Yu Chengjie. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (leaves 65-67). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgments --- p.ii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- A multiplicity estimate and applications --- p.5 / Chapter 2.1 --- A multiplicity estimate --- p.6 / Chapter 2.2 --- Sharp bounds for the dimensions of the spaces of holomorphic functions of polynomial growth --- p.14 / Chapter 2.3 --- Siegel's theorem on the fields of rational functions --- p.15 / Chapter 3 --- Quasi-embedding of complete Kahler manifolds --- p.21 / Chapter 3.1 --- The original map F0 --- p.21 / Chapter 3.2 --- Almost injectivity of F0 --- p.26 / Chapter 3.3 --- Almost surjectivity of F0 --- p.28 / Chapter 3.4 --- Weaker conditions for almost surjectivity --- p.41 / Chapter 3.5 --- Existence of quasi-embedding --- p.48 / Chapter 4 --- Desingularization of quasi-embeddings --- p.51 / Chapter 4.1 --- Normalization of a map with polynomial growth --- p.51 / Chapter 4.2 --- The method to desingularize a quasi-embedding --- p.54 / Chapter 4.3 --- The case of dimension two --- p.55 / Chapter 4.4 --- A uniformization theorem --- p.63 / Bibliography --- p.65

令X 為個帶有凱勒形式(Kähler form ω) 以及全純四形式( holomorphic four- form Ω )的四維緊致卡拉比丘空間(Calabi-Yau manifolds) 。在一些假設條件下，通過研究Donaldson- Thomas方程所決定的模空間，我們定義了四維Donaldson-Thomas不變量。我們也對四維局部卡拉比丘空間(local Calabi-Yau four-folds) 定義了四維Donaldson-Thomas 不變量，並且將之聯繫到三維Fano空間的Donaldson- Thomas 不變量。在一些情況下，我們還研究了DT/GW不變量對應。最后，我們在模空間光滑時計算了一些四維Donaldson- Thomas不變量。 / Let X be a complex four-dimensional compact Calabi-Yau manifold equipped with a Kahler form ω and a holomorphic four-form Ω. Under certain assumptions, we de ne Donaldson-Thomas type deformation invariants by studying the moduli space of the solutions of Donaldson-Thomas equations on the given Calabi-Yau manifold. We also study sheaves counting on local Calabi-Yau four-folds. We relate the sheaves countings over X = KY with the Donaldson- Thomas invariants for the associated compact three-fold Y . In some specialcases, we prove the DT/GW correspondence for X. Finally, we compute the Donaldson-Thomas invariants of certain Calabi-Yau four-folds when the moduli spaces are smooth. / Detailed summary in vernacular field only. / Cao, Yalong. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 100-105). / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- The *4 operator --- p.18 / Chapter 2.1 --- The *4 operator for bundles --- p.18 / Chapter 2.2 --- The *4 operator for general coherent sheaves --- p.20 / Chapter 3 --- Local Kuranishi structure of DT₄ moduli spaces --- p.22 / Chapter 4 --- Compactification of DT₄ moduli spaces --- p.34 / Chapter 4.1 --- Stable bundles compactification of DT₄ moduli spaces --- p.34 / Chapter 4.2 --- Attempted general compactification of DT₄ moduli spaces --- p.36 / Chapter 5 --- Virtual cycle construction --- p.39 / Chapter 5.1 --- Virtual cycle construction for DT₄ moduli spaces --- p.40 / Chapter 5.2 --- Virtual cycle construction for generalized DT₄ moduli spaces --- p.48 / Chapter 6 --- DT4 invariants for compactly supported sheaves on local CY₄ --- p.52 / Chapter 6.1 --- The case of X = KY --- p.52 / Chapter 6.2 --- The case of X = T*S --- p.57 / Chapter 7 --- DT₄ invariants on toric CY₄ via localization --- p.66 / Chapter 8 --- Computational examples --- p.70 / Chapter 8.1 --- DT₄=GW correspondence in some special cases --- p.71 / Chapter 8.1.1 --- The case of Hol(X) = SU(4) --- p.72 / Chapter 8.1.2 --- The case of Hol(X) = Sp(2) --- p.77 / Chapter 8.2 --- Some remarks on cosection localizations for hyper-kähler four-folds --- p.79 / Chapter 8.3 --- Li-Qin's examples --- p.80 / Chapter 8.4 --- Moduli space of ideal sheaves of one point --- p.83 / Chapter 9 --- Appendix --- p.85 / Chapter 9.1 --- Local Kuranishi models of Mc° --- p.85 / Chapter 9.2 --- Some remarks on the orientability of the determinant line bundles on the (generalized) DT₄ moduli spaces --- p.87 / Chapter 9.3 --- Seidel-Thomas twists --- p.90 / Chapter 9.4 --- A quiver representation of Mc --- p.92

