This dissertation proves several results for first passage percolation on graphs of polynomial growth.The class of limit shapes for first passage percolation with stationary weights on Cayley
graphs of virtually nilpotent groups is characterized.
Then strict monotonicity theorems for independent first passage percolation on graphs of
polynomial growth and quasi-trees...
The action of the automorphisms of a formal group on its deformation space is crucial to understanding periodic families in the homotopy groups of spheres and the unsolved Hecke orbit conjecture for unitary Shimura varieties. This action is called the Lubin-Tate action. We first show sufficient conditions for geometrically modelling...
The main topic of this thesis is generation in derived categories of coherent sheaves on smooth projective varieties. We develop a new approach that allows us to give a new proof of a recent result by Olander that powers of an ample line bundle generate the bounded derived category of...
In this thesis, we discuss classical and recent results around the damped wave equation on compact and noncompact manifolds. We firstly show that on asymptotically cylindrical and conic manifolds, the geometric control condition and the network control condition give exponential and logarithmic decay rates respectively. We then show that a...
We present both semiclassical asymptotics for the wave equation on a stationary Kaluza-Klein spacetime and an index theorem describing the difference of the positive-frequency
spectral projectors for two stationary regions in a globally hyperbolic spacetime. The
first result involves analyzing the restrictions of the wave trace to isotypic subspaces for...
In this thesis, we consider the categories of sheaves with singular support on certain Lagrangians and the categories of microlocal sheaves with support on certain Lagrangians obtained by microlocalization, and study properties of functors between these categories.First, we study one class of the microlocal restriction functor for open inclusions, namely...
A great deal of work has been done in recent years to construct algebraic invariants ofLegendrian knots, and their higher-dimensional analogues. Here we employ the diagram-
matic calculus developed in [3], in order to develop an iterative method for computing the
so-called vexillary functions of a class of Legendrian surfaces....
In the first part of the thesis we,given a dg-Lie algebra $\g$, a commutative
dg-algebra $A$ and a twisting cochain $\mu$ in
$A \otimes \g$, we construct a map
from the functor of curved deformations of $\g$ to the
functor of curved deformations of the twisted tensor product
$A \otimes_\mu...
The purpose of this thesis is to study the empirical measure of the t-distributed stochastic neighbor embedding algorithm (t-SNE) when the input is given by n independent, identically distributed inputs.We show that this sequence of measures converges to an equilibrium measure, which can be described as the solution of a...
In this thesis, we consider the spherical spin glass models at zero temperature. We first determine the structureof the Parisi measure at zero temperature for the spherical p+s spin glass model. We then
consider the spherical mixed p-spin model and show that for the spherical spin models with
n components,...
We consider constant scalar curvature K ̈ahler metrics on a smooth minimal model of general type in a neighborhood of the canonical class, which is the perturbation of the canonical class by a fixed K ̈ahler metric. We show that sequences of such metrics converge smoothly on compact subsets away...
We introduce empirical measures to study the $L^2$ norms of restrictions of quantum integrable eigenfunctions to the unique rotationally invariant geodesic $H$ on a convex surface of revolution. The weak* limit of these measures describes the dependence of their size on $H$ in terms of the angular momentum. The limit...
The purpose of this thesis is to derive three results in probability theory. The first is a proof that small powers of the normalized absolute characteristic polynomial of a random matrix sampled from either the Gaussian Orthogonal or Symplectic Ensemble converges in law to a Gaussian multiplicative chaos measure. The...
Even though a number of techniques have been developed for motion generation and perception, few of them focus on the computational efficiency and theoretical guarantees at the same time. Typically, improved guarantees come with increased complexity, making theoretically guaranteed methods challenging use in real-time applications. Thus, existing methods usually have...
In this paper we begin the study of the (dual) Steenrod algebra of the motivic Witt cohomology spectrum H_Wℤ by determining the algebra structure of H_Wℤ∗∗H_Wℤ over fields k of characteristic not 2 which are extensions of fields F with K^M_2(F)/2=0. For example, this includes all fields of odd characteristic,...
In this dissertation, we study the birational geometry of log pairs over the field of complex numbers, with an emphasis on the positivity properties of log pairs and their applications. First, we study the nonvanishing conjecture in the minimal model program. We prove the nonvanishing conjecture for uniruled log canonical...
This dissertation provides an introduction to the diffraction and scattering theory for the Aharonov--Bohm Hamiltonian with one or multiple poles on $\mathbf{R}^2$. It shows the propagation of diffractive singularities of the wave equation with the magnetic Hamiltonians with singular vector potential, which is related to the so-called Aharonov--Bohm effect. Based...
This thesis is composed of two major parts. The first major part includes the first two chapters. In Chapter 1, we go over the history and background of spin glass theory and introduce the Sherrington-Kirkpatrick (SK) model and the Ghatak-Sherrington (GS) model. In Chapter 2, we introduce the Thouless-Anderson-Palmer (TAP)...
We study the complexification of Laplace Eigenfunctions on the Grauert tube of a compact real analytic manifold. Our main results concern scaling asymptotics of Fourier coefficients of the Szego kernel on the Grauert tube boundary in a Heisenberg frequency scaled neighborhood of the geodesic flow. We show that in the...
Parshin's conjecture expects that the higher algebraic $K$-groups of smooth projective varieties over a finite field are torsion. In this thesis we prove that with the assumption of finite generation of higher \'etale algebraic $K$-theory of smooth projective varieties over a finite field, one can reduce Parshin's conjecture to the...