First published in the Bulletin of the American Mathematical Society in Vol.71, 1965, published by the American Mathematical Society

First published in the Bulletin of the American Mathematical Society in Vol.74, 1968, published by the American Mathematical Society

archives@tulane.edu / This thesis examines the geometry and combinatorics of Borel subgroup orbits in classical symmetric spaces G/L where G is complex linear algebraic group and L is a Levi factor of a maximal parabolic subgroup P in G. In these cases, known as symmetric spaces of Hermitian type, we show that the canonical projection map $\pi : G/L \to G/P$ has the structure of an affine bundle. This fact yields a cell decomposition of G/L as well as isomorphisms of the cohomology and Chow rings of G/L and G/P, and motivates the study of the Borel orbits of G/L in relation to their images under the equivariant map $\pi$. For all of the cases of interest (symmetric spaces of types AIII, CI, DIII and BDI), G/P is a Grassmannian variety with Borel orbits called Schubert cells. Borel orbits of most of these symmetric spaces are parametrized by combinatorial objects called clans. This thesis provides enumerative formulae for the orbits in type CI, DIII and BDI , and gives bijections between sets of clans and other families of objects such as (fixed-point free) partial involutions, rook placements, and set partitions. Clans come with a poset structure given by the closure containment relation of the corresponding Borel orbits, and we supply rank polynomials for these posets in types CI and DIII. We give a combinatorial description of the closure order relations in types AIII, CI, and DIII which allows us to resolve part of a conjecture of Wyser on the restriction of this order from type AIII to other types. In the course of this description, we identify the preimages of Schubert cells under the map \pi as collections of clans called “sects.” Our combinatorial description of the sects identifies Borel orbits whose closures generate the Chow ring of G/L and reveals additional structure in the closure poset of clans. In particular, the preimage of the largest Schubert cell coincides variously with well-known posets of matrix Schubert varieties and congruence Borel orbit closures. Furthermore, we show that in type AIII the closure order restricted to a given sect can be described combinatorially in terms of “rank tableaux.” / 0 / Aram Bingham

The thesis mainly studies two problems in Algebraic Geometry and Hodge Theory. The first problem deals with the geometric realizations of certain Hermitian symmetric domains as moduli space of algebraic varieties, notably the Abelian varieties and Calabi-Yau varieties. The study of the first problem occupies most of the thesis. In section 1.3; we study the second problem, namely, the L2 Higgs cohomology of polarized variation of Hodge structures over Hermitian symmetric domains. / Sheng Mao. / "December 2005." / Advisers: Shing-Tung Yau; Nai-Chung Conan Leung. / Source: Dissertation Abstracts International, Volume: 67-11, Section: B, page: 6442. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (p. 108-113). / 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.

In this thesis, we set up a framework to define knot invariants for each choice of a symmetric space. In order to address this task, we start by defining appropriate notions of singular bundles and singular connections for a given symmetric space. We can associate a moduli space to any singular bundle defined over a compact 4-manifold with possibly non-empty boundary. We study these moduli spaces and show that they enjoy nice properties. For example, in the case of the symmetric space SU(n)/SO(n) the moduli space can be perturbed to an orientable manifold. Although this manifold is not necessarily compact, we introduce a comapctification of it. We then use this moduli space for singular bundles defined over 4-manifolds of the form YxR to define knot invariants. In another direction we mimic the construction of Donaldson invariants to define polynomial invariants for closed 4-manifolds equipped with smooth action of Z/2Z. / Mathematics

Abstract is available from the print copy. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2011.

Thesis (diplomarbeit)--Bonn, 1988. / Includes bibliographical references.

Thesis (Ph.D.)--York University, 2005. Graduate Programme in Mathematics and Statistics. / Typescript. Includes bibliographical references (leaves 78-83). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://wwwlib.umi.com/cr/yorku/fullcit?pNR11588

First published in the Bulletin of the American Mathematical Society in Vol.79, 1973, published by the American Mathematical Society

Η θεωρία των συμμετρικών χώρων αποτελεί μια σπουδαία κλάση των ομογενών χώρων, με εφαρμογές σε πολλούς κλάδους των μαθηματικών όπως στην αλγεβρική και την διαφορική γεωμετρία. Σε αυτήν την εργασία θα δώσουμμε τον ορισμό των συμμετρικών χώρων, τα βασικά τους χαρακτηριστικά και την ταξινόμησή τους. Θα περιγράψουμε τους χώρους αυτούς κυρίως αλγεβρικά, οπότε δεν θεωρείται απαραίτητο από τον αναγνώστη να γνωρίζει εκτενώς την θεωρία της διαφορικής γεωμετρίας για να κατανοήσει πλήρως την εργασία. / The theory of symmetric spaces is an important class of homogeneous spaces, with applications in many branches of mathematics such as algebraic and differential geometry. In this work we will define the symmetric spaces, their key features and sort them. We will describe these spaces mainly algebraic, so it is not considered necessary by the reader to know in detail the theory of differential geometry to understand the work.