Schramm-Loewner evolution (SLE(kappa)) is an important contemporary tool for identifying critical scaling limits of two-dimensional statistical systems. The SLE(kappa) one-parameter family of processes can be viewed as a special case of a more general, two-parameter family of processes we denote SLE(kappa, mu). The SLE(kappa, mu) process is defined for kappa>0 and real numbers mu; it represents the solution of the chordal Loewner equations under special conditions on the driving function parameter which require that it is a Brownian motion with drift mu and variance kappa. We derive properties of this process by use of methods applied to SLE(kappa) and application of Girsanov's Theorem. In contrast to SLE(kappa), we identify stationary asymptotic behavior of SLE(kappa, mu). For kappa in (0,4] and mu > 0, we present a pathwise construction of a process with stationary temporal increments and stationary imaginary component and relate it to the limiting behavior of the SLE(kappa, mu) generating curve. Our main result is a spatial invariance property of this process achieved by defining a top-crossing probability for points in the upper half plane with respect to the generating curve.

The polar lobe of Ilyanassa is asymmetrically partitioned into the D lineage of cells. Two of these cells, 3D and 4d, induce proper axial cell fate patterning in the embryo. Based on known embryological data in Ilyanassa, I hypothesized that Notch signaling would be required for this induction. I found that Notch signaling is required for cell fates induced by 4d and is temporally required well after 4d induction. Based on these results, I hypothesize that Notch signaling is involved in a reciprocal induction between the micromeres and the macromeres (endoderm) resulting in the maintenance of micromere fate induction and endoderm specification.Loss of the polar lobe results in the loss of cell fate induction by 3D/4d. Therefore, I hypothesized that proteins are asymmetrically bequeathed to the inducing D lineage cells by the polar lobe. To test this hypothesis, I compared global protein differences between two cell stage intact embryos, lobeless embryos, and isolated polar lobes by 2-Dimensional Electrophoresis analysis. I found several (12) quantitative differences between these samples including four spots enriched in the polar lobe isolates. I identified voltage-dependent anion-selective channel (VDAC) as one of the candidate proteins enriched in polar lobe isolates. I propose that VDAC is asymmetrically distributed by the polar lobe to the D cell and that it may function in D cell induction and mesendoderm fate specification.Lastly, I identify an acetylated tubulin antigen as a marker for cilia. I describe the pattern of cilia differentiation in the developing larvae that results in the formation of two ciliary bands, the prototroch and the metatroch, required for locomotion and feeding. These ciliary bands are conserved among annelid and mollusc larvae. Interestingly, the metatroch is derived from third quartet derivatives in the annelid Polygordius and from second quartet derivatives in the mollusc Crepidula. I provide evidence that the metatroch is derived from the first quartet derivatives in the mollusc Ilyanassa. Thus while the larval metatroch is conserved, its clonal origin is not. Based on these results, I provide support for the hypothesis that the metatroch is not homologous between annelids and molluscs or even among molluscs.

The evolution of multicellularity is an example of an evolutionary transition in individuality, in which a group of lower-level biological units (cells, in this case) emerges as a higher-level unit (the multicellular organism) with its own fitness, heritability and individuality. The volvocine green algae are a model system for the transition to multicellularity and for the evolution of cellular differentiation. Some of the developmental changes that collectively make up this transition have occurred more than once in the volvocine lineage; others have reverted from derived to ancestral states. The transition from cells to multicellular organisms began over 200 million years ago in this lineage, and the subsequent changes have been sporadic, with several important changes occurring early in the transition and some body plans remaining largely unchanged over long evolutionary time scales. Two suites of characters that differ among species within the genus Volvox have each evolved convergently or in parallel in lineages that diverged at least 175 million years ago. This complex history suggests that other origins of multicellularity may have involved important roles for cooperation, conflict and conflict mediation; parallel evolution of some traits; sporadic rather than constant change; and long-term coexistence of forms with different levels of complexity. Data from one species, Pleodorina starrii, support motility as a major selective pressure driving the the origins of cellular differentiation. Optimization of the proportion of soma in this species appears to be prevented by a constraint that prevents independent change in colonies with different numbers of cells. Finally, P. starrii presents an exceptionally high level of phenotypic variability, suggesting that the genotype-phenotype map has not completely shifted from the cell to the colony and that the transition to a new, higher-level individual in this species is incomplete.

It has become increasingly clear that a complete picture of galaxy evolution requires a better understanding of the role of Active Galactic Nuclei (AGN). In particular, they could be responsible for regulating star formation and galaxy growth via feedback processes. There are also competing views about the main modes of stellar growth and supermassive black hole growth in galaxies that need to be resolved. With high infrared luminosities (thus star formation rates) and a frequent occurrence of AGN, galaxies selected in the far-infrared wavebands form an ideal sample to search for a connection between AGN and star formation. The first part of this thesis contains a detailed analysis of the molecular gas properties of nearby infrared luminous and ultraluminous infrared galaxies (LIRGs and ULIRGs). We find that the enhanced molecular gas density in the most IR-luminous systems can be explained by major galaxy mergers, and that AGN are more likely to reside in higher-density systems. While the frequent concurrence of AGN and galaxy mergers in ULIRGs was already established, this work provides a coherent framework that explains trends observed with five molecular gas tracers with a broad range of critical densities, and a comparison with simulations that reproduce observed molecular line ratios without invoking AGN-induced chemistry. The second part of the thesis presents an analysis of the AGN content of intermediate redshift galaxies (0.3<z<1). However, identifying complete AGN samples at these redshift is challenging because it is difficult to find X-ray weak or absorbed AGN. To alleviate this problem, we developed the Mass-Excitation (MEx) diagram, which is applicable out to redshift of 1 with existing optical spectra. It improves the overall AGN census by detecting AGN that are missed in even the most sensitive X-ray surveys. The new diagnostic was used to study the concurrence of star formation and AGN in 70 micron-selected galaxies from the Far-Infrared Deep Extragalactic Legacy survey. When multiple AGN diagnostics are combined, we find not only a high AGN fraction in FIR-selected galaxies (as high as for nearby FIR-selected galaxies), but a high incidence of X-ray absorbed AGN. These findings may have considerable implications for current views about the main mode of AGN growth.

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