2010 Ph.D Thesis Defenses
Title:Interval exchange transformations: Applications of Keane's construction and disjointness
Date: Tuesday, February 23, 2010
Thesis Advisor: Michael Boshernitzan
This thesis is divided into two parts. The first part uses a family of Interval Exchange Transformations constructed by Michael Keane to show that IETs can have some particular behavior including: (1) IETs can be topologically mixing. (2) A minimal IET can have an ergodic measure with Hausdorff dimension alpha for any alpha ∈ [0,1]. (3) The complement of the generic points for Lebesgue measure in a minimal non-uniquely ergodic IET can have Hausdorff dimension 0. Note that this is a dense Gdelta set. The second part shows that almost every pair of IETs are different. In particular, the product of almost every pair of IETs is uniquely ergodic. In proving this we show that any sequence of natural numbers of density 1 contains a rigidity sequence for almost every IET, strengthening a result of Veech.
Title: State cycles, quasipositive modification, and constructing H-thick knots in Khovanov homology
Thesis Advisor: Tim Cochran
We study Khovanov homology classes which have state cycle representatives, and examine how they interact with Jacobsson homomorphisms and Lee's map phi. As an application, we describe a general procedure, quasipositive modification, for constructing H-thick knots in rational Khovanov homology. Moreover, we show that specific families of such knots cannot be detected by Khovanov's thickness criteria. We also exhibit a sequence of prime links related by quasipositive modification whose width is increasing.
Title:Positive Lyapunov exponent for ergodic Schrodinger operators
Date: Friday, April 16, 2010
Thesis Advisor: David Damanik
The discrete Schrodinger equation describes the behavior of a 1-dimensional quantum particle in a disordered medium. The Lyapunov exponent L( E) describes the exponential behavior of solutions at an energy E. Positivity of the Lyapunov exponent in a set of energies is an indication of absence of transport for the Schrodinger equation. In this thesis, I will discuss methods based on multiscale analysis to prove positive Lyapunov exponent for ergodic Schrodinger operators. As an application, I prove positive Lyapunov exponent for operators whose potential is given by evaluating an analytic sampling function along the orbit of a skew-shift on a high dimensional torus. The first method is based only on ergodicity, but needs to eliminate a small set of energies. The second method uses recurrence properties of the skew-shift, combined with analyticity to prove a result for all energies.
Karoline Pershell Null
Title: Some conditions for recognizing a 3-manifold group
Thesis Advisor: John Hempel
In this work we ask when a group is a 3-manifold group, or more specifically, when does a group presentation come naturally from a Heegaard diagram for a 3-manifold? We will give some conditions for partial answers to this form of the Isomorphism Problem by addressing how the presentation associated to a diagram for a splitting is related to the fundamental group of a 3-manifold, still using diagrams as a tool to answer these questions. In the process, we determine an invariant of groups (by way of group presentations) for how far such presentations are from 3-manifolds.