**Title:** Centralizers and Conjugacy Classes in the Group of Interval Exchange Transformations **Date: **Wednesday, April 11, 2018**Thesis Advisor: **Michael Boshernitzan**Abstract: **

We study the group G of interval exchange transformations. Firstly, we will classify, up to conjugation in G, those interval exchange transformations which arise as towers over rotations. Secondly, expanding on previous work of Novak and Li, we will classify, up to conjugation in G, minimal interval exchange transformations which exhibit bounded discontinuity growth. As an application of this classification, we will prove that no infinite order interval exchange transformation could be conjugate in G to one of its proper powers. Thirdly, we will develop broadly applicable, generic conditions which ensure that an interval exchange transformation has no roots in G and that its centralizer is torsion-free. By combining these results

with a result of Novak, we will show that a typical interval exchange transformation does not commute with any interval exchange transformations other than its powers. Finally, we will completely describe the possible centralizers of a minimal three-interval exchange transformation.

**Title:** Anderson Localization for Discrete One-Dimensional Random Operators**Date: **Wednesday, April 4, 2018**Thesis Advisor: **David Damanik**Abstract: **

This thesis is concerned with the phenomenon of Anderson localization for one dimensional discrete Jacobi and Schroedinger operators acting on $\ell^2(\Z)$. Specifically, we prove dynamical and spectral localization at all energies for the discrete generalized Anderson model via the Kunz-Souillard approach to localization. This is an extension of the original Kunz-Souillard approach to localization for Schroedinger operators, to the case where a single random variable determines the potential on a block of an arbitrary, but fixed, size N. For this model, we also prove uniform positivity of the Lyapunov exponent. In fact, we prove a stronger statement where we also allow finitely supported distributions. We also show that for the generalized Anderson model, of any block size, there exists some finitely supported distribution $\nu$ for which the Lyapunov exponent will vanish for at least one energy. We also show that any Jacobi operator with bounded coefficients can be approximated, in operator norm, by Jacobi operators with slow-enough decaying diagonal entries and pure point spectrum.

**Title:** Transport Properties of One-dimensional Quantum Systems**Date: **Wednesday, April 4 2018**Thesis Advisor: **David Damanik**Abstract: **

We study quantum dynamical properties of discrete one-dimensional models of Schrodinger operators. The first goal of this thesis is to understand the time evolution of initial states supported on more than one site. We develop tools to bound the so-called transport exponents both from below and from above and apply them to several models. In particular we extend a group of results concerning Sturmain Hamiltonians, quasi-periodic Hamiltonians, substitution generated models and random polymer model. The second topic is a detailed analysis of the transport exponents in the case of a Sturmain model when the frequency is a quadratic irrational. In this case methods from hyperbolic dynamics are applied to study the trace map of the operator. We show that the wavepacket spreads out with the same polynomial rate on all possible timescales. The last model of interest is the Anderson model. We provide a new proof of the Anderson localization phenomenon and derive dynamical localization bounds for states supported on more than one site. This thesis contains joint work with Valmir Bucaj, David Damanik, Jake Fillman, Tom VandenBoom, Fengpeng Wang, and Zhenghe Zhang.

**Title:** Global Regularity for Euler Vortex Patch in Bounded Smooth Domains**Date: **Friday, April 13, 2018**Thesis Advisor: **Alexander Kiselev**Abstract: **

It is well known that the Euler vortex patch in two dimensional plane will remain regular if it is regular enough initially. In bounded domains, the regularity theory for patch solutions is less complete. In this thesis, I study the Euler vortex patch in a general smooth bounded domain. I prove global in time regularity by providing the upper bound of the growth on curvature of the patch boundary. For a special symmetric scenario, I construct an example of double exponential curvature growth, showing that our upper bound is qualitatively sharp.

**Title:** Cohomology classes responsible for Brauerâ€“Manin obstructions, with applications to rational and K3 surfaces**Date: **Friday, April 13, 2018**Thesis Advisor: **Anthony VĂˇrilly-Alvarado**Abstract: **

We study the classes in the Brauer group of varieties that never obstruct the Hasse principle for X. We prove that for a variety with a genus 1 fibration, if the generic fiber has a zero-cycle of degree d over the generic point, then the Brauer classes whose orders are prime to d do not play a role in the Brauer-Manin obstruction. As a result we show that the odd torsion Brauer classes never obstruct the Hasse principle for del Pezzo surfaces of degree 2, certain K3 surfaces, and Kummer varieties.

We also analyze the Brauer-Manin obstruction to rational points on the K3 surfaces over Q given by double covers of P2 ramied over a diagonal sextic. After finding an explicit set of generators for the geometric Picard group of such a surface, we find two types of infinite families of counterexamples to the Hasse principle explained by the algebraic Brauer-Manin obstruction. The first type of obstruction comes from a quaternion algebra, and the second type comes from a 3-torsion element of the Brauer group, which gives an affirmative answer to a question asked by Ieronymou and Skorobogatov.

**Title:** Isospectral dynamics of reflectionless Jacobi operators**Date: **Friday, March 30, 2018**Thesis Advisor: **David Damanik**Abstract: **

This thesis focuses on the isospectral torus of reflectionless Jacobi operators and the dynamics of its automorphisms. The novel perspective which it hopes to advertise is one of the joint utility of inverse spectral theoretic techniques and dynamical techniques to address direct and inverse spectral problems, respectively. Concretely, we use the former perspective to prove the reducibility of the shift cocycle for certain reflectionless Jacobi operators, and we use the latter perspective to prove spectral atypicality of discrete Schroedinger operators.

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