Let G be a semisimple linear algebraic group over an algebraically closed field of prime characteristic. In this thesis we outline the theory of such groups and their cohomology. We then concentrate on algebraic groups in rank 1 and 2, and prove some new results in their bounding cohomology.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1981. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Bibliography: leaves 100-102. / by Robert Alan Proctor. / Ph.D.

Thesis. 1977. Ph.D.--Massachusetts Institute of Technology. Dept. of Mathematics. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Vita. / Includes bibliographical references. / Ph.D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1981. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Bibliography: p. 135-138. / by James William Walker. / Ph.D.

The context of this thesis is derived categories in algebraic geometry and geo- metric quotients. Specifically, we prove the embedding of the derived category of a smooth curve of genus greater than one into the derived category of the moduli space of semistable pairs over the curve. We also describe closed cover conditions under which the composition of a pullback and a pushforward induces a fully faithful functor. To prove our main result, we give an exposition of how to think of general Geometric Invariant Theory quotients as quotients by the multiplicative group.

We provide a new relative trace formula approach to the theorem of Waldspurger on toric periods for GL₂, with possible applications to the global Gross-Prasad conjecture for orthogonal groups.

In this thesis, we report three preprints [Li17a] [Li17b] and [HL17] the author wrote (the last one was written jointly with D. Hansen) during his pursuing of PhD at Columbia. We study smooth proper rigid varieties which admit formal models whose special fibers are projective. The main theorem asserts that the identity components of the associated rigid Picard varieties will automatically be proper. Consequently, we prove that non-archimedean Hopf varieties do not have a projective reduction. The proof of our main theorem uses the theory of moduli of semistable coherent sheaves. Combine known structure theorems for the relevant Picard varieties, together with recent advances in p-adic Hodge theory, We then prove several related results on the low-degree Hodge numbers of proper smooth rigid analytic varieties over p-adic fields.

The goal of this thesis is to give an introduction to the geometric picture of bivariant K-theory developed by Emerson and Meyer building on the ideas Connes and Skandalis, and then to apply this machinery to give a geometric proof of a result of Emerson. We begin by giving an overview of topological K-theory, necessary for developing bivariant K-theory. Then we discuss Kasparov's analytic bivariant K-theory, and from there develop topological bivariant K-theory. In the final chapter we state and prove the result of Emerson. / Graduate

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As the second task of this thesis, we shall propose some mathematical model to simulate the movement of a floating bridge under some moving loadings. The floating bridge system consists of three parts, i.e. river (fluid), floating bridge (multibody system) and vehicles (load) which pass the bridge. Our objective is to find the motion and dynamical responses of the floating bridge with a truck or tracklayer passing on it. The floating bridge is a system of steel rectangular boxes which can be seen as rigid bodies connected by some kinematic joints. In fact, such system is a fluid-structure coupled system and one must treat the governing equations for the floating bridge and fluid, i.e. Euler-Lagrange equations and Navier-Stokes equations, simultaneously. In our work, we apply the one-leg method and operator splitting arbitrary Lagrangian-Eulerian method to solve the coupled system. / When performing dynamical analysis of a constrained mechanical system, a set of index-3 differential algebraic equations, i.e. Euler-Lagrange equations, are often needed to describe the time evolution of the mechanical system. In this thesis, we apply one-leg multi-step methods to integrate the DAEs directly. To overcome some difficulties leading to certain numerical instabilities, a velocity elimination technique is applied to generate a framework that the position and velocity profiles can be obtained in two separate stages: only the position variables and Lagrange multipliers take part in the convergent nonlinear iterations at each time step while the velocity is calculated by the multi-step formula directly without any iteration. The framework is constructed in a manner such that it satisfies all the constraints at the position level and involves variables as few as possible during the iteration. Some convergence analysis are presented and good stability and high efficiency can be seen through the experiments of some benchmark problems. / Zhao Yubo. / "July 2005." / Adviser: Zou Jun. / Source: Dissertation Abstracts International, Volume: 67-01, Section: B, page: 0310. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (p. 244-276). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.