Thesis (Ph. D.)--University of Oregon, 2000. / Includes vita and abstract. Includes bibliographical references (leaf 91). Also available for download via the World Wide Web; free to University of Oregon users.

Thesis (Ph. D.)--University of Oregon, 2000. / Title from title screen. Paging within document: vii, 91 p. Includes vita and abstract. Includes bibliographical references (p. 91).

Thesis (Ph. D.)--Ohio State University, 2008. / Title from first page of PDF file. Includes bibliographical references (p. 241-244).

The affine Grassmannian associated to a reductive group G is an affine analogue of the usual flag varieties. It is a rich source of Poisson varieties and their symplectic resolutions. These spaces are examples of conical symplectic resolutions dual to the Nakajima quiver varieties. In this work, we study their quantum connection. We use the stable envelopes of D. Maulik and A. Okounkov[MO2] to write an explicit formula for this connection. In order to do this, we construct a recursive relation for the stable envelopes in the G = PSL_2 case and compute the first-order correction in the general case. The computation of the purely quantum part of the multiplication is done based on the deformation approach of A. Braverman, D. Maulik and A. Okounkov[BMO]. For the case of simply-laced G, we identify the quantum connection with the trigonometric Knizhnik-Zamolodchikov equation for the Langlands dual group G^\vee.

We propose a coherent analogue of the non-archimedean case of Venkatesh's conjecture on the cohomology of locally symmetric spaces for Shimura varieties coming from unitary similitude groups. Let G be a unitary similitude group with an indefinite signature at at least one archimedean place. Let Π be an automorphic cuspidal representation of G whose archimedean component Π∞ is a non-degenerate limit of discrete series and let 𝑊 be an automorphic vector bundle such that Π contributes to the coherent cohomology of its canonical extension. We produce a natural action of the derived Hecke algebra of Venketesh with torsion coefficients via cup product coming from étale covers and show that under some standard assumptions this action coincides with the conjectured action of a certain motivic cohomology group associated to the adjoint representation Ad𝜌π of the Galois representation attached to Π. We also prove that if the rank of G is greater than two, then the classes arising from the \'etale covers do not admit characteristic zero lifts, thereby showing that previous work of Harris-Venkatesh and Darmon-Harris-Rotger-Venkatesh is exceptional.

Invariants of involutions on central simple algebras have been extensively studied. Many important results have been collected and extended by Knus, Merkurjev, Rost and Tignol in "The Book of Involutions" [BI]. Among those invariants are, for example, the (even) Clifford algebra for involutions of the first kind and the discriminant algebra for involutions of the second kind on an algebra of even degree. In his preprint "Triality, Cocycles, Crossed Products, Involutions, Clifford Algebras and Invariants" [S05], Saltman shows that the definition of the Clifford algebra can be generalized to Azumaya algebras and introduces a special cohomology, the so-called G-H cohomology, to describe its structure. In this dissertation, we prove analogous results about the discriminant algebra D(A; [tau]), which is the algebra of invariants under a special automorphism of order two of the [lambda]-power of an algebra A of even degree n = 2m with involution of the second kind, [tau]. In particular, we generalize its construction to the Azumaya case. We identify the exterior power algebra as defined in "Exterior Powers of Fields and Subfields" [S83] as a splitting subalgebra of the m-th [lambda]-power algebra and prove that a certain invariant subalgebra is a splitting subalgebra of the discriminant algebra. Assuming well-situatedness we show how this splitting subalgebra can be described as the fixed field of an S[subscript n] x C₂- Galois extension and that the corresponding subgroup is [Sigma] = S[subscript m] x S[subscript m] [mathematic symbol] C2. We give an explicit description of the corestriction map and define a lattice E that encodes the corestriction as being trivial. Lattice methods and cohomological tools are applied in order to define the group H²(G;E) which contains the cocycle that will describe the discriminant algebra as a crossed product. We compute this group to have order four and conjecture that it is the Klein 4-group and that the mixed element is the desired cocycle. / text

In this thesis, we establish a pair of systems of higher order cohomology operations that act on the Ζ_p-ordinary cohomology of spaces that are free of p-torsion. These "pyramids" of operations are generated by the p-divisibility of certain sums of "pseudo" primary cohomology operations that operate on the p-local cohomology of p-torsion-free spaces. The properties of these higher order operations allow us to prove theorems that either generalise or improve (in the sense of decreasing indeterminacy) several results in the literature.

Thesis (Ph.D.)--Tufts University, 2004. / Adviser: Loring W. Tu. Submitted to the Dept. of Mathematics. Includes bibliographical references (leaves 43-45). Access restricted to members of the Tufts University community. Also available via the World Wide Web;

Thesis (Ph. D.)--University of Texas at Austin, 2008. / Vita. Includes bibliographical references.

Thesis (Ph.D.) - University of Glasgow, 2008. / Ph.D. thesis submitted to the Faculty of Information and Mathematical Sciences, University of Glasgow, 2008. Includes bibliographical references. Print version also available.