Finding new examples of compact simply connected spaces admitting a Riemannian metric of positive sectional curvature is a fundamental problem in differential geometry. Likewise, studying topological properties of families of manifolds is very interesting to topologists. The Eschenburg spaces combine both of those interests: they are positively curved Riemannian manifolds whose topological classification is known. There is a second family consisting of the Witten manifolds: they are the examples of compact simply connected spaces admitting Einstein metrics of positive Ricci curvature. Thirdly, there is a notion of generalized Witten manifold as well. Topologically, all three families share the same cohomology ring. This common ring structure motivates the definition of a manifold of type r, where r is the order of the fourth cohomology group. In 1991, M. Kreck and S. Stolz classified manifolds M of type r up to homeomorphism and dieomorphism using invariants s̄[subscript i](M) and s[subscript i](M), for i = 1, 2, 3. This gave rise to many new examples of nondieomorphic but homeomorphic manifolds. In this dissertation, new versions of the homeomorphism and dieomorphism classification of manifolds of type r are proven. In particular, we can replace s̄₁ and s̄₃ by the first Pontrjagin class and the self-linking number in the homeomorphism classification of spin manifolds of type r. As the formulas of the two latter invariants are in general much easier to compute, this simplifies the classification of these manifolds up to homeomorphism significantly. / Graduation date: 2012

Thesis (Ph. D.)--University of Oregon, 2008. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 71-72). Also available online in Scholars' Bank; and in ProQuest, free to University of Oregon users.

Let g be a complex semisimple Lie algebra, Vλ an irreducible g-module with high weight λ, pI a standard parabolic subalgebra of g with Levi factor £I and nil radical nI, and H*(nI, Vλ) the cohomology group of Λn'I ⊗Vλ. We describe the decomposition of H*(nI, Vλ) into irreducible £1-modules.

We study the equivariant cohomology of moduli spaces of objects in the derived category of elliptic surfaces in order to find new examples of infinite dimensional quantum integrable systems and their geometric representation theoretic interpretation in enumerative geometry. This problem is related to a program to understand the cohomological and K-theoretic Hall algebras of holomorphic symplectic surfaces and to understand how it related to the Donaldson-Thomas theory of threefolds fibered in those surfaces. We use the theory of noncommutative deformations of Poisson surfaces and especially van den Berg’s noncommutative P1 bundles as well as Rains’s analysis of moduli theory for quasi-ruled noncommutative surfaces as well as the theory of Bridgeland stability conditions and their relative versions to understand equivariant deformations and birational transformations of Hilbert schemes of points on equivariant elliptic surfaces. The moduli spaces are closely related to elliptic versions of classical integrable systems. We also use these moduli spaces to construct vertex algebra representations of extensions of toroidal extended affine algebras on their equivariant cohomology, building on work of Eswara-Rao–Moody–Yokonuma, of Billig, and of Chen–Li–Tan on vertex representations of toroidal algebras, full toroidal algebras, and toroidal extended affine algebras. Using Fourier-Mukai transforms and their relative action on families of dg-categories we study the relationship between automorphisms of toroidal extended affine algebras and families of derived equivalences on dg categories, in particular finding a relativistic (difference) generalization of the Laumon-Rothstein deformation of the Fourier-Mukai duality. Finally, using the above analysis we extend the construction of Maulik–Okounkov’s stable envelopes to moduli of framed torsionfree sheaves on noncommutative surfaces in some cases and use this to study coproducts on associated algebras assigned to elliptic surfaces with applications to understanding new representation theoretic structures in the Donaldson-Thomas theory of local curves.