Homotopy Batalin-Vilkovisky algebras, trivializing circle actions, and moduli space
Author:
Gabriel C. Drummond-Cole
Year of Dissertation:
2010
This thesis comprises two main results, one topological, one algebraic. The topological result is that an action of the framed little disks operad and a trivialization of the circle action within it determine an action of the Deligne-Mumford compactification of the moduli space of genus zero curves. The algebraic result is a description of the structure of minimal homotopy Batalin-Vilkovisky algebras and the the theorem that in the case that the Batalin-Vilkovisky operator and its higher homotopies are trivial, the remaining algebraic structure is a minimal homotopy hypercommutative algebra. These results are related to one another because the algebraic structures involved are representations of the homology of, respectively, the framed little disks and the Deligne-Mumford compactification.
Cohomological aspects of complete reducibility of representations
Year of Dissertation:
2009
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On the Dynamics of Quasi-Self-Matings of Generalized Starlike Complex Quadratics and the Structure of the Mated Julia Sets
Year of Dissertation:
2009
It has been shown that, in many cases, Julia sets of complex polynomials can be "glued" together to obtain a new Julia set homeomorphic to a Julia set of a rational map; the dynamics of the two polynomials are reflected in the dynamics of the mated rational map. Here, I investigate the Julia sets of self-matings of generalized starlike quadratic polynomials, which enjoy relatively simple combinatorics. The points in the Julia sets of the mated rational maps are completely classified according to their topology. The presence and location of buried points in these Julia sets are addressed. The interconnections between complex dynamics, combinatorics, symbolic dynamics and Thurston's lamination theory are explored and utilized. The results are then extended to "quasi-self-matings".
Aspects of Supercompactness, HOD and Set Theoretic Geology
Year of Dissertation:
2009
In this thesis, we study HOD, primarily in the context of large cardinals and GCH. Chapter 1 contains our introductory comments and preliminary remarks. In Chapter 2, we extend a property of HOD-supercompactness due to Sargsyan to various models of set theory
The Witt Ring of a Smooth Curve with Good Reduction over a Local Field
Year of Dissertation:
2012
The modern study of bilinear forms has a rich history beginning with Witt's work over fields in the 1930's, when he defined a ring structure on the set of anisotropic forms over a field. It was revived, notably by Pfister, in the 1960's. With the advent of algebraic K-Theory, much of the theory of quadratic forms over fields was generalized to a theory of quadratic spaces over rings. In the 1960's and 1970's Knebusch, among others, formulated a compatible theory for quadratic forms over schemes in which a ring analogous to Witt's ring of anisotropic forms is prominent. Calculation of such "Witt rings" is a problem of interest in modern algebraic geometry.
Asymptotics for the parabolic, hyperbolic, and elliptic Eisenstein series through hyperbolic and elliptic degeneration
Year of Dissertation:
2009
Let $\Gamma$ be a Fuchsian group of the first kind
Divisible Groups in the K-theory Completion of SU(n)
Year of Dissertation:
2011
I use the results of Bendersky and Thompson for the E(1)-based E2 -term of S2n+1, K E2s,tS{2n+1}, and the results of Bendersky and Davis concerning the v1-periodic groups of SU(n) to compute the E(1)-based E2-term for X=SU(n) for all primes p. This computation is performed using the Bendersky Thompson spectral sequence for SU(n). For spaces like SU(n) this spectral sequence converges to homotopy groups of the K-theory completion of SU(n). Of particular interest is the existence of infinitely many divisible groups in the homotopy groups of the K-theory completion of SU which offers an example of how E-completion does not commute with direct limits.
Groups, Complexity, Cryptology
Year of Dissertation:
2012
Advisor:
Delaram Kahrobaei
The field of non-commutative group based cryptography has flourished in the past twelve years with the increasing need for secure public key cryptographic protocols. This has led to an active line of research called non-abelian group based cryptography.
On the rank of 2-primary part of Selmer group of certain elliptic curves
Year of Dissertation:
2012
Kolyvagin proved very remarkable results on Mordell-Weil groups and
The Admissible Dual of SL(2) of the Dyadic Numbers
Author:
Terence Kivran-Swaine
Year of Dissertation:
2011
The admissible dual of SL2(Q2) is constructed uniformly, based on a method adapted from the the theory of cuspidal types of GL2(F).