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Chan Kwok Wai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 78-81). / Abstracts in English and Chinese. / Chapter 1 --- Introduction to Toric Geometry --- p.7 / Chapter 1.1 --- Definitions of Toric Varieties --- p.7 / Chapter 1.2 --- Properties of Toric Varieties --- p.11 / Chapter 1.2.1 --- Smoothness --- p.12 / Chapter 1.2.2 --- Compactness --- p.12 / Chapter 1.2.3 --- Stratification --- p.13 / Chapter 1.3 --- Divisors on Toric Varieties --- p.14 / Chapter 1.3.1 --- Weil divisors --- p.14 / Chapter 1.3.2 --- Cartier divisors --- p.14 / Chapter 1.4 --- Polarized Toric Varieties --- p.16 / Chapter 2 --- Calabi-Yau Manifolds from Toric Varieties --- p.19 / Chapter 2.1 --- Toric Fano Varieties --- p.19 / Chapter 2.2 --- Calabi-Yau Hypersurf aces in Toric Fano Varieties --- p.23 / Chapter 2.3 --- Computation of Hodge Numbers of Zf --- p.28 / Chapter 2.3.1 --- The results of Danilov and Khovanskii --- p.29 / Chapter 2.3.2 --- "The Hodge number hn-2,1(Zf)" --- p.31 / Chapter 2.3.3 --- "The Hodge number hl,1(zf)" --- p.32 / Chapter 2.4 --- Calabi-Yau Complete Intersections in Toric Fano Va- rieties --- p.34 / Chapter 3 --- Calabi-Yau Manifolds by Quotients --- p.41 / Chapter 3.1 --- Free Group Actions --- p.41 / Chapter 3.2 --- Crepant Resolutions of Orbifolds --- p.44 / Chapter 3.3 --- Examples From Complex Tori --- p.49 / Chapter 3.4 --- Complex Multiplication and Calabi-Yau Threefolds --- p.51 / Chapter 4 --- Calabi-Yau Manifolds by Coverings --- p.56 / Chapter 4.1 --- Cyclic Coverings --- p.56 / Chapter 4.2 --- Admissible Blow-ups --- p.57 / Chapter 4.3 --- Double Covers of P3 Branched Along Octic Arrang- ments --- p.59 / Chapter 4.4 --- The Euler Number of X --- p.61 / Chapter 4.5 --- The Hodge Numbers of X --- p.65 / Chapter 4.6 --- K3-Fibrations and Modularity --- p.69 / Chapter 0 --- Bibliography --- p.78

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Zhang Yi. / "May 2003." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (p. 141-146). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

本論文給出了在鏡像對稱中對非拉格朗日A-膜的Fourier型的變換。 / SYZ構想斷言，鏡像對稱應該來自於一種在卡拉比-丘流形上逐纖維的Fourier-Mukai變換。在半平坦卡拉比-丘流形上的拉格朗日A-膜的情形下，這已經被實現。然而， Kapustin和Orlov指出，對於一些特殊的卡拉比-丘流形， A-膜範疇應該加上某些額外的物件。他們稱這些額外的物件為餘迷向A-膜。在半平坦卡拉比-丘流形的情況下，我們需要加入一些在每個纖維上是楊-米爾斯的A-膜以及B-膜。 / 我們首先推廣Nahm變換到環面上的楊-米爾斯叢。這也可以看作一種Fourier型的變換。然後我們在半平坦卡拉比-丘流形上實施逐纖維的這種Nahm變換。我們在一些半平坦卡拉比丘流形上構造了一些新的B-膜的例子。這些B-膜限制到每一個纖維環面上都是環面上的楊-米爾斯叢。並且我們驗證了在這種逐纖維的變換下，他們恰好就是Kapustin和Orlov所提出的餘迷向A 膜。 / This thesis gives the construction of Fourier type transformations in mirror symmetry for non-Lagrangian A-branes. / The SYZ proposal asserts that mirror symmetry should come from a fiberwise Fourier-Mukai transformation along torus fibrations on Calabi-Yau manifolds. This can be realized explicitly for Lagrangian A-branes in semi-flat case. However, Kapustin and Orlov pointed out that for certain Calabi-Yau manifolds some extra objects called coisotropic A-branes should be added into the category of A-branes. In semi-flat cases, we need to include A-and B-branes which are Yang-Mills along fibers. / We first generalize the Nahm transformation to Yang-Mills line bundles over tori which can also be regarded as a Fourier type transformation. Then we carry out a family version of this transformation for semi-flat Calabi-Yau manifolds. More precisely, we construct a new class of B-branes in semi-flat Calabi-Yau manifolds which are Yang-Mills line bundles when restricted to each fiber torus. And we show that this fiberwise transformation of these B-branes produce the coisotropic A-branes predicted by Kapustin and Orlov. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Zhang, Yi. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 61-62). / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Fourier-Mukai Transformation --- p.10 / Chapter 2.1 --- A torus case --- p.10 / Chapter 2.1.1 --- Moduli space of flat U(1) bundles over T --- p.11 / Chapter 2.1.2 --- Poincare line bundle P --- p.12 / Chapter 2.1.3 --- Definition of the Fourier-Mukai Transformation for a torus --- p.13 / Chapter 2.1.4 --- Some concrete computations --- p.14 / Chapter 2.2 --- Semi-flat Calabi-Yau case --- p.15 / Chapter 2.2.1 --- Semi-flat Calabi-Yau manifolds and semi-flat branes --- p.15 / Chapter 2.2.2 --- Fourier-Mukai transformation for semi-flat branes --- p.18 / Chapter 3 --- Coisotropic A-branes --- p.23 / Chapter 3.1 --- Why Lagrangian branes are not enough? --- p.23 / Chapter 3.2 --- An example --- p.27 / Chapter 4 --- Nahm transformation --- p.29 / Chapter 4.1 --- Spinor bundle and the Dirac operator --- p.30 / Chapter 4.1.1 --- Clifford algebra and spin group --- p.30 / Chapter 4.1.2 --- Spinor bundle --- p.33 / Chapter 4.1.3 --- Dirac operator --- p.36 / Chapter 4.2 --- Nahm transformation for a torus (T, g) --- p.38 / Chapter 4.3 --- Fourier-Mukai transformation for coisotropic A-branes --- p.53

