<p>Smith theory provides powerful tools for understanding the geometry of singular sets of group actions on spheres. In this thesis, tools from Smith theory and spectral sequences are brought together to study the singular sets of elementary abelian groups acting locally linearly on S4. It is shown that the singular sets of such actions are homeomorphic to the singular sets of linear actions. A short review of the literature on group actions on S4 is included.</p> / Master of Science (MSc)

<p>Let $p>5$ be a prime and $X_0$ a simply-connected $4$-manifold with boundary the Poincar\'e homology sphere $\Sigma(2,3,5)$ and even negative-definite intersection form $Q_=\text_8$ . We obtain restrictions on extending a free $\bZ/p$-action on $\Sigma(2,3,5)$ to a smooth, homologically-trivial action on $X_0$ with isolated fixed points. It is shown that for $p=7$ there is no such smooth extension. As a corollary, we obtain that there does not exist a smooth, homologically-trivial $\bZ/7$-equivariant splitting of $\#^8 S^2 \times S^2=E_8 \cup_ \overline$ with isolated fixed points. The approach is to study the equivariant version of Donaldson-Floer instanton-one moduli spaces for $4$-manifolds with cylindrical ends. These are $L^2$-finite anti-self dual connections which asymptotically limit to the trivial product connection.</p> / Doctor of Philosophy (PhD)

Persistent homology is a technique of topological data analysis that seeks to understand the shape of data. We study the effectiveness of a single-layer perceptron and gradient boosted classification trees in classifying perhaps the most well-known data set in machine learning, the MNIST-Digits, or MNIST. An alternative representation is constructed, called MNIST-PD. This construction captures the topology of the digits using persistence diagrams, a product of persistent homology. We show that the models are more effective when trained on MNIST compared to MNIST-PD. Promising evidence reveals that the topology is learned by the algorithms. / Thesis / Master of Science (MSc)

ITC/USA 2010 Conference Proceedings / The Forty-Sixth Annual International Telemetering Conference and Technical Exhibition / October 25-28, 2010 / Town and Country Resort & Convention Center, San Diego, California / The purpose of this paper is to explain the process to implementing a wireless sensor network in order to improve situational awareness in a dense urban environment. Utilizing a system of wireless nodes with Global Positioning System (GPS) and heart rate sensors, a system was created that was able to give both position and general health conditions. By linking the nodes in a mesh network line of sight barriers were overcome to allow for operation even in an environment full of obstruction.

In this work, several problems in the field of Hamiltonian dynamics are studied. Chapter 1 is a short review of some basic results in the theory of Hamiltonian dynamics. In chapter 2, we study the problem of computing the geometric monodromy of the torus bundle defined by integrable Hamiltonian systems. We show that for two degree of freedom systems near an isolated critical value of the energy momentum map, the monodromy group can be determined solely from the local data of the energy momentum map at the singularity. Along the way, we develop a simple method for computing the monodromy group which covers all the known examples that exhibit nontrivial monodromy. In chapter 3, we consider the topological aspects of the Kirchhoff case of the motion of a symmetric rigid body in an infinite ideal fluid. The bifurcation diagrams are constructed and the topology of all the invariant sets are determined. We show that this system has monodromy. We show also that this system undergoes a Hamiltonian Hopf bifurcation as the couple resultant passes through a certain value when the steady rotation of the rigid body about its symmetry axis changes stability. Chapter 4 is devoted to checking Kolmogorov's condition for the square potential pendulum. We prove, by essentially elementary methods, that Kolmogorov's condition is satisfied for all of the regular values of the energy momentum map. In chapter 5, we use Ziglin's theorem to prove rigorously that some of the generalized two degree of freedom Toda lattices are non-integrable.

No description available.

No description available.

No description available.

No description available.

No description available.