Thesis (Ph. D.)--University of Washington, 1999. / Vita. Includes bibliographical references (leaves [86]-88).

Thesis (M.S.)--Brigham Young University. Dept. of Mathematics, 2005. / Includes bibliographical references (leaves 112-113).

Thesis (Ph. D. in Mathematics)--Vanderbilt University, Aug. 2006. / Title from title screen. Includes bibliographical references.

Part I of this thesis concerns the question in which dimensions manifolds with higher orientability properties can have an odd Euler characteristic. In chapter 1 I prove that a k-orientable manifold (or more generally Poincare complex) has even Euler characteristic unless the dimension is a multiple of 2^k+1, where we call a manifold k-orientable if the i^th Stiefel-Whitney class vanishes for all 0 < i < 2^k (k ≥ 0). For k = 0, 1, 2, 3, k-orientable manifolds with odd Euler characteristic exist in all dimensions 2^k+1m, but whether there exist a 4-orientable manifold with an odd Euler characteristic is an open problem. In Chapter 2 I present calculations on the cohomology of the first two Rosenfeld planes, revealing that (O ⊗ C)P² is 2-orientable and (O ⊗ H)P² is at least 3-orientable. Part II discusses the homotopy type of spaces of nested manifolds. I prove that the space of d-dimensional manifolds with k-dimensional submanifolds inside Rⁿ has the homotopy type of a linearised model T_k<d, which can be thought of as a space of off-set d-planes inside Rⁿ with a (potentially empty) off-set k-plane inside of it, compactified with a point at infinity representing the empty set. Applying an induction I generalise this result to the case of higher nestings, establishing that the space Ψ_I (Rⁿ) of nested manifolds inside Rⁿ, for I a finite list of strictly increasing dimensions between 0 and n - 1, has the homotopy type of a linearised model space T_I.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Este trabalho consiste em duas partes. Na primeira parte, estudaremos hipersuperfÃcies compactas sem bordo imersas no espaÃo Euclidiano com o quociente das curvaturas mÃdias anisotrÃpicas constante. Provaremos que tais hipersuperfÃcies sÃo pontos crÃticos para um problema variacional de preservar uma combinaÃÃo linear da (k; F)-Ãrea e do (n+1)-volume determinado por M. Demostraremos que a hipersuperfÃcie à (r; k; a; b)-estÃvel se, e somente se, a menos de translaÃÃo e homotetia, ela à a Wulff shape de F (veja SeÃÃo 2.1), sob algumas condiÃÃes acerca de a; b â R. Na segunda parte desse trabalho, obtemos outras caracterizaÃÃes para a Wulff shape envolvendo as curvaturas mÃdias anisotrÃpicas de ordem superior de uma hipersuperfÃ- cie M em Rn+1 e o conjunto W = Rn+1 -UpâM Tp. Os resultados sÃo obtidos para hipersuperfÃcies compactas nÃo convexas satisfazendo W ╪ Ã. / This work consists of two parts. In the first part we deal with a compact hypersurface without boundary immersed in to the Euclidean space with the quotient of anisotropic mean curvatures constant. Such a hypersurface is a critical point for the variational problem preserving a linear combination of the (k; F)-area and (n + 1)-volume enclosed by M. We show that it is (r; k; a; b)-stable if, and only if, up to translations and homotheties, it is the Wulff shape, under some assumptions on a; b â R. In the second part we obtain further characterizations for the Wulff shape involving the anisotropic mean curvatures of higher order of a hypersurface M in Rn+1 and the set W = Rn+1-UpâM Tp. Results are obtained for non-convex compact hypersurfaces satisfying W ╪ Ã.

The problem of embedding a manifold in Euclidean space is considered. Manifolds are introduced in Chapter I along with other basic definitions and examples. Chapter II contains a proof of the Regular Value Theorem along with the "Easy" Whitney Embedding Theorem. In Chapter III, vector bundles are introduced and some of their properties are discussed. Chapter IV introduces the Stiefel-Whitney classes and the four properties that characterize them. Finally, in Chapter V, the Stiefel-Whitney classes are used to produce a lower bound on the dimension of Euclidean space that is needed to embed real projective space.

