The thesis consists of four chapters. First chapter is introductory. In Chapter 2, we recall some basic facts from the singularity theory of algebraic varieties (see Section 2.2) and the theory of minimal models (see Section 2.3), which will be used throughout the rest of the thesis. We also make some conventions on the notions and notation used in the thesis (see Section 2.1). Each Chapter 3 and 4 starts with some preliminary results (see Sections 3.1 and 4.1, respectively). Each Chapter 3 and 4 ends with some corollaries and conclusive remarks (see Sections 3.7 and 4.4, respectively). In Chapter 3, we prove Theorem 1.2.7, providing the complete description of Halphen pencils on a smooth projective quartic threefold X in P4. Let M be such a pencil. Firstly, we show that M ⊂ | − nKX | for some n ∈ N, and the pair (X,1n M) is canonical but not terminal. Further, if the set of not terminal centers CS(X, 1 ) (see Remark 2.2.8) does not contain points, we show that n = 1 (see Section 3.2). Finally, if there is a point P ∈ CS(X, n M), in Section 3.1 we show first that a general M ∈ M has multiplicity 2n at P (cf. Example 1.2.3). After that, analyzing the shape of the Hessian of the equation of X at the point P , we prove that n = 2 and M coincides with the exceptional Halphen pencil from Example 1.2.6 (see Sections 3.3-3.6). In Chapter 4, we prove Theorem 1.2.11, which shows, in particular, that a general smooth K3 surfaces of type R is an anticanonical section of the Fano threefold X with canonical Gorenstein singularities and genus 36. In Section 4.2, we prove that X is unique up to an isomorphism and has a unique singular point, providing the geometric quotient construction of the moduli space F in Section 4.3 (cf. Remark 1.2.12). Finally, in Section 4.3 we prove that the forgetful map F −→ KR is generically surjective.

Lau, Sui Ki. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 81). / Abstract also in Chinese. / Chapter 1 --- Basic Notions of Riemann Surfaces --- p.5 / Chapter 1.1 --- "Functions, Forms and Hurwitz's Formula" --- p.6 / Chapter 1.2 --- Divisors --- p.10 / Chapter 1.3 --- Plucker's Formula for a Smooth Projective Plane Curve --- p.14 / Chapter 1.4 --- Sheaves and Cohomology --- p.17 / Chapter 2 --- The Riemann-Roch Theorem and Algebraic Curves --- p.27 / Chapter 2.1 --- Finiteness Theorem --- p.27 / Chapter 2.2 --- Transcendence Degree of M(X) --- p.33 / Chapter 2.3 --- The Riemann-Roch Theorem and Serre Duality --- p.37 / Chapter 2.4 --- Holomorphic Embedding in a Projective Space --- p.44 / Chapter 2.5 --- Algebraic Curves --- p.50 / Chapter 3 --- Invertible Sheaves and Line Bundles --- p.55 / Chapter 3.1 --- Algebraic Sheaves --- p.55 / Chapter 3.2 --- Invertible Sheaves --- p.55 / Chapter 3.3 --- Line Bundles --- p.61 / Chapter 3.4 --- Isomorphic Representations of the Picard Group --- p.66 / Chapter 4 --- A Uniqueness Theorem for Algebraic Curves --- p.72 / Chapter 4.1 --- Associated Curves and Normal Forms --- p.72 / Chapter 4.2 --- Proof of a Uniqueness Theorem for Algebraic Curves --- p.76 / Bibliography --- p.81

Kwok, Tsz Chiu. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 43-45). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 1.1 --- Historical remarks and main results --- p.6 / Chapter 1.2 --- Structure of the thesis --- p.8 / Chapter 2 --- Basic properties --- p.10 / Chapter 2.1 --- Existence of infinite convolution --- p.10 / Chapter 2.2 --- Properties --- p.16 / Chapter 2.3 --- Law of pure type --- p.17 / Chapter 3 --- Some results related to pure singularity --- p.20 / Chapter 3.1 --- The Pisot-Vijayaraghavan numbers --- p.20 / Chapter 3.2 --- The Salem numbers --- p.22 / Chapter 3.3 --- The weak separation condition --- p.23 / Chapter 4 --- A proof of almost everywhere absolute continuity --- p.30 / Chapter 5 --- Other results and problems --- p.37 / Chapter 5.1 --- Entropy of Bernoulli convolutions --- p.37 / Chapter 5.2 --- Dimensions --- p.40 / Chapter 5.3 --- Non PV numbers with bad behavior --- p.41 / Chapter 5.4 --- Open problems --- p.41 / Bibliography --- p.43

Tropical polynomials are similar to classical polynomials, however addition and multiplication are replaced with tropical addition (minimums) and tropical multiplication (addition). Within this new construction, polynomials become piecewise linear curves with interesting behavior. All tropical polynomials are piecewise linear curves, and each linear component uniquely corresponds to a particular monomial. In addition, certain monomial in the tropical polynomial can be trivial due to the fact that tropical addition is the minimum operator. Therefore, it makes sense to consider a graph of connectivity of the monomials for any given tropical polynomial. We investigate tropical polynomials where all coefficients are chosen from a standard normal distribution, and ask what the distribution will be for the graphs of connectivity amongst the monomials. We present a rudimentary algorithm for analytically determining the probability and show a Monte Carlo based confirmation for our results. In addition, we will give a variety of different theorems comparing relative likelihoods of different types of tropical polynomials.

One of the restrictions used in all of the works done on phylogenetic invariants for group based models has been that the group be abelian. In my thesis, I aim to generalize the method of invariants for group-based models of DNA sequence evolution to include nonabelian groups. By using a nonabelian group to act one the nucleotides, one could capture the structure of the symmetric model for DNA sequence evolution. If successful, this line of research would unify the two separated strands of active research in the area today: Allman and Rhodes’s invariants for the symmetric model and Strumfels and Sullivant’s toric ideals of phylogenetic invariants. Furthermore, I want to look at the statistical properties of polynomial invariants to get a better understanding of how they behave when used with real, “noisy” data.

No description available.

The purpose of this thesis is to give a classification of anisotropic algebraic groups over number fields of higher real rank. This will complete the classification of algebraic groups over number fields of higher real rank, which was begun by V. Chernousov, L. Lifschitz and D.W. Morris in their paper "Almost-Minimal Non-Uniform Lattices of Higher Rank''. The classification of anisotropic groups of higher real rank is also used to provide a classification of uniform lattices of higher rank contained in semisimple Lie groups with no compact factors. In particular, it is shown that all such lattices sit inside Lie groups of type An. This thesis proceeds as follows: The first chapter provides motivation for the classification and introduces all the main results of the thesis. The second chapter provides relevant definitions and background material for the proof. The next chapters provide a proof of the classification theorem, with chapters 3-5 examining the absolutely simple groups and the final chapter examining the simple groups which are not absolutely simple. / Mathematics

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 1997. / Includes bibliographical references. Also available on the Internet.

In this work we study the space of group homomorphisms Hom(Γ,G) from a geometric and simplicial point of view. The case in which the source group is a free abelian group of rank n is studied in more detail since this space can be identified with the space of commuting n-tuples of elements from G. This latter case is of particular interest when the target is a Lie group. The simplicial approach allows us to to construct a family of spaces that filters the classifying space of a group by filtering group theoretical information of the given group. Namely, we use the lower central series of free groups to construct a family of simplicial subspaces of the bar construction of the classifying space of a group. The first layer of this filtration is studied in more detail for transitively commutative (TC) groups.

We construct some new spaces determined by SN functions based on the classical SN theory.Also we have some new result in the densness of algebraic elements in a C* algebra.