Mak, Kin Hei. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 48-50). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.2 / Chapter 2 --- Symplectic 4-Manifolds --- p.5 / Chapter 2.1 --- Basic Definitions --- p.5 / Chapter 2.2 --- Simple Examples of Symplectic Manifolds --- p.6 / Chapter 2.3 --- A Theorem of Thurston --- p.8 / Chapter 2.4 --- Lefschetz Pencils --- p.13 / Chapter 3 --- Classification of Lefschetz Fibrations --- p.17 / Chapter 3.1 --- Definitions --- p.17 / Chapter 3.2 --- Handlebody Decomposition --- p.19 / Chapter 3.3 --- Genus 1 --- p.29 / Chapter 3.4 --- Genus 2 --- p.36 / Chapter 3.5 --- Genus g≥3 --- p.43 / Bibliography --- p.48

Ma, Ding. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (p. 59-60). / Abstracts in English and Chinese. / Chapter 1 --- Generalized complex structures --- p.8 / Chapter 1.1 --- Maximal isotropic subspaces of V+ V* --- p.8 / Chapter 1.2 --- Courant bracket --- p.12 / Chapter 1.3 --- Dirac structures --- p.17 / Chapter 1.4 --- Linear generalized complex structures and almost gen- eralized complex structures --- p.19 / Chapter 1.5 --- Integrability conditions --- p.23 / Chapter 2 --- L∞-algebra --- p.25 / Chapter 2.1 --- Original definition of L∞-algebra in terms of lots of brackets --- p.25 / Chapter 2.2 --- Reformulation in terms of differential coalgebra --- p.27 / Chapter 2.3 --- Reformulation in terms of differential algebra --- p.28 / Chapter 2.4 --- Associating T+T* a Lie 2-algebra --- p.29 / Chapter 3 --- Generalized symplectic structures --- p.33 / Chapter 3.1 --- Linear generalized symplectic structures --- p.33 / Chapter 3.2 --- Generalized almost symplectic structures --- p.39 / Chapter 3.3 --- Generalized exterior derivatives and integrability con- ditions --- p.42 / Chapter 3.4 --- Generalized Darboux theorem --- p.45 / Chapter 3.5 --- Generalized Lagrangian submanifolds --- p.46 / Chapter 3.6 --- Generalized moment maps --- p.50 / Chapter 4 --- Generalized Kahler structures --- p.55 / Chapter 4.1 --- Definitions and integrability conditions --- p.55 / Bibliography --- p.59

In recent years symplectic geometry and symplectic topology have grown to large subbranches in mathematics and had a great impact on other areas in mathematics. When interested in geometry, a geometer always considers geometric structures that arise on immersed submanifolds. In symplectic geometry there is a distinguished class of immersions, known as Lagrangian submanifolds . In particular, minimal Lagrangian submanifolds, called special Lagrangians, are very important in mirror symmetry. Lagrangian mean curvature flow is an important example of Lagrangian deformation. From which we can get the special Lagrangian submanifolds. In recent years, there have been many papers about this subject and the result by K.Smoczyk and Mu-Tao Wang [WS] is very important and beautiful. Our main purpose in this article is to give a new proof for the main result in [WS] from the viewpoint of fully nonlinear partial differential equations.

No description available.

published_or_final_version / Civil and Structural Engineering / Doctoral / Doctor of Philosophy

Kwong, Kwok Kun. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 105-109). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgements --- p.iii / Introduction --- p.1 / Chapter 1 --- Morse Theory --- p.4 / Chapter 1.1 --- Introduction --- p.4 / Chapter 1.2 --- Morse Homology --- p.11 / Chapter 2 --- Symplectic Fixed Points and Arnold Conjecture --- p.24 / Chapter 2.1 --- Introduction --- p.24 / Chapter 2.2 --- The Variational Approach --- p.29 / Chapter 2.3 --- Action Functional and Moduli Space --- p.30 / Chapter 2.4 --- Construction of Floer Homology --- p.42 / Chapter 3 --- Fredholm Theory --- p.46 / Chapter 3.1 --- Fredholm Operator --- p.47 / Chapter 3.2 --- The Linearized Operator --- p.48 / Chapter 3.3 --- Maslov Index --- p.50 / Chapter 3.4 --- Fredholm Index --- p.57 / Chapter 4 --- Floer Homology --- p.75 / Chapter 4.1 --- Transversality --- p.75 / Chapter 4.2 --- Compactness and Gluing --- p.76 / Chapter 4.3 --- Floer Homology --- p.88 / Chapter 4.4 --- Invariance of Floer Homology --- p.90 / Chapter 4.5 --- An Isomorphism Theorem --- p.98 / Chapter 4.6 --- Further Applications --- p.103 / Bibliography --- p.105

<html> <head> <meta http-equiv="Content-Type" content="text/html;charset=iso-8859-1"> </head> Let &Sigma;<em>_g</em> be a closed Riemann surface of genus <em>g</em>. Generalizing Ivan Smith's construction, for each <em>g</em> &ge; 1 and <em>h</em> &ge; 0 we construct an infinite set of infinite families of homotopic but pairwise non-isotopic symplectic surfaces inside the product symplectic manifold &Sigma;<em>_g</em> ×&Sigma;<em>_h</em>. In particular, we achieve all positive genera from these families, providing first examples of infinite families of homotopic but pairwise non-isotopic symplectic surfaces of even genera inside &Sigma;<em>_g</em> ×&Sigma;<em>_h</em>.

<html> <head> <meta http-equiv="Content-Type" content="text/html;charset=iso-8859-1"> </head> Let &Sigma;<em>_g</em> be a closed Riemann surface of genus <em>g</em>. Generalizing Ivan Smith's construction, for each <em>g</em> &ge; 1 and <em>h</em> &ge; 0 we construct an infinite set of infinite families of homotopic but pairwise non-isotopic symplectic surfaces inside the product symplectic manifold &Sigma;<em>_g</em> ×&Sigma;<em>_h</em>. In particular, we achieve all positive genera from these families, providing first examples of infinite families of homotopic but pairwise non-isotopic symplectic surfaces of even genera inside &Sigma;<em>_g</em> ×&Sigma;<em>_h</em>.

Thesis (Ph. D.)--Ohio State University, 2004. / Title from first page of PDF file. Document formatted into pages; contains v, 88 p. Includes bibliographical references (p. 87-88). Available online via OhioLINK's ETD Center

Thesis (Ph. D.)--University of Hong Kong, 1998. / Includes bibliographical references (leaf 135-143).