Modeling the impact of large-scale solid mass transfers on the gravity field

The focus of this thesis is to explore and quantify the response of large-scale solid mass transfer events on satellite-based gravity observations. The gravity signature of large-scale solid mass transfers has not been deeply explored yet; mainly due to the lack of significant events during dedicated satellite gravity missions‘ lifespans. In light of the next generation of gravity missions, the feasibility of employing satellite gravity observations to detect submarine and surface mass transfers is of importance for geoscience (improves the understanding of geodynamic processes) and for geodesy (improves the understanding of the dynamic gravity field). The aim of this thesis is twofold and focuses on assessing the feasibility of using satellite gravity observations for detecting large-scale solid mass transfers and on modeling the impact on the gravity field caused by these events. A methodology that employs 3D forward modeling simulations and 2D wavelet multiresolution analysis is suggested to estimate the impact of solid mass transfers on satellite gravity observations. The gravity signature of various submarine and subaerial events that occurred in the past was estimated. Case studies were conducted to assess the sensitivity and resolvability required in order to observe gravity differences caused by solid mass transfers. Simulation studies were also employed in order to assess the expected contribution of the Next Generation of Gravity Missions for this application.

Irreducibility of Random Hilbert Schemes

We prove that a random Hilbert scheme that parametrizes the closed subschemes with a fixed Hilbert polynomial in some projective space is irreducible and nonsingular with probability greater than $0.5$. To consider the set of nonempty Hilbert schemes as a probability space, we transform this set into a disjoint union of infinite binary trees, reinterpreting Macaulay's classification of admissible Hilbert polynomials. Choosing discrete probability distributions with infinite support on the trees establishes our notion of random Hilbert schemes. To bound the probability that random Hilbert schemes are irreducible and nonsingular, we show that at least half of the vertices in the binary trees correspond to Hilbert schemes with unique Borel-fixed points.

Optical properties of chiral thin films and microparticles

In this work I study the optical properties of helical particles and chiral sculptured thin films, using computational modeling (discrete dipole approximation, Berreman calculus), and experimental techniques (glancing angle deposition, ellipsometry, scatterometry, and non-linear optical measurements). The first part of this work focuses on linear optics, namely light scattering from helical microparticles. I study the influence of structural parameters and orientation on the optical properties of particles: circular dichroism (CD) and optical rotation (OR), and show that as a consequence of random orientation, CD and OR can have the opposite sign, compared to that of the oriented particle, potentially resulting in ambiguity of measurement interpretation. Additionally, particles in random orientation scatter light with circular and elliptical polarization states, which implies that in order to study multiple scattering from randomly oriented chiral particles, the polarization state of light cannot be disregarded. To perform experiments and attempt to produce particles, a newly constructed multi stage thin film coating chamber is calibrated. It enables the simultaneous fabrication of multiple sculptured thin film coatings, each with different structure. With it I successfully produce helical thin film coatings with Ti and TiO_{2}. The second part of this work focuses on non-linear optics, with special emphasis on second-harmonic generation. The scientific literature shows extensive experimental and theoretical work on second harmonic generation from chiral thin films. Such films are expected to always show this non-linear effect, due to their lack of inversion symmetry. However no experimental studies report non-linear response of chiral sculptured thin films. In this work I grow films suitable for a second harmonic generation experiment, and report the first measurements of non-linear response.