Yiu, Fa Wai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 55-58). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Elementary properties and notations of Hyperbolic space --- p.9 / Chapter 3 --- Poisson kernel and Conformal densities --- p.16 / Chapter 3.1 --- Poisson kernel --- p.17 / Chapter 3.2 --- Conformal densities --- p.19 / Chapter 4 --- Patterson construction and decomposition --- p.27 / Chapter 4.1 --- Patterson construction --- p.27 / Chapter 4.2 --- Patterson decomposition --- p.33 / Chapter 5 --- Bonahon surfaces and Grided surfaces --- p.39 / Chapter 5.1 --- Bonahon surfaces --- p.40 / Chapter 5.2 --- Grided surfaces --- p.46 / Chapter 6 --- Margulis number of Hyperbolic Manifolds --- p.51 / Margulis Number for Hypcrbolic 3-manifolds --- p.5 / Chapter 6.1 --- Gcomertrically finite groups --- p.51 / Chapter 6.2 --- Margulis number of Closed Hyperbolic Manifolds --- p.53 / Bibliography --- p.55

The stable envelope for symplectic resolutions, constructed by Maulik and Okounkov, is a key ingredient in their work on quantum cohomology and quantum K-theory of Nakajima quiver varieties. In this thesis, we study the various aspects of the cohomological stable basis for the cotangent bundle of flag varieties. We compute its localizations, use it to calculate the quantum cohomology of the cotangent bundles, and relate it to the Chern--Schwartz--MacPherson class of Schubert cells in the flag variety.

This thesis consists of three applications of Ranicki's algebraic theory of surgery to the topology of manifolds. The common theme is a decomposition of a global algebraic object into simple local pieces which models the decomposition of a global topological object into simple local pieces. Part I: Algebraic reconstruction of 4-manifolds. We extend the product and glueing constructions for symmetric Poincaré complexes, pairs and triads to a thickening construction for a symmetric Poincaré representation of a quiver. Gay and Kirby showed that, subject to certain conditions, the fold curves and fibres of a Morse 2-function F : M4 → Ʃ 2 determine a quiver of manifold and glueing data which allows one to reconstruct M and F up to diffeomorphism. The Gay-Kirby method of reconstructing M glues the pre-images of disc neighbourhoods of cusps and crossings with thickenings of regular fibres and thickenings of cobordisms between regular fibres. We use our thickening construction for a symmetric Poincaré representation of a quiver to give an algebraic analogue of the Gay-Kirby result to reconstruct the symmetric Poincaré complex (C(M); ϕ M) of M from a Morse 2-function. Part II: The L-theory of triangular matrix rings. We construct a chain duality on the category of left modules over a triangular matrix ring A = (A1;A2;B) where A1;A2 are rings with involution and B is an (A1;A2)-bimodule. We describe the resulting L-theory of A and relate it to the L-theory of A1;A2 and to the change of rings morphism B ⊗A2 − : A2-Mod → A1-Mod. By examining algebraic surgery over A we define a relative algebraic surgery operation on an (n+1)-dimensional symmetric Poincaré pair with data an (n+2)-dimensional triad. This gives an algebraic model for a half-surgery on a manifold with boundary. We then give an algebraic analogue of Borodzik, Némethi and Ranciki's half-handle decomposition of a relative manifold cobordism and show that every relative Poincaré cobordism is homotopy equivalent to a union of traces of elementary relative surgeries. Part III: Seifert matrices of braids with applications to isotopy and signatures. Let β be a braid with closure ^β a link. Collins developed an algorithm to find the Seifert matrix of the canonical Seifert surface Ʃ of ^ β constructed by Seifert's algorithm. Motivated by Collins' algorithm and a construction of Ghys, we define a 1-dimensional simplicial complex K(β) and a bilinear form λβ : C1(K(β);Z)×C1(K(β);Z) → Z[ 1/2 ] such that there is an inclusion K(β) ~ → Ʃ which is a homotopy equivalence inducing an isomorphism H1(Ʃ;Z) ≅ H1(K(β);Z) such that [λβ] : H1(K(β);Z) × H1(K(β);Z) → Z ⊂ Z[ 1/2 ] is the Seifert form of Ʃ. We show that this chain level model is additive under the concatenation of braids and then verify that this model is chain equivalent to Banchoff's combinatorial model for the linking number of two space polygons and Ranicki's surgery theoretic model for a chain level Seifert pairing. We then define the chain level Seifert pair (λβ; d β) of a braid β and equivalence relations, called A and Â-equivalence. Two n-strand braids are isotopic if and only if their chain level Seifert pairs are A-equivalent and this yields a universal representation of the braid group. Two n-strand braids have isotopic link closures in the solid torus D2 ×S1 if and only if their chain level Seifert pairs are  A-equivalent and this yields a representation of the braid group modulo conjugacy. We use the first representation to express the ω signature of a braid β in terms of the chain level Seifert pair (λ β; d β).

