**Title:** Two mod-p Johnson filtrations**Date: **Friday, April 11, 2014**Thesis Advisor: **Andrew Putman**Abstract: **

We consider two mod-p central series of the free group given by Stallings and Zassenhaus. Applying these series to definitions of Dennis Johnson's filtration of the mapping class group we obtain two mod-p Johnson filtrations. Further, we adapt the definition of the Johnson homomorphisms to obtain mod-p Johnson homomorphisms. We calculate the image of the first of these homomorphisms. We give generators for the kernels of these homomorphisms as well. We restrict the range of our mod-p Johnson homomorphisms using work of Morita. We finally prove the announced result of Perron that a rational homology 3-sphere may be given as a Heegaard splitting with gluing map coming from certain members of our mod-p Johnson filtrations.

**Title: **Spectral Regularity in Some Models of Aperiodic Order**Date: **Tuesday, April 22, 2014**Thesis Advisor: **David Damanik**Abstract: **

This thesis deals with the spectral theory of Schroedinger and CMV operators. Specifically, it concentrates on regularity properties of the spectrum, and on operators that arise from irrational rotations of the circle. Joint work with David Damanik, Darren Ong, and William Yessen is contained herein.

**Title:** Spectral characteristics of aperiodic CMV and Schroedinger operators**Date:** Tuesday, April 1, 2014**Thesis Advisor: **David Damanik**Abstract:**

This PhD. thesis consists of theorems concerning the spectral theory of CMV and Schroedinger operators that are either almost periodic, derived from low-complexity sequence subshifts, or perturbations of periodic operators.

**Title:** Casson towers and filtrations of the smooth knot concordance group**Date:** Tuesday, April 8, 2014**Thesis Advisor: **Tim Cochran**Abstract:**

The study of knots in 3-space is intimately connected with the study of manifolds. The four-dimensional equivalence relation of 'concordance' (smooth or merely topological) gives a group structure on the set of knots. The n-solvable filtration of the smooth knot concordance group (denoted C), due to Cochran-Orr-Teichner, has been instrumental in the study of knot concordance in recent years. Part of its significance is due to the fact that certain geometric characterizations of a knot imply membership in various levels of the filtration. We show the counterpart of this fact for two new filtrations of C due to Cochran-Harvey-Horn, the positive and negative filtrations. The positive and negative filtrations have definitions similar to the n-solvable filtration, but have the ability (unlike the n-solvable filtration) to distinguish between smooth and topological concordance. Our geometric counterparts for the positive and negative filtrations of C are defined in terms of 'Casson towers', four-dimensional objects which approximate disks in a precise manner. We establish several relationships between these new Casson tower filtrations and the various previously known filtrations of C, such as the n-solvable, positive, negative, and 'grope' filtrations. These relationships allow us to draw connections between some well-known open questions in the field.

**Title:** Deformations of Hilbert Schemes of Points on K3 Surfaces and Representation Theory**Date: **Tuesday, March 25, 2014**Thesis Advisor: **Brendan Hassett**Abstract:**

We study the cohomology rings of Kaehler deformations X of Hilbert schemes of points on *K3* surfaces by representation theory. We compute the graded character formula of the Mumford - Tate group representation on the cohomology ring of X. Furthermore, we also study the Hodge structure of *X*, and find the generating series for deducting the number of canaonical Hodge classes in the middle cohomology.