We show that any Borel action on a standard Borel space of a group which is topologically isomorphic to the sum of a countable abelian group with a countable sum of lines and circles induces an orbit equivalence relation which is hypersmooth. We also show that any Borel action of a second countable locally compact abelian group on a standard Borel space induces an orbit equivalence relation which is essentially hyperfinite, generalizing a result of Gao and Jackson for the countable abelian groups.

The Lie algebra Vec(ℝ) of polynomial vector fields on the line acts naturally on ℂ[]. This action has a one-parameter family of deformations called the tensor density modules F_λ. The bounded indecomposable modules of Vec(ℝ) of length 2 composed of tensor density modules have been classified by Feigin and Fuchs. We present progress towards describing the annihilators of the unique indecomposable extension of F_λ by F_(λ+2) in the non-resonant case λ ≠ -½. We give the intersection of the annihilator and the subalgebra of lowest weight vectors of the universal enveloping algebra (Vec(ℝ)) of Vec(ℝ). This result is found by applying structural descriptions of the lowest weight vectors of (Vec(ℝ)).

First, we show that the Rothberger and 2-Rothberger games are equivalent. Then we adjust the former proof and introduce another game, the restricted Menger game, in order to obtain a broader result. This provides an answer in the context of Hausdorff spaces for an open question posed by Aurichi, Bella, and Dias.

Spreading models have become fundamental to the study of asymptotic geometry in Banach spaces. The existence of spreading models in every Banach space, and the so-called good sequences which generate them, was one of the first applications of Ramsey theory in Banach space theory. We use Ramsey theory and other techniques from infinitary combinatorics to examine some old and new questions concerning spreading models and good sequences. First, we consider the lp spreading model problem which asks whether a Banach space contains lp provided that every spreading model of a normalized block basic sequence of the basis is isometrically equivalent to lp. Next, using the Hindman-Milliken-Taylor theorem, we prove a new stabilization theorem for spreading models which produces a basic sequence all of whose normalized constant coefficient block basic sequences are good. When the resulting basic sequence is semi-normalized, all the spreading models generated by the above good sequences must be uniformly equivalent to lp or c0. Finally, we investigate the assumption that every normalized block tree on a Banach space has a good branch. This turns out to be a very strong assumption and is equivalent to the space being 1-asymptotic lp. We also show that the stronger assumption …

In prophet problems, two players with different levels of information make decisions to optimize their return from an underlying optimal stopping problem. The player with more information is called the "prophet" while the player with less information is known as the "gambler." In this thesis, as in the majority of the literature on such problems, we assume that the prophet is omniscient, and the gambler does not know future outcomes when making his decisions. Certainly, the prophet will get a better return than the gambler. But how much better? The goal of a prophet problem is to find the least upper bound on the difference (or ratio) between the prophet's return, M, and the gambler's return, V. In this thesis, we present new prophet problems where we seek the least upper bound on M-V when there is a fixed cost per observations. Most prophet problems in the literature compare M and V when prophet and gambler buy (or sell) one asset. The new prophet problems presented in Chapters 3 and 4 treat a scenario where prophet and gambler optimize their return from selling two assets, when there is a fixed cost per observation. Sharp bounds for the problems on small …

The study of Borel equivalence relations on Polish spaces has become a major area of focus within descriptive set theory. Primarily, work in this area has been carried out using the standard methods of descriptive set theory. In this work, however, we develop a model-theoretic framework suitable for the study of Borel equivalence relations, introducing a class of objects we call Borel structurings. We then use these structurings to examine conditions under which marker sets for Borel equivalence relations can be concluded to exist or not exist, as well as investigating to what extent the Compactness Theorem from first-order logic continues to hold for Borel structurings.