Leung, Wai Sing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 61-62). / Abstracts in English and Chinese. / Chapter 0.1 --- Introduction --- p.6 / Chapter 1 --- Sphere Theorems --- p.8 / Chapter 1.1 --- Rauch-Berger-Klingenberg Sphere Theorem --- p.8 / Chapter 1.2 --- Maximal Diameter Theorem --- p.15 / Chapter 1.3 --- Minimal Diameter Theorem --- p.17 / Chapter 2 --- A Differentiable Sphere Theorem --- p.27 / Chapter 2.1 --- Definitions --- p.27 / Chapter 2.2 --- Preliminary results not related to curvature --- p.28 / Chapter 2.3 --- Preliminary result related to the curvature --- p.33 / Chapter 2.4 --- Differentiable Sphere Theorem --- p.35 / Chapter 3 --- The fundamental equations of Riemannian submer- sions --- p.43 / Chapter 3.1 --- Definitions --- p.43 / Chapter 3.2 --- The fundamental tensors T and A --- p.44 / Chapter 3.3 --- Covariant derivatives of T and A --- p.47 / Chapter 3.4 --- Fundamental equations and O'Neill's formulas --- p.49 / Chapter 4 --- A study on exotic spheres --- p.52 / Chapter 4.1 --- Construction of Milnor sphere --- p.52 / Chapter 4.2 --- Construction of GM-sphere (Σ7) --- p.53 / Chapter 4.3 --- The curvature of Σ7 --- p.54 / Chapter 4.4 --- Some recent results and open questions --- p.59 / Bibliography --- p.61

