Given an embedding of a closed k-dimensional manifold M into N-dimensional Euclidean space R^N, we aim to perform negative gradient flow of a penalty function P that acts on the space of all smooth embeddings of M into R^N to find an ideal manifold embedding. We study the computation of the gradient for a penalty function that contains both a curvature and distance term. We also find a lower bound for how long an embedding will remain in the space of embeddings when moving in a fixed, normal gradient direction. Finally, we study the distance penalty function in a special case in which we can prove short time existence of the negative gradient flow using the Cauchy-Kovalevskaya Theorem.

Basic concepts of differential geometry and Lie theory are introduced. Lie transformation groups are applied to linear systems of differential equations and the problem of describing rigid body orientation. Linear Hamiltonian systems are then treated as a Lie system of differential equations. This theory is applied to a particular Hamiltonian system arising from a problem in control theory, the linear state regulator problem.

We survey standard topics in elementary differential geometry and complex analysis to build up the necessary theory for studying applications of univalent harmonic function theory to minimal surfaces. We then proceed to consider convex combination harmonic mappings of the form f=sf_1+(1-s) f_2 and give conditions on when f lifts to a one-parameter family of minimal surfaces via the Weierstrauss-Enneper representation formula. Finally, we demand two minimal surfaces M and M' be locally isometric, formulate a system of partial differential equations modeling this constraint, and calculate their symmetry group. The group elements generate transformations that when applied to a prescribed harmonic mapping, lift to locally isometric minimal surfaces with varying graphs embedded in mathbb{R}^3.

Much progress has been made in understanding of plasmas through the use of the MHD equations and newer models such as Hall MHD and electron MHD. As with most equations of fluid behavior, these equations are nonlinear, and no general solutions can be found. The use of invariant structures allows limited predictions of fluid behavior without requiring a full solution of the underlying equations. The use of gauge transformation can allow the creation of new invariants, while differential geometry offers useful tools for constructing additional invariants from those that are already known. Using these techniques, new geometric, integral and topological invariants are constructed for Hall and electron MHD models. Both compressible and incompressible models are considered, where applicable. An application of topological invariants to magnetic reconnection is provided. Finally, a particular geometric invariant, which can be interpreted as the fluid impulse density, is studied in greater detail, its nature and invariance in plasma models is demonstrated, and its behavior is predicted in particular geometries under different models.

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For a particular equilibrium point, the local performance is determined by the partial derivatives of the control law evaluated at the equilibrium point. For a linear controller, the derivatives are equal to the state-feedback gains and the gains on the external inputs. These gains can be changed to vary the local performance of the system. An extended-linear controller links together the desired local controllers of various equilibrium points producing a nonlinear controller with the desired characteristics in a neighborhood of the equilibrium curve. Global performance is the behavior of the system away from the equilibrium curve. Although the extended-linear controller has good local performance, the global performance might be poor or even unstable. This thesis uses cubic spline techniques to investigate the coupling of global information into local controllers without affecting the local performance. Although “stand-alone” interpolative spline structures do not give the desired local performance, global information can be splined into linear and extended-linear controllers to provide both good global and good local performance. / Master of Science

We study log varieties, over a field of characteristic zero, which are generically logarithmically smooth and fs in the Kummer normally log étale topology. As an application, we prove an analog of Abramovich-Temkin-Wlodarczyk’s log resolution of singularities of fs log schemes in the Kummer fs setting.<br>

[pt] Nesse trabalho descrevemos superfícies mínimas mergulhadas em espaços produtos M x R, onde M = R2, S2 e H2 que são folheadas por geodésicas (superfícies regradas ) e curvas de curvatura constante de M (supefícies cíclicas ). Em R2xR, ou seja, em R3 vamos demonstrar que só existem duas superfícies mínimas cíclicas, que são o catenóide e o exemplo de Riemann. Em seguida caracterizamos as superfícies mínimas cíclicas em S2 x R que formam uma família a dois parâmetros e por fim exibimos três famílias de dois parâmetros de superfícies mínimas cíclicas em H2 x R. / [en] In this work we describe minimal surfaces embedded in product spaces M x R, where M = R2, S2 and H2 which are foliated by geodesics (ruled surfaces) and curves of M with constant curvature (cyclic surfaces). In R2 x R, i.e. R3, we shall prove that there exist only two minimal cyclic surfaces which are the catenoid and the Riemann example. Then we characterize minimal cyclic surfaces in S2 x R; they form a two-parameter family. Finally we exhibit three two-parameter families of minimal cyclic surfaces in H2 x R.

The bending of elastic strips and rods is a field of research that continues to offer new possibilities for exploration. This dissertation focuses on two distinct problems within this context. These are the search for the equilibrium shape of thin inextensible elastic strips, such as a M�öbius strip made out of paper, and the optimal shape of tapered columns that are stable against buckling. A theoretical approach based on the principle of virtual work is used to investigate both problems. This produces novel governing non-linear differential equations that describe both equilibrium and form. In order to discover the equilibrium shapes, numerical algorithms are developed that are based on Dynamic Relaxation. There are two ways in which they are used, one as an explicit form-finding tool, and the other as a way of solving differential equations. Results are provided that extend current theoretical models. The numerical schemes produce three-dimensional shapes for strips, going beyond the canonical Möbius strip, and solution shapes for tapered columns made from non-linear elastic materials. With the aid of analytical and numerical tools, finding the form of the M�öbius strip and the tallest possible column are interesting challenges in the search for new shapes that are driven by physical and material rules. These have applicability in structural engineering, architecture, nano-technology and even artistic endeavour.

Let (M,g) be a quasi-Sasaki manifold with Reeb vector field xi. Our goal is to understand the structure of M when g is an Einstein metric. Assuming that the S^1 action induced by xi is locally free or assuming a certain non-negativity condition on the transverse curvature, we prove some rigidity results on the structure of (M,g). Naturally associated to a quasi-Sasaki metric g is a transverse Kahler metric g^T. The transverse Kahler-Ricci flow of g^T is the normalized Ricci flow of the transverse metric. Exploiting the transverse Kahler geometry of (M,g), we can extend results in Kahler-Ricci flow to our transverse version. In particular, we show that a deep and beautiful theorem due to Perleman has its counterpart in the quasi-Sasaki setting. We also consider evolving a Sasaki metric g by Ricci flow. Unfortunately, if g(0) is Sasaki then g(t) is not Sasaki for t>0. However, in some instances g(t) is quasi-Sasaki. We examine this and give some qualitative results and examples in the special case that the initial metric is eta-Einstein.