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

Given a symplectic manifold, it is of interest how Lie group actions, their orbit spaces look like and what are some topological requirements on the existence of such actions. In this thesis we present the work of Ono, giving some sufficient conditions for non-existence of circle actions on symplectic manifolds and work of Li, describing the fundamental groups of symplectic reductions of circle actions.

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.

Thesis (M.Phil.)--Hong Kong University of Science and Technology, 2008. / Includes bibliographical references (leaves 35). Also available in electronic version.

We introduce an $A_\infty$ map from the cubical chain complex of the based loop space of Lagrangian submanifolds with Legendrian boundary in a Liouville Manifold $C_{*}(\Omega_{L} \Lag)$ to wrapped Floer cohomology of Lagrangian submanifold $\CW^{-*}(L,L)$. In the case of a cotangent bundle and a Lagrangian co-fiber, the composition of our map with the map from $\CW^{-*}(L,L) \to C_{*}(\Omega_q Q) $ as defined in \cite{Ab12} shows that this map is split surjective.

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We apply branched covering techniques to construct minimal simply-connected symplectic 4-manifolds with small χ_h values. We also use these constructions to provide an alternate proof that for each s ≥ 0, there exists a positive integer λ(s) such that each pair (j,8j+s) with j ≥ λ(s) is realized as (χ_h(M),c_1^2(M)) for some minimal simply-connected symplectic M. The smallest values of λ(s) currently known to the author are also explicitly computed for 0 ≤ s ≤ 99. Our computations in these cases populate 19 952 points in the (χ,c)-plane not previously realized in the existing literature.

We apply branched covering techniques to construct minimal simply-connected symplectic 4-manifolds with small χ_h values. We also use these constructions to provide an alternate proof that for each s ≥ 0, there exists a positive integer λ(s) such that each pair (j,8j+s) with j ≥ λ(s) is realized as (χ_h(M),c_1^2(M)) for some minimal simply-connected symplectic M. The smallest values of λ(s) currently known to the author are also explicitly computed for 0 ≤ s ≤ 99. Our computations in these cases populate 19 952 points in the (χ,c)-plane not previously realized in the existing literature.

In this project we introduce the general idea of geometric quantization and demonstratehow to apply the process on a few examples. We discuss how to construct a line bundleover the symplectic manifold with Dirac’s quantization conditions and how to determine if we are able to quantize a system with the help of Weil’s integrability condition. To reducethe prequantum line bundle we employ real polarization such that the system does notbreak Heisenberg’s uncertainty principle anymore. From the prequantum bundle and thepolarization we construct the sought after Hilbert space.