathcal{O}_{\mathbb{P}^N} (−1)^s {M \atop \rightarrow} \mathcal{O}^{s+r}_{\mathbb{P}^N} \rightarrow E \rightarrow 0\), where the map \(M\) is general. We complete the classification of slopes of semistable Steiner bundles on \(\mathbb{P}^N\) by showing every admissible slope is realized by a bundle which restricts to a balanced bundle on a rational curve. The proof involves a basic question about multiplication of polynomials on \(\mathbb{P}^1\) which is interesting in its own right. / Mathematics

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 1999. / Includes bibliographical references. Also available on the Internet.

Thesis (Ph. D.)--University of Chicago, Department of Mathematics, June 2000. / Includes bibliographical references. Also available on the Internet.