Suen, Chun Kit Anthony. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (leaves 89-92). / Abstracts in English and Chinese. / Abstract --- p.iii / Abstract --- p.iv / Acknowledgements --- p.v / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Morse Homology --- p.5 / Chapter 2.1 --- Introduction --- p.5 / Chapter 2.2 --- Classical Morse Theory and Morse Functions --- p.5 / Chapter 2.3 --- Handlebody Decomposition for 3-manifold --- p.7 / Chapter 2.4 --- Stable manifold and Unstable manifold --- p.10 / Chapter 2.5 --- Trajectory flows and the Morse-Smale-Witten Complex --- p.11 / Chapter 3 --- Lagrangian Floer Homology --- p.22 / Chapter 3.1 --- Introduction --- p.22 / Chapter 3.2 --- Preliminaries on Symplectic Geometry --- p.23 / Chapter 3.2.1 --- Basic Definitions --- p.23 / Chapter 3.2.2 --- The Symplectic Group --- p.26 / Chapter 3.2.3 --- Maslov index for non-degenerate paths in Sp(2n) --- p.28 / Chapter 3.2.4 --- Maslov index - the analytic aspect --- p.35 / Chapter 3.3 --- Definition of Floer Homology --- p.37 / Chapter 3.4 --- Some Remarks --- p.41 / Chapter 4 --- Heegaard Floer Homology --- p.43 / Chapter 4.1 --- Introduction --- p.43 / Chapter 4.2 --- Basic Set-Up --- p.43 / Chapter 4.3 --- Topological Preliminaries --- p.44 / Chapter 4.3.1 --- Symmetric Product --- p.44 / Chapter 4.3.2 --- The Tori Tα and Tβ --- p.47 / Chapter 4.3.3 --- Intersection Points and Disks --- p.48 / Chapter 4.3.4 --- Domains --- p.52 / Chapter 4.3.5 --- Spinc Structures --- p.54 / Chapter 4.3.6 --- Holomorphic Disks and Maslov Index --- p.63 / Chapter 4.4 --- Definition of Heegaard Floer Homology --- p.65 / Chapter 4.4.1 --- The chain complex CF --- p.66 / Chapter 4.4.2 --- The chain complex CF∞ --- p.67 / Chapter 4.4.3 --- The chain complexes CF+ and CF- --- p.68 / Chapter 4.4.4 --- Some Remarks --- p.70 / Chapter 5 --- Examples and Applications --- p.72 / Chapter 5.1 --- Introduction --- p.72 / Chapter 5.2 --- The homology three-spheres --- p.72 / Chapter 5.2.1 --- The sphere S3 --- p.72 / Chapter 5.2.2 --- The Poincare sphere and the Brieskorn spheres --- p.74 / Chapter 5.2.3 --- Long exact surgery sequence and the absolutely graded Hee- gaard Floer homology --- p.78 / Chapter 5.3 --- More Application --- p.84 / Chapter 5.3.1 --- Knot Floer homology --- p.84 / Chapter 5.3.2 --- Invariants on 4-manifolds --- p.86 / Chapter 5.4 --- Further developments --- p.87 / Bibliography --- p.89

published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

We introduce the naive notion of a relative open book decomposition for contact 3-manifolds with torus boundary. We then use this to construct nice, minimal genus open book decompositions compatible with all of the universally tight contact structures (as well as a few others) on torus-bundles over S¹, following Honda's classification. In an accurate sense, we find Stein fillings of 'half' of the torus bundles. In addition, these give the first examples of open books compatible with the universally tight contact structures on circle bundles over higher genus surfaces, as well, following a pattern introduced by a branched covering of B⁴. Some interesting examples of open books without positive monodromy are emphasized.

A quaternion-Hermitian manifold, of dimension at least 12, with closed fundamental 4-form is shown to be quaternionic Kähler. A similar result is proved for 8-manifolds. HyperKähler metrics are constructed on the fundamental quaternionic line bundle (with the zero-section removed) of a quaternionic Kähler manifold (indefinite if the scalar curvature is negative). This construction is compatible with the quaternionic Kähler and hyperKähier quotient constructions and allows quaternionic Kähler geometry to be subsumed into the theory of hyperKähler manifolds. It is shown that the hyperKähler metrics that arise admit a certain type of SU(2)- action, possess functions which are Kähler potentials for each of the complex structures simultaneously and determine quaternionic Kähler structures via a variant of the moment map construction. Quaternionic Kähler metrics are also constructed on the fundamental quaternionic line bundle and a twistor space analogy leads to a construction of hyperKähler metrics with circle actions on complex line bundles over Kähler-Einstein (complex) contact manifolds. Nilpotent orbits in a complex semi-simple Lie algebra, with the hyperKähler metrics defined by Kronheimer, are shown to give rise to quaternionic Kähler metrics and various examples of these metrics are identified. It is shown that any quaternionic Kähler manifold with positive scalar curvature and sufficiently large isometry group may be embedded in one of these manifolds. The twistor space structure of the projectivised nilpotent orbits is studied.