This thesis focuses on convex sets and convex cones defined using hyperbolic polynomials. We first review some of the theory of convex sets in $\R^d$ in general. We then review some classical algebraic theorems concerning polynomials in a single variable, as well as presenting a few more modern results about them. We then discuss the theory of hyperbolic polynomials in several variables and their associated hyperbolicity cones. We survey various ways to build and decompose hyperbolic cones and we prove that every nontrivial hyperbolic cone is the intersection of its derivative cones. We conclude with a brief discussion of the set of extreme rays of a hyperbolic cone.

This thesis explores the implications of non-human primates’ propensity to hyperbolically discount the future. Hyperbolic discounting occurs when small, near-term rewards are preferred over larger rewards that are realized at a future point in time, but these preferences do not hold when the choice between long term and short term rewards is made at a time far removed from when the choice produces rewards-- i.e., at a time when the payoff of the choice is distant. I discuss two mutually exclusive models that attempt to explain why non-human primates hyperbolically discount: the cognitivist and the behaviorist model. I then present evidence that supports the cognitivist model and undermines the behaviorist model. I then argue that a “War of Interests” (WOI) occurs within the non-human primate mind. I explain this WOI model, discuss its philosophical implications, and then conclude with a general theory of the non-human primate mind.

This thesis focuses on convex sets and convex cones defined using hyperbolic polynomials. We first review some of the theory of convex sets in $\R^d$ in general. We then review some classical algebraic theorems concerning polynomials in a single variable, as well as presenting a few more modern results about them. We then discuss the theory of hyperbolic polynomials in several variables and their associated hyperbolicity cones. We survey various ways to build and decompose hyperbolic cones and we prove that every nontrivial hyperbolic cone is the intersection of its derivative cones. We conclude with a brief discussion of the set of extreme rays of a hyperbolic cone.

Thesis (Ph. D.)--University of Texas at Austin, 2002. / Vita. Includes bibliographical references. Available also from UMI Company.

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This dissertation concerns the study of canonical components of the SL(2, C) character varieties of hyperbolic 3-manifolds. Although character varieties have proven to be a useful tool in studying hyperbolic 3-manifolds, very little is known about their structure. Chapter 1 provides background on this subject. Chapter 2 is dedicated to the canonical component of the Whitehead link. We provide a projective model and show that this model is isomorphic to P^2 blown up at 10 points. The Whitehead link can be realized as 1/1 Dehn surgery on one cusp of both the Borromean rings and the 3-chain link. In Chapter 3 we examine the canonical components for the two families of hyperbolic link complements obtained by 1/n Dehn filling on one component of both the Borromean rings and the 3-chain link. These examples extend the work of Macasieb, Petersen and van Luijk who have studied the character varieties associated to the twist knot complements. We conjecture that the canonical components for the links obtained by 1/n Dehn filling on one component of the 3-chain link are all rational surfaces isomorphic to P^2 blown up at 9n + 1 points. A major goal is to understand how the algebro-geometric structure of these varieties reflects the topological structure of the associated manifolds. At the end of Chapter 3 we discuss common features of these examples and explain how our results lend insight into the affect Dehn surgery has on the character variety. We conclude, in Chapter 4, with a description of possible directions for future research. / text

According to the work of Dennis Sullivan, there exists a smooth flow on the 5-sphere all of whose orbits are periodic although there is no uniform bound on their periods. The question addressed in this thesis is whether such an example can occur in the partially hyperbolic context. That is, does there exist a partially hyperbolic diffeomorphism of a compact manifold such that all the leaves of its center foliation are compact although there is no uniform bound for their volumes. We will show that the answer to the previous question under the very mild hypothesis of dynamical coherence is no. The thesis is organized as follows. In the first chapter we give the necessary background and results in partially hyperbolic dynamics needed for the rest of the work, studying in particular the geometry of the center foliation. Chapter two is devoted to a general discussion of compact foliations. We give proof or sketches of all the relevant results used. Chapter three is the core of the thesis, where we establish the non existence of Sullivan's type of examples in the partially hyperbolic domain, and generalize to diffeomorphisms whose center foliation has arbitrary dimension. The last chapter is devoted to applications of the results of chapter three, where in particular it is proved that if the center foliation of a dynamically coherent partially hyperbolic diffeomorphism is compact and without holonomy, then it is plaque expansive.

A Smale space is a chaotic dynamical system with canonical coordinates of contracting and expanding directions. The basic sets for Smale’s Axiom A systems are a key class of examples. R.F. Williams considered the special case where the basic set had a totally disconnected contracting set and a Euclidean expanding one. He provided a construction using inverse limits of such examples and also proved that (under appropriate hyptotheses) all such basic sets arose from this construction. We will be working in the metric setting of Smale spaces, but the goal is to extend Williams’ results by removing all hypotheses on the unstable sets. We give criteria on a stationary inverse limit which ensures the result is a Smale space. We also prove that any irreducible Smale space with totally disconnected local stable sets is obtained through this construction. / Graduate