This thesis consists of two parts. In the first part, we study certain Ricci flow invariant nonnegative curvature conditions as given by B. Wilking. We begin by proving that any such nonnegative curvature implies nonnegative isotropic curvature in the Riemannian case and nonnegative orthogonal bisectional curvature in the K¨ahler case. For any closed AdSO(n,C) invariant subset S so(n, C) we consider the notion of positive curvature on S, which we call positive S- curvature. We show that the class of all such subsets can be naturally divided into two subclasses: The first subclass consists of those sets S for which the following holds: If two Riemannian manifolds have positive S- curvature then their connected sum also admits a Riemannian metric of positive S- curvature. The other subclass consists of those sets for which the normalized Ricci flow on a closed Riemannian manifold with positive S-curvature converges to a metric of constant positive sectional curvature. In the second part of the thesis, we study the behavior of Ricci flow for a manifold having positive S - curvature, where S is in the first subclass. More specifically, we study the Ricci flow for a special class of metrics on Sp+1 x S1 , p ≥ 4, which have positive isotropic curvature.

In this paper, we prove, for a certain class of open billiard dynamical systems, the existence of a family of smooth probability measures on the leaves of the dynamical system's unstable manifold. These measures describe the conditional asymptotic behavior of forward trajectories of the system. Furthermore, properties of these families are proven which are germane to the PYC programme for these systems. Strong sufficient conditions for the uniqueness of such families are given which depend upon geometric properties of the system's phase space. In particular, these results hold for a fairly nonrestrictive class of triangular configurations of scatterers.

Computer simulations for civil and mechanical engineering that efficiently leverage computational resources to solve boundary value problems have pervasive impacts on many aspects of civilization, including manufacturing, communication, transportation, medicine, and defense. Conventionally, a solver that predicts the mechanical behaviors of solids requires constitutive laws that represent mechanisms not directly derived from balance principles. These mechanisms are often characterized by mathematical models validated and tested via tabulated data, organized in grids or, more broadly, within normed Euclidean space (e.g., principle stress space, Mohr circle). These mathematical models often involved mapping between/among normed vector spaces that adhere to physical constraints. This methodology has manifested frameworks such as hyperelastic energy functional, elastoplasticity models with evolving internal variables, cohesive zone models for fracture, etc. However, the geometry of material data plays a crucial role in the efficiency, accuracy, and robustness of predictions. This thesis introduces a collection of mathematical models, tools, algorithms, and frameworks that, when integrated, may unleash the potential of leveraging data geometry to advance solid mechanics modeling. In the first part of the thesis, we introduce the concept of treating constitutive data as a manifold. This idea leads to a novel data-driven paradigm called “Manifold Embedding Data-Driven Mechanics,” which incorporates the manifold structure of data into the distance minimization model-free method. By training an invertible artificial neural network (ANN) to embed nonlinear constitutive data onto a hyperplane, we replace the costly combinatoric optimization necessary for the classical model-free paradigm with a projection and, as a result, significantly improve the efficiency and robustness of the model-free approach with a distance measure consistent with the data geometry. This method facilitates consistent interpolation on the manifold, which improves the accuracy when data is limited. To handle noisy data, we relax the invertibility constraint of the designed ANN and construct the desired embedding space via a geometric autoencoder. Unlike the classical autoencoder, which compresses data by reducing the data dimensionality in the latent space, our design focuses on reducing the dimensionality of the data by imposing constraints. This technique enables us to learn a noise-free embedding through a simple projection by assuming the orthogonality between the data and noise. To improve the interpretability and, ultimately, the trustworthiness of machine learning-derived constitutive models, we abandon the design of the fully connected neural networks and instead introduce polynomials in feature space that enable us to turn neural network parametrized black-box models back into mathematical models understandable by engineers. We present geometrically inspired structures in a feature space spanned by univariate ANNs and then learn a sparse representation of the data using these acquired features. Our divide-and-conquer scheme takes advantage of the learned univariate functions to perform parallel symbolic regression, ultimately extracting human-readable equations for material modeling. Our approach mitigates the well-known computational burden associated with symbolic regression for high-dimensional data and data that must adhere to physical constraints. We demonstrate the interpretability, accuracy, and computational efficiency of our algorithm in discovering constitutive models for hyperelastic materials and plastic yield surfaces.

Let Γ be a crystallographic group acting on the n-dimensional Euclidean space. In this dissertation, the surgery obstruction groups of Γ are computed in terms of certain sheaf homology groups defined by F. Quinn, when Γ has no 2-torsion. The main theorem is : Theorem : If a crystallographic group Γ has no 2-torsion, there is a natural isomorphism a : H_*(Rⁿ /Γ; L(p)) → L_*^-∞(Γ). / Ph. D.