Flag varieties of reductive Lie groups and their subvarieties play a central role in representation theory. In the early 1980s, V. Deodhar introduced a decomposition of the flag variety which was then used to study the Kazdan-Lusztig polynomials. A Deodhar-type decomposition of the product of the flag variety with itself, referred to as the double flag variety, was introduced in 2007 by B. Webster and M. Yakimov, and each piece of the decomposition was shown to be coisotropic with respect to a naturally defined Poisson structure on the double flag variety. The work of Webster and Yakimov was partially motivated by the theory of cluster algebras in which Poisson structures play an important role. The Deodhar decomposition of the flag variety is better understood in terms of a cell decomposition of Bott-Samelson varieties, which are resolutions of Schubert varieties inside the flag variety. In the thesis, double Bott-Samelson varieties were introduced and cell decompositions of a Bott-Samelson variety were constructed using shuffles. When the sequences of simple reflections defining the double Bott-Samelson variety are reduced, the Deodhar-type decomposition on the double flag variety defined by Webster and Yakimov was recovered. A naturally defined Poisson structure on the double Bott-Samelson variety was also studied in the thesis, and each cell in the cell decomposition was shown to be coisotropic. For the cells that are Poisson, coordinates on the cells were also constructed and were shown to be log-canonical for the Poisson structure. / published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

Thesis (Ph.D.)--Auburn University, 2007. / Abstract. Vita. Includes bibliographic references (ℓ.42-43)

Thesis (Ph. D.)--Virginia Polytechnic Institute and State University, 1992. / Vita. Abstract. Includes bibliographical references (leaves 99-105). Also available via the Internet.

Thesis (M.S.)--Virginia Polytechnic Institute and State University, 1992. / Vita. Abstract. Includes bibliographical references (leaves 44-48). Also available via the Internet.

Thesis (Ph. D.)--University of Oregon, 2008. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 121-125). Also available online in Scholars' Bank; and in ProQuest, free to University of Oregon users.

Thesis (B.A.)--Haverford College, Dept. of Mathematics, 2002. / Includes bibliographical references.

Thesis (Ph. D.)--Michigan State University. Dept. of Mathematics, 2008. / Title from PDF t.p. (viewed on July 23, 2009) Includes bibliographical references (p. 106-108). Also issued in print.

Thesis (M.S.)--Ohio University, March, 2006. / Title from PDF t.p. Includes bibliographical references (p. 47-48)

This thesis discusses the problem of decomposing rectilinear regions, with or without holes, into a minimum number of rectangles. There are two different types of decomposition considered here : decomposing a figure into non-overlapping parts, called partitioning, and decomposing a figure into possibly overlapping parts, called covering. A method is outlined and proved for solving the above two problems, and algorithms for the solutions of these problems are presented. The partitioning problem can be solved in time O(n⁵ ²), where n is the number of vertices of the figure, whereas the covering problem is exponential in its time complexity. / M.S.

In an exploratory approach to data analysis, it is often useful to consider the observations as generated from a set of latent generators or 'sources' via a generally unknown mapping. Reconstructing sources from their mixtures is an extremely ill-posed problem in general. However, solutions to such inverse problems can, in many cases, be achieved by incorporating prior knowledge about the problem, captured in the form of constraints. This setting is a natural candidate for the application of the Bayesian method- ology, allowing us to incorporate "soft" constraints in a natural manner. This Thesis proposes the use of sparse statistical decomposition methods for ex- ploratory analysis of datasets. We make use of the fact that many natural signals have a sparse representation in appropriate signal dictionaries. The work described in this Thesis is mainly driven by problems in the analysis of large datasets, such as those from functional magnetic resonance imaging of the brain for the neuro-scientific goal of extracting relevant 'maps' from the data. We first propose Bayesian Iterative Thresholding, a general method for solv- ing blind linear inverse problems under sparsity constraints, and we apply it to the problem of blind source separation. The algorithm is derived by maximiz- ing a variational lower-bound on the likelihood. The algorithm generalizes the recently proposed method of Iterative Thresholding. The probabilistic view en- ables us to automatically estimate various hyperparameters, such as those that control the shape of the prior and the threshold, in a principled manner. We then derive an efficient fully Bayesian sparse matrix factorization model for exploratory analysis and modelling of spatio-temporal data such as fMRI. We view sparse representation as a problem in Bayesian inference, following a ma- chine learning approach, and construct a structured generative latent-variable model employing adaptive sparsity-inducing priors. The construction allows for automatic complexity control and regularization as well as denoising. The performance and utility of the proposed algorithms is demonstrated on a variety of experiments using both simulated and real datasets. Experimental results with benchmark datasets show that the proposed algorithms outper- form state-of-the-art tools for model-free decompositions such as independent component analysis.