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...
We prove the uniqueness of equilibrium states for certain potentials satisfying the Bowen property for two flows related to geodesic flows on surfaces with sufficient hyperbolicity. Our first result is the uniqueness of equilibrium states for Hölder continuous potentials and the geometric potential for products of geodesic flows of rank...
The BV Laplacian Δ, first introduced by Batalin and Vilkovisky, is a second-order differential operator that appears in the quantum master equation for quantizing gauge theories. The geometric framework for the BV formalism was later recognized by Schwarz as the setting of odd symplectic geometry and Khudaverdian showed that Δ...
In this thesis, we study the homological mirror symmetry for the Gross-Siebert program of local mirror symmetry. We construct a pair of mirror objects by lifting a tropical curve in the integral tropical manifold of the Gross-Siebert program. Furthermore, we evaluate the central charges of the mirror objects and show...
In this thesis we study the geometric limits under degree growth of Julia sets and filled Julia sets for complex polynomials with a unique critical point at $z = 0$. Specifically, for $c \in \mathbb{S}^1$, we are interested in the limit of the associated sequence of Julia sets $J(f_{n,c})$ in...
We prove the Rigidity Conjecture of Goette and Igusa, which states that, after rationalizing, there are no stable exotic smoothings of manifold bundles with closed even dimensional fibers. The key ingredients of the proof are fiberwise Poincaré–Hopf theorems generalizing earlier such results about the Becker–Gottlieb transfer. These theorems show how...
In this dissertation, we study and prove formulas of Hodge ideals for three types of isolated hypersurface singularities, including quasihomogeneous singularities, Newton non-degenerate singularities and analytically irreducible plane curve singularities.
For quasihomogeneous singularities, we provide
a formula of Hodge ideals of $D=\alpha Z$ which is a Q-divisor.
A direct consequence...
We study Hermitian metrics with constant Chern scalar curvature on a compact complex manifold. In the first part of the thesis, we prove a priori estimates for metrics of constant Chern scalar curvature on a compact complex manifold conditional on an upper bound on the entropy, extending a recent result...
The topic of this dissertation is the coherent-constructible correspondence (CCC). CCC is a version of homological mirror symmetry for toric varieties. It equates the derived category of coherent sheaves on a toric variety and the category of constructible sheaves on a torus whose singular support lies in the specific conical...
We study plurisubharmonic functions and their applications to K\"ahler geometry. We begin by studying regularity of envelopes of plurisubharmonic functions, particularly when the reference form is degenerate. This is then applied to show regularity of geodesic of K\"ahler metrics on singular varieties, as well as regularity of certain geodesic rays....
In this thesis, we study applications of the theory of perverse sheaves and their enhancements to problems in birational geometry. In the first application, we give positive results towards a conjecture of Batyrev about the nonnegativity of stringy Hodge numbers. In particular, we prove the nonnegativity of $(p,1)$-stringy Hodge numbers...
The variation of entropy in a family of dynamical systems is a natural indication of the bifurcations that the family undergoes. In the context of one-dimensional dynamics, Milnor's monotonicity of entropy conjecture (now a theorem of Bruin and van Strien) asserts that for polynomial interval maps with real critical points...
In this dissertation we study the connections over principal bundles in dimension four with bounded Yang-Mills energy, and present a new result on the existence a global Coulomb gauge with estimate in optimal space. To be precise, let $A$ be a $W^{1,2}$-connection on a principal $\text{SU}(2)$-bundle $P$ over a smooth...
We study the symmetry groups with respect to various equivalence relations defined on subshifts, and more generally, on Cantor systems. Two basic notions of equivalence for dynamical systems are conjugacy and flow equivalence. In this dissertation, we focus on the well-studied automorphism group, which is the group of self-conjugacies, and...
This thesis provides an introduction to decay rates for the damped wave equation on compact manifolds. It also gives a proof of a sharp decay rate for solutions to the damped wave equation on the torus with damping of a particular polynomial form. Finally it gives a proof of a...
In this thesis, we study the geometry of planar shapes and their harmonic caps. Specifically, given a compact continuum $P$, we are interested in constructing a planar cap $\hat P$ such that $P$ and $\hat P$ can be glued together along their boundary to form a topological sphere with prescribed...
We study modular forms, Jacobi forms, and hermitian formal Fourier-Jacobi series over imaginary quadratic fields. In the first section, we prove that the ring of classical Jacobi forms of a fixed genus g, varying index m and weight k is generated by theta functions. From this result we show that...
Arising from the study of multiple ergodic averages, nilsequences and multiple correlation sequences lie at the crossroads of ergodic theory, combinatorics and number theory. We study these types of sequences along various subsequences of integers, and provide applications to ergodic theory and harmonic analysis. Our first result involves multiple correlation...
We address the problem of efficient maintenance of the answer to a new type of query: Continuous Maximizing Range-Sum (Co-MaxRS) for moving objects trajectories. The traditional static/spatial MaxRS problem finds a location for placing the centroid of a given (axes-parallel) rectangle $R$ so that the sum of the weights of...
In this paper, we study the basic locus in the fiber at $p$ of a certain unitary Shimura variety with a certain parahoric level structure. The basic locus $\widehat{\CM^{ss}}$ is uniformized by a formal scheme $\CN$ which is called Rapoport-Zink space. We show that the irreducible components of the induced...
The Picard group is an important invariant of the $K(n)$-local category. If the prime $p$ is relatively large compared to the height $n$, the Picard group of the $K(n)$-local category is purely algebraic. In \cref{chapter:finitetype}, we describe the necessary and sufficient numerical condition when an element $X$ in the Picard...
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...
Let X,Y be algebraic varieties defined over the reals. Assume Y is smooth and X is Gorenstein. Suppose f:X -> Y is a flat R-morphism such that all the fibers have rational singularities. We show that the pushforward of any smooth, compactly supported measure on X has a continuous density...
We compare two different methods to compute the mod 2 homology of an infinite loop space. One method is to approximate the infinite loop functor using functor calculus. The other is to approximate the spectrum using an Adams resolution. We show that these two ways lead to isomorphic spectral sequences....
