Integrals of characteristic classes of tautological sheaves on the Hilbert scheme of points on a surface frequently arise in enumerative problems. We use the K-theoretic Donaldson-Thomas theory of certain toric Calabi-Yau threefolds to study K-theoretic variants of such expressions. We study limits of the K-theoretic Donaldson-Thomas partition function of a toric Calabi-Yau threefold under certain one-parameter subgroups called slopes, and formulate a condition under which two such limits coincide. We then explicitly compute the limits of components of the partition function under so-called preferred slopes, obtaining explicit combinatorial expressions related to the refined topological vertex of Iqbal, Kos\c{c}az and Vafa. Applying these results to specific Calabi-Yau threefolds, we deduce dualities satisfied by a generating function built from tautological bundles on the Hilbert scheme of points on $\C^2$. We then use this duality to study holomorphic Euler characteristics of exterior and symmetric powers of tautological bundles on the Hilbert scheme of points on a general surface.

We give an explicit formula for the arithmetic intersection number of CM cycles on Lubin-Tate spaces for all levels. We prove our formula by formulating the intersection number on the infinite level. Our CM cycles are constructed by choosing two separable quadratic extensions K1, K2/F of non-Archimedean local fields F . Our formula works for all cases, K1 and K2 can be either the same or different, ramify or unramified. As applications, this formula translate the linear Arithmetic Fundamental Lemma (linear AFL) into a comparison of integrals. This formula can also be used to recover Gross and Keating’s result on lifting endomorphism of formal modules.

No description available.

No description available.

This thesis is concerned with the spectral theory of the Laplacian on non-Euclidean spaces and its intimate links with harmonic analysis and the theory of special functions. More specifically, it studies the spectral theory of the Laplacian on the quotients M = Γ\G/K and X = G/K, where G is a connected semisimple Lie group, K is a maximal compact subgroup of G and Γ is a discrete subgroup of G.

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

This thesis uses algebraic and combinatorial methods to study subsets of the Desarguesian plane IIq = PG(2, q). Emphasis, in particular, is given to complete (k, n)-arcs and plane projective curves. Known Diophantine equations for subsets of PG(2, q), no more than n of which are collinear, have been applied to k-arcs of arbitrary degree. This yields a new lower bound for complete (k, n)-arcs in PG(2, q) and is a generalization of a classical result of Barlotti. The bound is one of few known results for complete arcs of arbitrary degree and establishes new restrictions upon the parameters of associated projective codes. New results governing the relationship between (k, 3)-arcs and blocking sets are also provided. Here, a sufficient condition ensuring that a blocking set is induced by a complete (k, 3)-arc in the dual plane q is established and shown to complement existing knowledge of relationships between k-arcs and blocking sets. Combinatorial techniques analyzing (k, 3)-arcs in suitable planes are then introduced. Utilizing the numeric properties of non-singular cubic curves, plane (k, 3)-arcs satisfying prescribed incidence conditions are shown not to attain existing upper bounds. The relative sizes of (k, 3)-arcs and non-singular cubic curves are also considered. It is conjectured that m3(2, q), the size of the largest complete (k, 3)-arc in PG(2, q), exceeds the number of rational points on an elliptic curve. Here, a sufficient condition for its positive resolution is given using combinatorial analysis. Exploiting its structure as a (k, 3)-arc, the elliptic curve is then considered as a method of constructing cubic arcs and results governing completeness are established. Finally, classical theorems relating the order of the plane q to the existence of an elliptic curve with a specified number of rational points are used to extend theoretical results providing upper bounds to t3(2, q), the size of the smallest possible complete (k, 3)-arc in PG(2, q).

No description available.

In the projective space PG(k−1; q) over Fq, the finite field of order q, an (n; r)-arc K is a set of n points with at most r on a hyperplane and there is some hyperplane meeting K in exactly r points. An arc is complete if it is maximal with respect to inclusion. The arc K corresponds to a projective [n; k;n − r]q-code of length n, dimension k, and minimum distance n − r; if K is a complete arc, then the corresponding projective code cannot be extended. In this thesis, the n-sets in PG(1; 19) up to n = 10 and the n-arcs in PG(2; 19) for 4 B n B 20 in both the complete and incomplete cases are classified. The set of rational points of a non-singular, plane cubic curve can be considered as an arc of degree three. Over F19, these curves are classified, and the maximum size of the complete arc of degree three that can be constructed from each such incomplete arc is given.

A binary low-density parity-check (LDPC) code is a linear block code that is defined by a sparse parity-check matrix H, that is H has a low density of 1's. LDPC codes were originally presented by Gallager in his doctoral dissertation [9], but largely overlooked for the next 35 years. A notable exception was [29], in which Tanner introduced a graphical representation for LDPC codes, now known as Tanner graphs. However, interest in these codes has greatly increased since 1996 with the publication of [22] and other papers, since it has been realised that LDPC codes are capable of achieving near-optimal performance when decoded using iterative decoding algorithms. LDPC codes can be constructed randomly by using a computer algorithm to generate a suitable matrix H. However, it is also possible to construct LDPC codes explicitly using various incidence structures in discrete mathematics. For example, LDPC codes can be constructed based on the points and lines of finite geometries: there are many examples in the literature (see for example [18, 28]). These constructed codes can possess certain advantages over randomly-generated codes. For example they may provide more efficient encoding algorithms than randomly-generated codes. Furthermore it can be easier to understand and determine the properties of such codes because of the underlying structure. LDPC codes have been constructed based on incidence structures known as partial geometries [16]. The aim of this research is to provide examples of new codes constructed based on structures known as semipartial geometries (SPGs), which are generalisations of partial geometries. Since the commencement of this thesis [19] was published, which showed that codes could be constructed from semipartial geometries and provided some examples and basic results. By necessity this thesis contains a number of results from that paper. However, it should be noted that the scope of [19] is fairly limited and that the overlap between the current thesis and [19] is consequently small. [19] also contains a number of errors, some of which have been noted and corrected in this thesis.