Mathematicshttp://hdl.handle.net/1903/22612019-01-18T06:39:31Z2019-01-18T06:39:31ZMotivic Cohomology of Groups of Order p^3Black, Rebeccahttp://hdl.handle.net/1903/214062018-09-16T03:24:01Z2018-01-01T00:00:00ZMotivic Cohomology of Groups of Order p^3
Black, Rebecca
In this thesis we compute the motivic cohomology ring (also known as Bloch's higher Chow groups) with finite coefficients for the two nonabelian groups of order $27$, thought of as affine algebraic groups over $\mathbb{C}$. Specifically, letting $\tau$ denote a generator of the motivic cohomology group $H^{0,1}(BG,\Z/3) \cong \Z/3$ where $G$ is one of these groups, we show that the motivic cohomology ring contains no $\tau$-torsion, and so can be computed as a weight filtration on the ordinary group cohomology. In the case of a prime $p > 3$, there are also two nonabelian groups of order $p^3$. We make progress toward computing the motivic cohomology in the general case as well by reducing the question to understanding the $\tau$-torsion on the motivic cohomology of a $p$-dimensional variety; we also compute the motivic cohomology of $BG$ for general $p$ modulo the $\tau$-torsion classes.
2018-01-01T00:00:00ZHigher order asymptotics for the Central Limit Theorem and Large Deviation PrinciplesAkurugodage, Buddhima Kasun Fernandohttp://hdl.handle.net/1903/214022018-09-16T03:23:47Z2018-01-01T00:00:00ZHigher order asymptotics for the Central Limit Theorem and Large Deviation Principles
Akurugodage, Buddhima Kasun Fernando
First, we present results that extend the classical theory of Edgeworth expansions to independent identically distributed non-lattice discrete random variables. We consider sums of independent identically distributed random variables whose distributions have (d+1) atoms and show that such distributions never admit an Edgeworth expansion of order d but for almost all parameters the Edgeworth expansion of order (d-1) is valid and the error of the order (d-1) Edgeworth expansion is typically O(n^{-d/2}) but the O(n^{-d/2}) terms have wild oscillations.
Next, going a step further, we introduce a general theory of Edgeworth expansions for weakly dependent random variables. This gives us higher order asymptotics for the Central Limit Theorem for strongly ergodic Markov chains and for piece-wise expanding maps. In addition, alternative versions of asymptotic expansions are introduced in order to estimate errors when the classical expansions fail to hold. As applications, we obtain Local Limit Theorems and a Moderate Deviation Principle.
Finally, we introduce asymptotic expansions for large deviations. For sufficiently regular weakly dependent random variables, we obtain higher order asymptotics (similar to Edgeworth Expansions) for Large Deviation Principles. In particular, we obtain asymptotic expansions for Cramer's classical Large Deviation Principle for independent identically distributed random variables, and for the Large Deviation Principle for strongly ergodic Markov chains.
2018-01-01T00:00:00ZSome Applications of Set Theory to Model TheoryUlrich, Douglas Samuelhttp://hdl.handle.net/1903/213842018-09-16T02:54:50Z2018-01-01T00:00:00ZSome Applications of Set Theory to Model Theory
Ulrich, Douglas Samuel
We investigate set-theoretic dividing lines in model theory. In particular, we are interested in Keisler's order and Borel complexity.
Keisler's order is a pre-order on complete countable theories $T$, measuring the saturation of ultrapowers of models of $T$. In Chapter~\ref{SurveyChapter}, we present a self-contained survey on Keisler's order. In Chapter~\ref{KeislerNew}, we uniformize and sharpen several ultrafilter constructions of Malliaris and Shelah. We also investigate the model-theoretic properties detected by Keisler's order among the simple unstable theories.
Borel complexity is a pre-order on sentences of $\mathcal{L}_{\omega_1 \omega}$ measuring the complexity of countable models. In Chapter~\ref{ChapterURL}, we describe joint work with Richard Rast and Chris Laskowski on this order. In particular, we connect the Borel complexity of $\Phi \in \mathcal{L}_{\omega_1 \omega}$ with the number of potential canonical Scott sentences of $\Phi$. In Chapter~\ref{ChapterSB}, we introduce the notion of thickness; when $\Phi$ has class-many potential canonical Scott sentences, thickness is a measure of how quickly this class grows in size. In Chapter~\ref{ChapterTFAG}, we describe joint work with Saharon Shelah on the Borel complexity of torsion-free abelian groups.
2018-01-01T00:00:00ZPROBLEMS ORIGINATING FROM THE PLANNING OF AIR TRAFFIC MANAGEMENT INITIATIVESEstes, Alexanderhttp://hdl.handle.net/1903/213112018-09-13T03:41:22Z2018-01-01T00:00:00ZPROBLEMS ORIGINATING FROM THE PLANNING OF AIR TRAFFIC MANAGEMENT INITIATIVES
Estes, Alexander
When weather affects the ability of an airport to accommodate flights, a ground delay program is used to control the rate at which flights arrive at the airport. This prevents excessive congestion at the airport. In this thesis, we discuss several problems arising from the planning of these programs. Each of these problems provides insight that can be applied in a broader setting, and in each case we develop generalizations of these results in a wider context. We show that a certain type of greedy policy is optimal for planning a ground delay program when no air delays are allowed. More generally, we characterize the conditions under which policies are optimal for a dynamic stochastic transportation problem. We also provide results that ensure that certain assignments are optimal, and we apply these results to the problem of matching drivers to riders in an on-demand ride service.
When flights are allowed to take air delays, then a greedy policy is no longer optimal, but flight assignments can be produced by solving an integer program. We establish the strength of an existing formulation of this problem, and we provide a new, more scalable formulation that has the same strength properties. We show that both of these methods satisfy a type of equity property. These formulations are a special case of a dynamic stochastic network flow problem, which can be modeled as a deterministic flow problem on a hypergraph. We provide strong formulations for this general class of hypergraph flow problems.
Finally, we provide a method for summarizing a dataset of ground delay programs. This summarization consists of a small subset of the original data set, whose elements are referred to as "representative" ground delay programs. More generally, we define a new class of data exploration methods, called "representative region selection" methods. We provide a framework for evaluating the quality of these methods, and we demonstrate statistical properties of these methods.
2018-01-01T00:00:00Z