We give a homotopy theoretic characterization of stacks on a site C which allows one to think of stacks as the homotopy sheaves of groupoids on C. We use this characterization to construct a model category, that is a formal ...

In this thesis, I consider hyper-Kähler manifolds of complex dimension 4 which are fibrations. It is known that the fibers are abelian varieties and the base is P2. We assume that the general fiber is isomorphic to a product ...

In this thesis, we effectively construct a predecessor function for the type definitions in the raw hierarchy for any weakly-scattered theory. Using this predecessor function, we improve a recent bounding result by Sacks ...

In this thesis, I investigate the index of Callias type operators on asymptotically conic manifolds (also known as asymptotically locally Euclidean manifolds or scattering manifolds) and give an application to the moduli ...

In recent work of Baas-Dundas-Richter-Rognes, the authors introduce the notion of the K- theory of a bimonoidal category R, and show that it is equivalent to the algebraic K-theory space of the ring spectrum KR. In this ...

The central focus of my thesis is to study the instability jets of various complexity by analyzing the linear and nonlinear dynamics. We applied this methodology to four different situations in order to learn the following. ...

A new continuum model has been proposed by Singh, Ceder, and Bazant for the ion intercalation dynamics in a single crystal of rechargeable-battery electrode materials. It is based on the Cahn-Hilliard equation coupled to ...