Thesis (Ed.M.)--Boston University / An enrichment activity in non-metric geometry for sixth grade children academically talented in mathematics.

Thesis (Ed.M.)--Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / 2031-01-01

This thesis presents two main results on analytic and topological aspects of scalar curvature. The first is a gluing theorem for scalar-flat manifolds with vanishing mean curvature on the boundary. Our methods involve tools from conformal geometry and perturbation techniques for nonlinear elliptic PDE. The second part studies bordisms of positive scalar curvature metrics. We present a modification of the Schoen-Yau minimal hypersurface technique to manifolds with boundary which allows us to prove a hereditary property for bordisms of positive scalar curvature metrics. The main technical result is a convergence theorem for stable minimal hypersurfaces with free boundary in bordisms with long collars which may be of independent interest.

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

by Chen Bing-Long. / "August 2003." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (p. 68-71). / 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.

This thesis presents through a number of applications a self-contained and robust methodology for exploring mathematical models of pattern formation from the perspective of a dynamical system. The contents of this work applies the methodology to investigate the influence of the domain-size and geometry on the evolution of the dynamics modelled by reaction-diffusion systems (RDSs). We start with deriving general RDSs on evolving domains and in turn explore Arbitrary Lagrangian Eulerian (ALE) formulation of these systems. We focus on a particular RDS of activator-depleted class and apply the detailed framework consisting of the application of linear stability theory, domain-dependent harmonic analysis and the numerical solution by the finite element method to predict and verify the theoretically proposed behaviour of pattern formation governed by the evolving dynamics. This is achieved by employing the results of domain-dependent harmonic analysis on three different types of two-dimensional convex and non-convex geometries consisting of a rectangle, a disc and a flat-ring.

The projective plane, PG(2;q), over a Galois field Fq is an incidence structure of points and lines. A (k;n)-arc K in PG(2;q) is a set of k points such that no n+1 of them are collinear but some n are collinear. A (k;n)-arc K in PG(2;q) is called complete if it is not contained in any (k+1;n)-arc. The existence of arcs for particular values of k and n pose interesting problems in finite geometry. It connects with coding theory and graph theory, with important applications in computer science. The main problem, known as the packing problem, is to determine the largest size mn(2;q) of K in PG(2;q). This problem has received much attention. Here, the work establishes complete arcs with a large number of points. In contrast, the problem to determine the smallest size tn(2;q) of a complete (k;n)-arc is mostly based on the lower bound arising from theoretical investigations. This thesis has several goals. The first goal is to classify certain (k;4)-arcs for k = 6,...,38 in PG(2;13). This classification is established through an approach in Chapter 2. This approach uses a new geometrical method; it is a combination of projective inequivalence of (k;4)-arcs up to k = 6 and certain sdinequivalent (k;4)-arcs that have sd-inequivalent classes of secant distributions for k = 7,...,38. The part related to projectively inequivalent (k;4)-arcs up to k=6 starts by fixing the frame points f1;2;3;88g and then classify the projectively inequivalent (5;4)-arcs. Among these (5;4)-arcs and (6;4)-arcs, the lexicographically least set are found. Now, the part regarding sd-inequivalent (k;4)-arcs in this method starts by choosing five sd-inequivalent (7;4)-arcs. This classification method may not produce all sd-inequivalent classes of (k;4)-arcs. However, it was necessary to employ this method due to the increasing number of (k;4)-arcs in PG(2;13) and the extreme computational difficulty of the problem. It reduces the constructed number of (k;4)-arcs in each process for large k. Consequently, it reduces the executed time for the computation which could last for years. Also, this method decreases the memory usage needed for the classification. The largest size of (k;4)-arc established through this method is k = 38. The classification of certain (k;4)-arcs up to projective equivalence, for k = 34,35,36,37,38, is also established. This classification starts from the 77 incomplete (34;4)-arcs that are constructed from the sd-inequivalent (33;4)-arcs given in Section 2.29, Table 2.35. Here, the largest size of (k;4)-arc is still k = 38. In addition, the previous process is re-iterated with a different choice of five sd-inequivalent (7;4)-arcs. The purpose of this choice is to find a new size of complete (k;4)-arc for k > 38. This particular computation of (k;4)-arcs found no complete (k;4)-arc for k > 38. In contrast, a new size of complete (k;4)-arc in PG(2;13) is discovered. This size is k = 36 which is the largest complete (k;4)-arc in this computation. This result raises the second largest size of complete (k;4)-arc found in the first classification from k = 35 to k = 36. The second goal is to discuss the incidence structure of the orbits of the groups of the projectively inequivalent (6;4)-arcs and also the incidence structures of the orbits of the groups other than the identity group of the sd-inequivalent (k;4)-arcs. In Chapter 3, these incidence structures are given for k = 6,7,8,9,10,11,12,13,14,38. Also, the pictures of the geometric configurations of the lines and the points of the orbits are described. The third goal is to find the sizes of certain sd-inequivalent complete (k;4)-arcs in PG(2;13). These sizes of complete (k;4)-arcs are given in Chapter 4 where the smallest size of complete (k;4)-arc is at most k = 24 and the largest size is at least k = 38. The fourth goal is to give an example of an associated non-singular quartic curve C for each complete (k;4)-arc and to discuss the algebraic properties of each curve in terms of the number I of inflexion points, the number jC \K j of rational points on the corresponding arc, and the number N1 of rational points of C . These curves are given in Chapter 5. Also, the algebraic properties of complete arcs of the most interesting sizes investigated in this thesis are studied. In addition, there are two examples of quartic curves C (g0 1) and C (g0 2) attaining the Hasse-Weil- Serre upper bound for the number N1 of rational points on a curve over the finite field of order thirteen. This number is 32. The fifth goal is to classify the (k;4)-arcs in PG(2;13) up to projective inequivalence for k < 10. This classification is established in Chapter 6. It starts by fixing a triad, U1, on the projective line, PG(1;13). Here, the number of projectively inequivalent (k;4)-arcs are tested by using the tool given in Chapter 2. Then, among the number of the projectively inequivalent (10;4)-arcs found, the classification of sd-inequivalent (k;4)-arcs for k = 10 is made. The number of these sd-inequivalent arcs is 36. Then, the 36 sd-inequivalent arcs are extended. The aim here is to investigate if there is a new size of sd-inequivalent (k;4)-arc for k > 38 that can be obtained from these arcs. The largest size of sd-inequivalent (k;4)-arc in this process is the same as the largest size of the sd-inequivalent (k;4)-arc established in Chapter 2, that is, k = 38. In addition, the classification of (k;n)-arcs in PG(2;13) is extended from n = 4 to n = 6. This extension is given in Chapter 7 where some results of the classification of certain (k;6)-arcs for k = 9; : : : ;25 are obtained using the same method as in Chapter 2 for k = 7,...,38. This process starts by fixing a certain (8;6)-arc containing six collinear points in PG(2;13).

