A norm ||⋅|| on a Banach space X is Fréchet differentiable at x ∈ X if there is a functional ∫∈ X* such that [Formula Omitted] This concept reflects the smoothness characteristics of X. A dual Banach space X* has the Radon-Nikodym Property (RNP) if whenever C ⊂ X* is weak*-compact and convex, and ∈ > 0, there is an x ∈ X and an ⍺ > 0 such that diameter [Formula Omitted] this property reflects the convexity characteristics of X*. Culminating several years of work by many researchers, the following theorem established a strong connection between the smoothness of X and the convexity of X*: Every equivalent norm on X is Fréchet differentiable on a dense set if and only if X* has the RNP. A more general measure of convexity has been recently receiving a great deal of attention: A dual Banach space X* has the weak* Convex Point-of-Continuity Property (C*PCP) if whenever ɸ ≠ C ⊂ X* is weak*-compact and convex, and ∈ > 0, there is a weak*-open set V such that V ⋂ C ≠ ɸ and diam V ⋂ C < ∈. In this thesis, we develop the corresponding smoothness properties of X which are dual to C*PCP. For this, a new type of differentiability, called cofinite Fréchet differentiability, is introduced, and we establish the following theorem: Every equivalent norm on X is cofinitely Fréchet differentiable everywhere if and only if X* has the C*PCP. Representing joint work with R. Deville, G. Godefroy and V. Zizler, an alternate approach is developed in the case when X is separable. We show that if X is separable, then every equivalent norm on X which has a strictly convex dual is Fréchet differentiable on a dense set if and only if X* has the C*PCP, if and only if every equivalent norm on X which is Gâteaux differentiable (everywhere) is Fréchet differentiable on a dense set. This result is used to show that if X* does not have the C*PCP, then there is a subspace Y of X such that neither Y* nor (X/Y)* have the C*PCP, yet both Y and X/Y have finite dimensional Schauder decompositions. The corresponding result for spaces X* failing the RNP remains open. / Science, Faculty of / Mathematics, Department of / Graduate

Thesis (M.Phil.)--Chinese University of Hong Kong. / Bibliography: leaves 38-39.

This thesis studies the Bar-Natan skein module of the solid torus with a particular boundary curve system, and in particular a diagrammatic presentation of it due to Russell. This module has deep connections to topology and categorification: it is isomorphic to both the total homology of the (n,n)-Springer variety and the 0th Hochschild homology of the Khovanov arc ring H^n. We can also view the Bar-Natan--Russell skein module from a representation-theoretic viewpoint as an extension of the Frenkel--Khovanov graphical description of the Lusztig dual canonical basis of the nth tensor power of the fundamental U_q(sl_2)-representation. One of our primary results is to extend a dualization construction of Khovanov using Jones--Wenzl projectors from the Lusztig basis to the Russell basis. We also construct and explore several deformations of the Russell skein module. One deformation is a quantum deformation that arises from embedding the Russell skein module in a space that obeys Kauffman--Lins diagrammatic relations. Our quantum version recovers the original Russell space when q is specialized to -1 and carries a natural braid group action that recovers the symmetric group action of Russell and Tymoczko. We also present an equivariant deformation that arises from replacing the TQFT algebra A used in the construction of the rings H^n by the equivariant homology of the two-sphere with the standard action of U(2) and taking the 0th Hochschild homology of the resulting deformed arc rings. We show that the equivariant deformation has the expected rank. Finally, we consider the Khovanov two-functor F from the category of tangles. We show that it induces a surjection from the space of cobordisms of planar (2m, 2n)-tangles to the space of (H^m, H^n)-bimodule homomorphisms and give an explicit description of the kernel. We use our result to introduce a new quotient of the Russell skein module.

by Leung Chi Kwan. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1996. / Includes bibliographical references (leaves 133-135). / Preface --- p.ii / Chapter 1 --- Introduction to Module Equivalence --- p.1 / Chapter 1.1 --- Introduction and Preliminaries --- p.1 / Chapter 2 --- Some Classical Results --- p.12 / Chapter 2.1 --- Morita Theorem --- p.12 / Chapter 2.2 --- Puller Theorem --- p.13 / Chapter 2.3 --- The Equivalence Mod-A ~Im(TP) --- p.29 / Chapter 2.4 --- The Equivalence Im(HP)~Im(Tp) --- p.33 / Chapter 3 --- *-modules and Tilting Modules --- p.39 / Chapter 3.1 --- The Equivalence Cogen(KA)~Gen(PR) --- p.39 / Chapter 3.2 --- Torsion Theories and *-modules --- p.56 / Chapter 3.3 --- The Structure of *-modules --- p.60 / Chapter 3.4 --- Characterizations of Tilting Modules --- p.65 / Chapter 4 --- Equivalences and Dualities --- p.85 / Chapter 4.1 --- The Equivalence PA~IR --- p.85 / Chapter 4.2 --- The Equivalence FGP-A ~FCI-R --- p.93 / Chapter 5 --- Torsion Theories Induced by Tilting Modules --- p.100 / Chapter 5.1 --- The Tilting Theorem --- p.100 / Chapter 5.2 --- Tilting Torsion Theories --- p.113 / Chapter 5.3 --- Isomorphisms of Endomorphism Rings of Tilting Modules --- p.122 / References --- p.133

Thesis. 1977. Ph.D.--Massachusetts Institute of Technology. Dept. of Mathematics. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Bibliography: leaves 270-272. / by Stephen Kinyon Young. / Ph.D.

Thesis (Ph. D.)--University of Washington, 2003. / Vita. Includes bibliographical references (p. 79-80).

The Kardar-Parisi-Zhang (KPZ) equation is a stochastic PDE describing various objects in statistical mechanics such as random interface growth, directed polymers, interacting particle systems. We study large deviations of the KPZ equation, both in the short time and long time regime. We prove the first short time large deviations for the KPZ equation and detects a Gaussian - 5/2 power law crossover in the lower tail rate function. In the long-time regime, we study the upper tail large deviations of the KPZ equation starting from a wide range of initial data and explore how the rate function depends on the initial data. The KPZ equation plays a role as the weak scaling limit of various models in the KPZ universality class. We show the stochastic higher spin six vertex model, a class of models which sit on top of the KPZ integrable systems, converges weakly to the KPZ equation under certain scaling. This extends the weak universality of the KPZ equation. On the other hand, we show that under a different scaling, the stochastic higher spin six vertex model converges to a hyperbolic stochastic PDE called stochastic telegraph equation. One key tool behind the proof of these two stochastic PDE limits is a property called Markov duality.

Thesis: M.S., Massachusetts Institute of Technology, Department of Mathematics, 1979 / Bibliography: leaf 61. / by Paulo Arruda Villela. / M.S. / M.S. Massachusetts Institute of Technology, Department of Mathematics

Thesis (Ph. D.)--Florida State University, 2004. / Advisor: Dr. Daniel M. Oberlin, Florida State University, College of Arts and Sciences, Dept. of Mathematics. Title and description from dissertation home page (June 18, 2004). Includes bibliographical references.

Thesis (Ph. D.)--State University of New York at Binghamton, of Department of Mathematical Sciences, 2007. / Includes bibliographical references.