In this thesis we develop the notion of LA-Courant algebroids, the infinitesimal analogue of multiplicative Courant algebroids. Specific applications include the integration of q- Poisson (d, g)-structures, and the reduction of Courant algebroids. We also introduce the notion of pseudo-Dirac structures, (possibly non-Lagrangian) subbundles W ⊆ E of a Courant algebroid such that the Courant bracket endows W naturally with the structure of a Lie algebroid. Specific examples of pseudo-Dirac structures arise in the theory of q-Poisson (d, g)-structures.

In this thesis we develop the notion of LA-Courant algebroids, the infinitesimal analogue of multiplicative Courant algebroids. Specific applications include the integration of q- Poisson (d, g)-structures, and the reduction of Courant algebroids. We also introduce the notion of pseudo-Dirac structures, (possibly non-Lagrangian) subbundles W ⊆ E of a Courant algebroid such that the Courant bracket endows W naturally with the structure of a Lie algebroid. Specific examples of pseudo-Dirac structures arise in the theory of q-Poisson (d, g)-structures.

<p>In this dissertation we study scalar curvature rigidity phenomena for the upper hemisphere, and subsets thereof. In particular, we are interested in Min-Oo's conjecture that there exist no metrics on the upper hemisphere having scalar curvature greater or equal to that of the standard spherical metric, while satisfying certain natural geometric boundary conditions.</p><p>While the conjecture as originally stated has recently been disproved, there are still many interesting modications to consider. For instance, it has been shown that Min-Oo's rigidity conjecture holds on sufficiently small geodesic balls contained in the upper hemisphere, for metrics sufficiently close to the spherical metric. We show that this local rigidity phenomena can be extended to a larger class of domains in the hemisphere, in particular finding that it holds on larger geodesic balls, and on certain domains other than geodesic balls (which necessarily have more complicated boundary geometry). We discuss a possible method for finding the largest possible domain on which the local rigidity theorem is true, and give a Morse-theoretic interpretation of the problem.</p><p>Another interesting open question is whether or not such a rigidity statement holds for metrics that are not close to the spherical metric. We find that a scalar curvature rigidity theorem can be proved for metrics on sufficiently small geodesic balls in the hemisphere, provided certain additional geometric constraints are satisfied.</p> / Dissertation

We propose the study of robustness measures for signal detection in non-stationary noise using differential geometric tools in conjunction with empirical distribution analysis. Our approach shows that the gradient can be viewed as a random variable and therefore used to generate sample densities allowing one to draw conclusions regarding the robustness. As an example, one can apply the geometric methodology to the detection of time varying deterministic signals in imperfectly known dependent nonstationary Gaussian noise. We also compare stationary to non-stationary noise and prove that robustness is barely reduced by admitting non-stationarity. In addition, we show that robustness decreases with larger sample sizes, but there is a convergence in this decrease for sample sizes greater than 14. We then move on to compare the effect on robustness for signal detection between non-Gaussian tail effects and residual dependency. The work focuses on robustness as applied to tail effects for the noise distribution, affecting discrete-time detection of signals in independent non-stationary noise. This approach makes use of the extension to the generalized Gaussian case allowing the comparison in robustness between the Gaussian and Laplacian PDF. The obtained results are contrasted with the influence of dependency on robustness for a fixed tail category and draws consequences on residual dependency versus tail uncertainty.

Thesis (Ph. D.)--University of Chicago, 1933. / Vita. Lithoprinted. "Private edition, distributed by the University of Chicago libraries, Chicago, Illinois."

Thesis (Ph. D.)--University of Texas at Austin, 2000. / Vita. Includes bibliographical references (leaves 113-119). Available also in a digital version from Dissertation Abstracts.

In this thesis, we develop two methods for constructing Lie groupoids. The first method is a blow-up construction, corresponding to the elementary modification of a Lie algebroid along a subalgebroid over some closed hypersurface. This construction may be specialized to the Poisson groupoids and Lie bialgebroids. We then apply this method to three cases. The first is the adjoint Lie groupoid integrating the Lie algebroid of vector fields tangent to a collection of normal crossing hypersurfaces. The second is the adjoint symplectic groupoid of a log symplectic manifold. The third is the adjoint Lie groupoid integrating the tangent algebroid of a Riemann surface twisted by a divisor. The second method is a gluing construction, whereby Lie groupoids defined on the open sets of an appropriate cover may be combined to obtain global integrations. This allows us to construct and classify the Lie groupoids integrating the given Lie algebroid. We apply this method to the aforementioned cases, albeit with small differences, and characterize the category of integrations in each case.

In this thesis, we develop two methods for constructing Lie groupoids. The first method is a blow-up construction, corresponding to the elementary modification of a Lie algebroid along a subalgebroid over some closed hypersurface. This construction may be specialized to the Poisson groupoids and Lie bialgebroids. We then apply this method to three cases. The first is the adjoint Lie groupoid integrating the Lie algebroid of vector fields tangent to a collection of normal crossing hypersurfaces. The second is the adjoint symplectic groupoid of a log symplectic manifold. The third is the adjoint Lie groupoid integrating the tangent algebroid of a Riemann surface twisted by a divisor. The second method is a gluing construction, whereby Lie groupoids defined on the open sets of an appropriate cover may be combined to obtain global integrations. This allows us to construct and classify the Lie groupoids integrating the given Lie algebroid. We apply this method to the aforementioned cases, albeit with small differences, and characterize the category of integrations in each case.

Thesis (Ph. D.)--University of Chicago, 1932. / Vita. Lithoprinted. "Private edition, distributed by the University of Chicago libraries, Chicago, Illinois." Bibliography: p. 55-56.

Thin elastic objects, including leaves, flowers, plastic sheets and sails, are ubiquitous in nature and their technological applications are growing with the introduction of hydrogel thin-films, flexible electronics and environmentally responsive gels. The intricate rippling and buckling patterns are postulated to be the result of minimizing an elastic energy. In this dissertation, we investigate the role of regularity in minimizing the elastic energy. Though there exist smooth isometric immersions of arbitrarily large subsets of H2 into R3, we show that the introduction of weakly-irregular singularities, of smoothness class C^{1,1}, significantly reduces the energy; we provide numerical evidence supporting an upper bound on the asymptotic scaling of the minimum energy over C^{1,1} isometries which is an exponentially large improvement as compared to the conjectured lower bound over C2 surfaces. This work provides insight into the quantitative nature of the Hilbert-Efimov theorem. The introduction of such singularities is energetically inexpensive, and so too is their relocation. Therefore, isometries are "floppy" or easily-deformable, motivating a shift in focus from finding the exact minimizers of the elastic energy in favor of understanding the statistical mechanics of the collection of zero-stretching immersions.