On the Mumford-Tate conjecture for Abelian varieties with reduction conditions

In this thesis we study Galois representations corresponding to abelian varieties with certain reduction conditions. We show that these conditions force the image of the representations to be "big," so that the Mumford-Tate conjecture (:= MT) holds. We also prove that the set of abelian varieties satisfying these conditions is dense in a corresponding moduli space.

The main results of the thesis are the following two theorems.

Theorem A: Let A be an absolutely simple abelian variety, End° (A) = k : imaginary quadratic field, g = dim(A). Assume either dim(A) ≤ 4, or A has bad reduction at some prime ϕ, with the dimension of the toric part of the reduction equal to 2r, and gcd(r,g) = 1, and (r,g) ≠ (15,56) or (m -1, m(m+1)/2). Then MT holds.

Theorem B: Let M be the moduli space of abelian varieties with fixed polarization, level structure and a k-action. It is defined over a number field F. The subset of M(Q) corresponding to absolutely simple abelian varieties with a prescribed stable reduction at a large enough prime ϕ of F is dense in M(C) in the complex topology. In particular, the set of simple abelian varieties having bad reductions with fixed dimension of the toric parts is dense.

Besides this we also established the following results:

(1) MT holds for some other classes of abelian varieties with similar reduction conditions. For example, if A is an abelian variety with End° (A) = Q and the dimension of the toric part of its reduction is prime to dim( A), then MT holds.

(2) MT holds for Ribet-type abelian varieties.

(3) The Hodge and the Tate conjectures are equivalent for abelian 4-folds.

(4) MT holds for abelian 4-folds of type II, III, IV (Theorem 5.0(2)) and some 4-folds of type I.

(5) For some abelian varieties either MT or the Hodge conjecture holds.

Manganese: Minerals, Microbes, and the Evolution of Oxygenic Photosynthesis

Oxygenic photosynthesis fundamentally transformed our planet by releasing molecular oxygen and altering major biogeochemical cycles, and this exceptional metabolism relies on a redox-active cubane cluster of four manganese atoms. Not only is manganese essential for producing oxygen, but manganese is also only oxidized by oxygen and oxygen-derived species. Thus the history of manganese oxidation provides a valuable perspective on our planet’s environmental past, the ancient availability of oxygen, and the evolution of oxygenic photosynthesis. Broadly, the general trends of the geologic record of manganese deposition is a chronicle of ancient manganese oxidation: manganese is introduced into the fluid Earth as Mn(II) and it will remain only a trace component in sedimentary rocks until it is oxidized, forming Mn(III,IV) insoluble precipitates that are concentrated in the rock record. Because these manganese oxides are highly favorable electron acceptors, they often undergo reduction in sediments through anaerobic respiration and abiotic reaction pathways.

The following dissertation presents five chapters investigating manganese cycling both by examining ancient examples of manganese enrichments in the geologic record and exploring the mineralogical products of various pathways of manganese oxide reduction that may occur in sediments. The first chapter explores the mineralogical record of manganese and reports abundant manganese reduction recorded in six representative manganese-enriched sedimentary sequences. This is followed by a second chapter that further analyzes the earliest significant manganese deposit 2.4 billon years ago, and determines that it predated the origin of oxygenic photosynthesis and thus is supporting evidence for manganese-oxidizing photosynthesis as an evolutionary precursor prior to oxygenic photosynthesis. The lack of oxygen during this early manganese deposition was partially established using oxygen-sensitive detrital grains, and so a third chapter delves into what these grains mean for oxygen constraints using a mathematical model. The fourth chapter returns to processes affecting manganese post-deposition, and explores the relationships between manganese mineral products and (bio)geochemical reduction processes to understand how various manganese minerals can reveal ancient environmental conditions and biological metabolisms. Finally, a fifth chapter considers whether manganese can be mobilized and enriched in sedimentary rocks and determines that manganese was concentrated secondarily in a 2.5 billion-year-old example from South Africa. Overall, this thesis demonstrates how microbial processes, namely photosynthesis and metal oxide-reducing metabolisms, are linked to and recorded in the rich complexity of the manganese mineralogical record.

The interactions of basic proteins and DNA

I. ELECTROPHORESIS OF THE NUCLEIC ACIDS

A zone electrophoresis apparatus using ultraviolet optics has been constructed to study nucleic acids at concentrations less than 0.004%. Native DNA has a mobility about 15% higher than denatured DNA over a range of conditions. Otherwise, the electrophoretic mobility is independent of molecular weight, base composition or source. DNA mobilities change in the expected way with pH but the fractional change in mobility is less than the calculated change in charge. A small decrease in mobility accompanies an increase in ionic strength. RNA’s from various sources have mobilities slightly lower than denatured DNA except for s-RNA which travels slightly faster. The important considerations governing the mobility of nucleic acids appear to be the nature of the hydrodynamic segment, and the binding of counterions. The differences between electrophoresis and sedimentation stem from the fact that all random coil polyelectrolytes are fundamentally free draining in electrophoresis.

