Microtubules are critically important cytoskeletal elements. Together with microtubule associated proteins (MAPs), they form the latticework on which eukaryotic life exists. Simply put, microtubules are polymers of tubulin heterodimers, which are composed of the globular proteins alpha and beta tubulin. In vivo, these monomers associate with one another to form heterodimers, which then polymerize to form microtubules. In mammals, microtubule polymerization is a temperature-dependent process with an optimum of 37°C (Detrich et al., 2000). If temperatures exceed this optimal temperature by even a few degrees, the microtubule will begin to dissemble due to denaturation of the tubulin subunit and permanent loss of both shape and function will occur. This thermal barrier seems to be consistent in most eukaryotic organisms. Two exceptions are the thermophilic red algae, Cyanidium caldarium and Cyanidioschyzon merolae. These thermophilic acidophiles have been discovered in volcanic vents around the globe from Yellow Stone Park to Italy and grow at optimal temperatures of around 55°C. These organisms have been primarily studied in the context of evolutionary biology because of their primitive characteristics. Very little is known about the molecular biology of these organisms, and certainly nothing is known about how the biochemistry of these organisms brings about the ability to survive the harsh conditions of their environment. Currently, my hypothesis concerning the thermostable tubulin expressed within these organisms is that there may be key amino acid differences in the tubulin's primary structure that confer enhanced thermostability. I am testing this hypothesis by sequencing the alpha and beta tubulin genes of Cyanidium caldarium and Cyanidioschyzon merolae, generating homology models of the tubulin dimers, and comparing these models to a known mesophilic tubulin heterodimer structure in order to identify potential structural differences. / Master of Science

In this thesis we present a slight generalisation of the Pearl complex or relative quantum homology to some non monotone Lagrangian submanifolds. First we develop the theory for the so called almost monotone Lagrangian submanifolds, We apply it to uniruling problems as well as estimates for the relative Gromov width. In the second part we develop the theory for toric fiber in toric Fano manifolds, recovering previous computaional results of Floer homology .

<p>This dissertation places intersection homology and local homology within the framework of persistence, which was originally developed for ordinary homology by Edelsbrunner, Letscher, and Zomorodian. The eventual goal, begun but not completed here, is to provide analytical tools for the study of embedded stratified spaces, as well as for high-dimensional and possibly noisy datasets for which the number of degrees of freedom may vary across the parameter space. Specifically, we create a theory of persistent intersection homology for a filtered stratified space and prove several structural theorems about the pair groups asso- ciated to such a filtration. We prove the correctness of a cubic algorithm which computes these pair groups in a simplicial setting. We also define a series of intersec- tion homology elevation functions for an embedded stratified space and characterize their local maxima in dimension one. In addition, we develop a theory of persistence for a multi-scale analogue of the local homology groups of a stratified space at a point. This takes the form of a series of local homology vineyards which allow one to assess the homological structure within a one-parameter family of neighborhoods of the point. Under the assumption of dense sampling, we prove the correctness of this assessment at a variety of radius scales.</p> / Dissertation

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Given a Liouville manifold, symplectic cohomology is defined as the Hamiltonian Floer homology for the symplectic action functional on the free loop space. In this thesis, we propose two versions of periodic S^1-equivariant homology or S^1-equivariant Tate homology for the natural S^1-action on the free loop space. The first version is called periodic symplectic cohomology. We prove that there is a localization theorem or a fix point property for periodic symplectic cohomology. The second version is called the completed periodic symplectic cohomology which satisfies Goodwillie's excision isomorphism. Inspired by the work of Seidel and Solomon on the existence of dilations on symplectic cohomology, we formulate the notion of Liouville manifolds admitting higher dilations using Goodwillie's excision isomorphism on the completed periodic symplectic cohomology. As an application, we derive obstructions to existence of certain exact Lagrangian immersions in Liouville manifolds admitting higher dilations.

Li Changzheng. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (leaves 43-44). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Preliminaries --- p.3 / Chapter 2.1 --- Clifford Algebras --- p.3 / Chapter 2.2 --- Spin Representation and Spinor Bundles --- p.7 / Chapter 2.3 --- Normed Division Algebras --- p.11 / Chapter 3 --- Associated Representations on /\. V* --- p.15 / Chapter 3.1 --- Exterior Forms and Spinors --- p.15 / Chapter 3.2 --- Direct Calculations --- p.16 / Chapter 3.3 --- "u(l,l,K) Action on V + V*" --- p.24 / Chapter 3.4 --- "su(l,l,K)´so(R1´ة0+K)" --- p.30 / Chapter 4 --- Some Applications to Geometry --- p.35 / Chapter 4.1 --- Holonomy Representations and Spinor Bundles --- p.35 / Chapter 4.2 --- The Lefschetz Action: Kahler Case --- p.37 / Chapter 4.3 --- The Lefschetz Action: HyperKahler Case --- p.41 / Bibliography --- p.43

The stable envelope for symplectic resolutions, constructed by Maulik and Okounkov, is a key ingredient in their work on quantum cohomology and quantum K-theory of Nakajima quiver varieties. In this thesis, we study the various aspects of the cohomological stable basis for the cotangent bundle of flag varieties. We compute its localizations, use it to calculate the quantum cohomology of the cotangent bundles, and relate it to the Chern--Schwartz--MacPherson class of Schubert cells in the flag variety.