The structural aspects of biological systems are tightly paired with their functions. This understanding has been demonstrated over a broad range of length scales, spanning the ultrastructure of a cell to the macroscopic architecture of organs. Connecting structure and function relies on the integration of physical and biological sciences to...
We use Goerss-Hopkins theory to show that if E is a p-local Landweber exact homology theory of height n and p > n^2 + n + 1, then there exists an equivalence hSpE ≃ hD(E∗E) between homotopy categories of E-local spectra and differential E∗E-comodules, generalizing Bousfield’s and Franke’s results to...
This thesis contains results in mathematical quantum ergodicity in a probabilistic or a complex analytic setting. For the former, we show that a random orthonormal basis of spherical harmonics is almost surely quantum ergodic, in which the randomness is induced by the generalized Wigner ensemble. For the latter, we show...
This dissertation concerns the probabilistic aspects of diffusion processes generated by a family of differential operators, which is similar to the family of hypoelliptic Laplacian operators, acting on the tangent bundle of a compact Riemannian manifold. By lifting the processes to the product of the frame bundle and the euclidean...
In this thesis, we study pushforwards of canonical and log-pluricanonical bundles on projective log canonical pairs over the complex numbers. We partially answer a Fujita-type conjecture proposed by Popa and Schnell in the log canonical setting. Built on Kawamata’s result for morphisms that are smooth outside a simple normal crossing...
The Brink-Schwarz superparticle is a one-dimensional analogue of the Green-Schwarz superstring. In this thesis, we use the Batalin-Vilkovisky formalism to study the superparticle. After proving a vanishing result for its Batalin-Vilkovisky cohomology, we explain the sense in which the superparticle exhibits general covariance in the world-line. Using techniques from rational...
This dissertation addresses the property of amenability of discrete groups and their actions. In Chapter 2, following the introduction, all necessary definitions are given to introduce amenable groups, elementary amenable groups, random walks, topological full groups, Thompson's group $F$ and to show connections between them. The chapter also briefly covers...
The spatial autoregressive model has been widely applied in science, in areas such as economics, public finance, political science, agricultural economics, environmental studies and transportation analyses. The classical spatial autoregressive model is a linear model for describing spatial correlation. In this work, we expand the classical model to include time...