II. THE CYTOCHROME C/DNA COMPLEX

The basic protein, cytochrome c, has been complexed to DNA. Up to a cytochrome:DNA mass ratio of 2, a single type of complex is formed. Dissociation of this complex occurs between 0.05F and 0.1F NaCl. The complexing of cytochrome to DNA causes a slight increase in the melting temperature of the DNA, and a reduction of the electrophoretic mobility proportional to the decrease in net charge. Above a cytochrome:DNA mass ratio of 2.5, a different type of complex is formed. The results suggest that complexes such as are formed in the Kleinschmidt technique of electron microscopy would not exist in bulk solution and are exclusively film phenomena.

III. STUDIES OF THE ELECTROPHORESIS AND MELTING BEHAVIOUR OF NUCLEOHISTONES

Electrophoresis studies on reconstituted nucleohistones indicate that the electrophoretic mobility for these complexes is a function of the net charge of the complex. The mobility is therefore dependent on the charge density of the histone complexing the DNA, as well as on the histone/DNA ratio. It is found that the different histones affect the transition from native to denatured DNA in different ways. It appears that histone I is exchanging quite rapidly between DNA molecules in 0.01 F salt, while histone II is irreversibly bound. Histone III-IV enhances the capacity of non-strand separated denatured DNA to reanneal. Studies on native nucleoproteins indicate that there are no gene-sized uncomplexed DNA regions in any preparations studied.

IV. THE DISSOCIATION OF HISTONE FROM CALF THYMUS CROMATIN

Calf thymus nucleoprotein was treated with varying concentrations of NaCl. The identity of the histones associated and dissociated from the DNA at each salt concentration was determined by gel electrophoresis. It was found that there is no appreciable histone dissociation below 0.4 F NaCl. The lysine rich histones dissociate between 0.4 and 0.5 F NaCl. Their dissociation is accompanies by a marked increase in the solubility of the chromatin. The moderately lysine rich histones dissociate mainly between 0.8 and 1.1 F NaCl. There are two arginine rich histone components: the first dissociates between 0.8 F and 1.1 F NaCl, but the second class is the very last to be dissociated from the DNA (dissociation beginning at 1.0 F NaCl). By 2.0 F NaCl, essentially all the histones are dissociated.

The properties of the extracted nucleoprotein were studied. The electrophoretic mobility increases and the melting temperature decreases as more histones are dissociated from the DNA. A comparison with the dissociation of histones from DNA in NaClO₄ shows that to dissociate the same class of histones, the concentration of NaCl required is twice that of NaClO₄.

I. Foehn winds of southern California. II. Foehn wind cyclo-genesis. III. Weather conditions associated with the Akron disaster. IV. The Los Angeles storm of December 30, 1933 to January 1, 1934

I. Foehn winds of southern California.
An investigation of the hot, dry and dust laden winds occurring in the late fall and early winter in the Los Angeles Basin and attributed in the past to the influences of the desert regions to the north revealed that these currents were of a foehn nature. Their properties were found to be entirely due to dynamical heating produced in the descent from the high level areas in the interior to the lower Los Angeles Basin. Any dust associated with the phenomenon was found to be acquired from the Los Angeles area rather than transported from the desert. It was found that the frequency of occurrence of a mild type foehn of this nature during this season was sufficient to warrant its classification as a winter monsoon. This results from the topography of the Los Angeles region which allows an easy entrance to the air from the interior by virtue of the low level mountain passes north of the area. This monsoon provides the mild winter climate of southern California since temperatures associated with the foehn currents are far higher than those experienced when maritime air from the adjacent Pacific Ocean occupies the region.

II. Foehn wind cyclo-genesis.
Intense anticyclones frequently build up over the high level regions of the Great Basin and Columbia Plateau which lie between the Sierra Nevada and Cascade Mountains to the west and the Rocky Mountains to the east. The outflow from these anticyclones produce extensive foehns east of the Rockies in the comparatively low level areas of the middle west and the Canadian provinces of Alberta and Saskatchewan. Normally at this season of the year very cold polar continental air masses are present over this territory and with the occurrence of these foehns marked discontinuity surfaces arise between the warm foehn current, which is obliged to slide over a colder mass, and the Pc air to the east. Cyclones are easily produced from this phenomenon and take the form of unstable waves which propagate along the discontinuity surface between the two dissimilar masses. A continual series of such cyclones was found to occur as long as the Great Basin anticyclone is maintained with undiminished intensity.

III. Weather conditions associated with the Akron disaster.
This situation illustrates the speedy development and propagation of young disturbances in the eastern United States during the spring of the year under the influence of the conditionally unstable tropical maritime air masses which characterise the region. It also furnishes an excellent example of the superiority of air mass and frontal methods of weather prediction for aircraft operation over the older methods based upon pressure distribution.