Mak, Kit Ho. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 78-81). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.7 / Chapter 2 --- Preliminaries on Number Theory --- p.10 / Chapter 2.1 --- Finite Fields --- p.10 / Chapter 2.2 --- p-adic Numbers --- p.13 / Chapter 2.3 --- The Teichmuller Representatives --- p.16 / Chapter 2.4 --- Character Theory --- p.18 / Chapter 3 --- Basic Calabi-Yau Geometry --- p.26 / Chapter 3.1 --- Definition and Basic Properties of Calabi-Yau Manifolds --- p.26 / Chapter 3.2 --- Calabi-Yau Manifolds of Low Dimensions --- p.29 / Chapter 3.3 --- Constructions of Calabi-Yau Manifolds --- p.32 / Chapter 3.4 --- Importance of Calabi-Yau Manifolds in Physics --- p.35 / Chapter 4 --- Number of Points on Calabi-Yau Manifolds over Finite Fields --- p.39 / Chapter 4.1 --- The General Method --- p.39 / Chapter 4.2 --- The Number of Points on a Family of Calabi-Yau Varieties over Finite Fields --- p.45 / Chapter 4.2.1 --- The Case ψ = 0 --- p.45 / Chapter 4.2.2 --- The Case ψ ß 0 --- p.50 / Chapter 4.3 --- The Number of Points on the Affine Mirrors over Finite Fields --- p.55 / Chapter 4.3.1 --- The Case ψ = 0 --- p.55 / Chapter 4.3.2 --- The Case ψ ß 0 --- p.56 / Chapter 4.4 --- The Number of points on the Projective Mirror over Finite Fields --- p.59 / Chapter 4.5 --- Summary of the Results and Related Conjectures --- p.61 / Chapter 5 --- The Relation Between Periods and the Number of Points over Finite Fields modulo q --- p.67 / Chapter 5.1 --- Periods of Calabi-Yau Manifolds --- p.67 / Chapter 5.2 --- The Case for Elliptic Curves --- p.69 / Chapter 5.2.1 --- The Periods of Elliptic Curves --- p.69 / Chapter 5.2.2 --- The Number of Fg-points on Elliptic Curves Modulo q --- p.70 / Chapter 5.3 --- The Case for a Family of Quintic Threefolds --- p.73 / Chapter 5.3.1 --- The Periods of Xψ --- p.73 / Chapter 5.3.2 --- The Number of F9-points on Quintic Three- folds Modulo q --- p.75 / Bibliography --- p.78

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Thesis (Ph. D.)--University of Texas at Austin, 2004. / Supervisor: Jacques Distler. Vita. Includes bibliographical references. Also available from UMI.

Thesis (Ph. D.)--University of Texas at Austin, 2002. / Vita. Includes bibliographical references. Available also from UMI Company.