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Filling Essential Laminations, Michael Hamm

All Theses and Dissertations (ETDs)

Thurston and, later, Calegari-Dunfield found superlaminations in certain laminated 3-manifolds, the existence of which implies inclusions into Homeo S1 of the fundamental groups of those manifolds. The present paper extends the construction of the superlamination, and finds an infinite class of manifolds to which the extension does not yield such an inclusion of groups. Specifically, Calegari and Dunfield's proof of the existence of such an inclusion into Homeo S1 depended on their filling lemma, which states that essential laminations with solid torus guts can have leaves added to them to yield essential laminations with solid torus complementary regions.: Roughly, a …

The Nonexistence Of Shearlet-Like Scaling Multifunctions That Satisfy Certain Minimally Desirable Properties And Characterizations Of The Reproducing Properties Of The Integer Lattice Translations Of A Countable Collection Of Square Integrable Functions, Robert Houska

All Theses and Dissertations (ETDs)

In Chapter 1, we introduce three varieties of reproducing systems—Bessel systems, frames, and Riesz bases—within the Hilbert space context and prove a number of elementary results, including qualitative characterizations of each and several results regarding the combination and partitioning of reproducing systems.

In Chapter 2, we characterize when the integer lattice translations of a countable collection of square integrable functions forms a Bessel system, a frame, and a Riesz basis.

In Chapter 3, we introduce composite wavelet systems and generalize several well-known classical wavelet system results—including those regarding pointwise values of the Fourier transform of the wavelet and scaling function …

Connections Between Floer-Type Invariants And Morse-Type Invariants Of Legendrian Knots., Michael Henry

All Theses and Dissertations (ETDs)

We investigate existing Legendrian knot invariants and discover new connections between the theory of generating families, normal rulings and the Chekanov-Eliashberg differential graded algebra: CE-DGA). Given a Legendrian knot $\sK$ with generic front projection $\sfront$, we define a combinatorial/algebraic object on $\sfront$ called a \emph{Morse complex sequence}, abbreviated MCS. An MCS encodes a finite sequence of Morse homology complexes. Every suitably generic generating family for $\sfront$ admits an MCS and every MCS has a naturally associated graded normal ruling. In addition, every MCS has a naturally associated augmentation of the CE-DGA of the Ng resolution $\sNgres$ of the front $\sfront$. …