IV. The Los Angeles storm of December 30, 1933 to January 1, 1934.
This discussion points out some of the fundamental interactions occurring between air masses of the North Pacific Ocean in connection with Pacific Coast storms and the value of topographic and aerological considerations in predicting them. Estimates of rainfall intensity and duration from analyses of this type may be made and would prove very valuable in the Los Angeles area in connection with flood control problems.

Heteroepitaxy of Group IV and Group III-V semiconductor alloys for photovoltaic applications

Photovoltaic energy conversion represents a economically viable technology for realizing collection of the largest energy resource known to the Earth -- the sun. Energy conversion efficiency is the most leveraging factor in the price of energy derived from this process. This thesis focuses on two routes for high efficiency, low cost devices: first, to use Group IV semiconductor alloy wire array bottom cells and epitaxially grown Group III-V compound semiconductor alloy top cells in a tandem configuration, and second, GaP growth on planar Si for heterojunction and tandem cell applications.

Metal catalyzed vapor-liquid-solid grown microwire arrays are an intriguing alternative for wafer-free Si and SiGe materials which can be removed as flexible membranes. Selected area Cu-catalyzed vapor-liquid solid growth of SiGe microwires is achieved using chlorosilane and chlorogermane precursors. The composition can be tuned up to 12% Ge with a simultaneous decrease in the growth rate from 7 to 1 μm/min^-1. Significant changes to the morphology were observed, including tapering and faceting on the sidewalls and along the lengths of the wires. Characterization of axial and radial cross sections with transmission electron microscopy revealed no evidence of defects at facet corners and edges, and the tapering is shown to be due to in-situ removal of catalyst material during growth. X-ray diffraction and transmission electron microscopy reveal a Ge-rich crystal at the tip of the wires, strongly suggesting that the Ge incorporation is limited by the crystallization rate.

Tandem Ga_1-xIn_xP/Si microwire array solar cells are a route towards a high efficiency, low cost, flexible, wafer-free solar technology. Realizing tandem Group III-V compound semiconductor/Si wire array devices requires optimization of materials growth and device performance. GaP and Ga_1-xIn_xP layers were grown heteroepitaxially with metalorganic chemical vapor deposition on Si microwire array substrates. The layer morphology and crystalline quality have been studied with scanning electron microscopy and transmission electron microscopy, and they provide a baseline for the growth and characterization of a full device stack. Ultimately, the complexity of the substrates and the prevalence of defects resulted in material without detectable photoluminescence, unsuitable for optoelectronic applications.

Coupled full-field optical and device physics simulations of a Ga_0.51In_0.49P/Si wire array tandem are used to predict device performance. A 500 nm thick, highly doped "buffer" layer between the bottom cell and tunnel junction is assumed to harbor a high density of lattice mismatch and heteroepitaxial defects. Under simulated AM1.5G illumination, the device structure explored in this work has a simulated efficiency of 23.84% with realistic top cell SRH lifetimes and surface recombination velocities. The relative insensitivity to surface recombination is likely due to optical generation further away from the free surfaces and interfaces of the device structure.

Finally, GaP has been grown free of antiphase domains on Si (112) oriented substrates using metalorganic chemical vapor deposition. Low temperature pulsed nucleation is followed by high temperature continuous growth, yielding smooth, specular thin films. Atomic force microscopy topography mapping showed very smooth surfaces (4-6 Å RMS roughness) with small depressions in the surface. Thin films (~ 50 nm) were pseudomorphic, as confirmed by high resolution x-ray diffraction reciprocal space mapping, and 200 nm thick films showed full relaxation. Transmission electron microscopy showed no evidence of antiphase domain formation, but there is a population of microtwin and stacking fault defects.

Earth-Abundant Zinc-IV-Nitride Semiconductors

This investigation is motivated by the need for new visible frequency direct bandgap semiconductor materials that are abundant and low-cost to meet the increasing demand for optoelectronic devices in applications such as solid state lighting and solar energy conversion. Proposed here is the utilization of zinc-IV-nitride materials, where group IV elements include silicon, germanium, and tin, as earth-abundant alternatives to the more common III-nitrides in optoelectronic devices. These compound semiconductors were synthesized under optimized conditions using reactive radio frequency magnetron sputter deposition. Single phase ZnSnN₂, having limited experimental accounts in literature, is validated by identification of the wurtzite-derived crystalline structure predicted by theory through X-ray and electron diffraction studies. With the addition of germanium, bandgap tunability of ZnSn_xGe_1-xN₂ alloys is demonstrated without observation of phase separation, giving these materials a distinct advantage over In_xGa_1-xN alloys. The accessible bandgaps range from 1.8 to 3.1 eV, which spans the majority of the visible spectrum. Electron densities, measured using the Hall effect, were found to be as high as 10²² cm⁻³ and indicate that the compounds are unintentionally degenerately doped. Given these high carrier concentrations, a Burstein-Moss shift is likely affecting the optical bandgap measurements. The discoveries made in this thesis suggest that with some improvements in material quality, zinc-IV-nitrides have the potential to enable cost-effective and scalable optoelectronic devices.