Lee, King Leung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leave 95). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 1.1 --- Organization --- p.10 / Chapter 2 --- Semigroups and Cones --- p.13 / Chapter 2.1 --- Relation between Semigroups and Cones --- p.13 / Chapter 2.2 --- Subadditive Functions on Semigroups --- p.23 / Chapter 2.3 --- Relation between Cones and Bases --- p.29 / Chapter 3 --- General Theories of Okounkov Bodies --- p.33 / Chapter 3.1 --- Okounkov Bodies and Volumes --- p.33 / Chapter 3.2 --- Relation of Subadditive Functions on Semigroups and Okounkov Bodies --- p.39 / Chapter 3.3 --- Convex Functions on Okounkov Bodies --- p.47 / Chapter 4 --- Okounkov Bodies and Complex Geometry --- p.55 / Chapter 4.1 --- Holomorphic Line Bundles --- p.55 / Chapter 4.2 --- Chebyshev Transform --- p.65 / Chapter 4.3 --- Bernstein-Markov Norms --- p.74 / Chapter 5 --- Applications of Okounkov Bodies --- p.81 / Chapter 5.1 --- Relative Energy of Weights --- p.81 / Chapter 5.2 --- Computational Methods and Some Examples --- p.89 / Bibliography --- p.95

In this dissertation we study tropicalization curves which have a theta characteristic with large rank. This fits in the more general framework of studying the limit linear series on a curve which degenerates to a singular curve. We explore this when the singular curve is not of compact type. In particular we investigate the case when dual graph of the degenerate curve is a chain of g-loops. The fundamental object under consideration is a family of curves over a complete discrete valuation ring. In the first half of the dissertation we study geometry of such a family. In the third chapter we study metric graphs and divisors on them. This could be a thought of as the theory of limit linear series on a curve of non-compact type. In the fourth chapter we make this connection via tropicalization. We consider a family of curves with smooth generic fiber X η of genus g such that the dual graph of the special fiber is a chain of g loops. The main theorem we prove is that if X η has a theta characteristic of rank r then there are at least r linear relations on the edge lengths of the dual graph.

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