The luminosity function of galaxies, the measurement of the space density as a function of luminosity, is an important test of cosmology, galaxy formation and evolution. Unfortunately, there is a factor of two variation in recent measurements of the luminosity function. Most of this variation is due to systematic errors, caused by various selection effects. With two large new surveys, the Two degree Field Galaxy Redshift Survey and the Sloan Digital Sky Survey, underway it is important to recognise and eliminate these selection effects if we are going to improve our measurement of the luminosity function and fully utilise these surveys. By measuring the space density of galaxies as a function of surface brightness as well as luminosity, a bivariate brightness distribution, we can comprehend many of the selection effects such as light loss, incompleteness and the visibility of galaxies. Since galaxies have a variety of shapes and sizes, a distribution in luminosity and surface brightness helps to separate out different types of galaxy. Correlations between the luminosity and surface brightness place extra constraints on models of galaxy formation and evolution. When we analyse our results, we find that recent surveys that have not taken into account surface brightness selection effects underestimate the luminosity of the bright end by 5-10%. Using the bivariate brightness distribution, we can constrain the luminosity density to a range that varies by < 20% rather than by a factor of 2. We find that the luminosity function is flat over the range -19.5 < M < -17 and then rises sharply as late-type spiral galaxies begin to dominate. The space density does not vary with surface brightness with the result that low surface brightness galaxies are at least as common as normal galaxies. However, low surface brightness galaxies are also intrinsically faint, following the luminosity-surface brightness correlation for spirals, so they do not contribute significantly to the luminosity density.

The formation of stars from interstellar gas is the cornerstone of galaxy evolution. This thesis represents work undertaken in order to characterise the role of cool interstellar gas, and its relation to star formation, in galaxy evolution across cosmic time. In particular, it concentrates on star forming galaxies at the extremes of the galaxy assembly spectrum - extremely faint dwarfs, and extremely luminous starbursts - in an attempt to test the limits of galaxy evolution models. The thesis falls into two complimentary halves, addressing topics in the low redshift and high redshift Universe respectively. In the low redshift Universe, I discuss multi-wavelength studies of large samples of z rv O galaxies, which include extremely faint dwarf galaxies in the Local Volume. Using these samples, it is possible to derive a multitude of physical parameters (including star formation rates, stellar masses, and gas masses) which allow the interrelationship between star formation and gas content to be assessed in a statistically significant manner. In particular, modern wide field surveys (combined with deep, volume-limited data) allow trends to be analysed across many orders of magnitude in galaxy mass and star formation rate, shedding light on the global properties of galaxies in the local Universe. Moving to higher redshift, I discuss targeted observations of molecular gas in extreme star forming galaxies in the early Universe. These 'sub-millimetre' galaxies number amongst the most luminous objects ever discovered, and molecular gas observations have the power to uncover many of their physical properties, including their morphologies, kinematics, and star formation behaviour. I begin by presenting high-resolution observations of a small number of these galaxies at z rv 2, and discussing the implications for galaxy evolution studies. The final chapter of this thesis consists of the results of a survey for molecular gas in sub-millimetre galaxies conducted over the last decade, which represents the largest single study of molecular gas in the early Universe to date.

Lenticular and elliptical galaxies, collectively referred to as "early-type galaxies" (ETGs), are commonly thought to represent the end-points of galaxy evolution. Lying in the red sequence of galaxies, these objects are defined by their mostly old stellar populations and by their "red and dead" appearance in optical observations. Much progress in understanding these objects has been made with integral-field spectroscopy in recent years, with results repeatedly pointing to a link between early-type galaxies and high-redshift spiral galaxies. However, the exact nature of this link remains unclear, with a wide variety of evolution scenarios likely required to fully explain the range of observed early-type galaxy properties. In my study, I analysed observations of twelve early-type galaxies taken with the Mitchell Integral-Field Spectrograph at McDonald Observatory, Texas. These galaxies have previously been found to contain detectable quantities of neutral hydrogen gas, with ten out of the twelve displaying large-scale hydrogen disks. I extracted line-of-sight kinematics of the stellar and ionised gas components of these galaxies, and I used various modelling approaches to constrain their stellar population parameters as well as their three-dimensional mass structure in terms of both dark and visible components. An important feature of this study is the wide field of view of the spectroscopic observations, which reach beyond two half-light radii for almost all of the sample; this remains rare for integral-field unit (IFU) studies of ETGs, and so sets this study apart from most earlier works. The gas-rich nature of the sample is likewise novel. I find all aspects of my analysis to yield a consistent view of these galaxies' evolution, in which one or more gaseous interaction events served to shape them into their observed forms. I find these galaxies to contain low dark matter fractions on average within the inner half-light radius, and I also find mass modelling to favour near-isothermal total density profiles over much of the sample.

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A spectroscopic analysis of 10 intergalactic star-forming objects (ISFOs) and a photometric analysis of 67 ISFOs in a sample of 14 interacting systems is presented. The majority of the ISFOs have relative polycyclic aromatic hydrocarbon (PAH) band strengths similar to those of nearby spiral and starburst galaxies. In contrast to what is observed in blue compact dwarfs (BCDs) and local giant H II regions in the Milky Way (NGC 3603) and the Magellanic Clouds (30 Doradus and N 66), the relative PAH band strengths in ISFOs correspond to models with a significant PAH ion fraction (<50%) and bright emission from large PAHs (∼100 carbon atoms). The [Ne III]/[Ne II] and [S IV]/[S III] line flux ratios indicate moderate levels of excitation with an interstellar radiation field that is harder than the majority of the Spitzer Infrared Nearby Galaxies Survey and starburst galaxies, but softer than BCDs and local giant H II regions. The ISFO neon line flux ratios are consistent with a burst of star formation ∼6 million years ago. Most of the ISFOs have ∼106 M ⊙ of warm H2 with a likely origin in photo-dissociation regions (PDRs). Infrared Array Camera photometry shows the ISFOs to be bright at 8 μm, with one-third having [4.5]-[8.0] > 3.7, i.e., enhanced non-stellar emission, most likely due to PAHs, relative to normal spirals, dwarf irregulars, and BCD galaxies. The relative strength of the 8 μm emission compared to that at 3.6 μm or 24 μm separates ISFOs from dwarf galaxies in Spitzer two-color diagrams. The infrared power in two-thirds of the ISFOs is dominated by emission from grains in a diffuse interstellar medium. One in six ISFOs have significant emission from PDRs, contributing ∼30%-60% of the total power. ISFOs are young knots of intense star formation.

The presented study gives a comprehensive overview of the theory and the evidence for a systematically varying stellar initial mass function (IMF). Then we focus on the impact of this paradigm change, that is, from the universal invariant IMF to a variable IMF, on galaxy chemical evolution (GCE) studies. For this aim, we developed the first GCE code, GalIMF, that is able to incorporate the empirically calibrated environment-dependent IMF variation theory, the integrated galactic initial mass function (IGIMF) theory. In this theory, the galaxy-wide IMF is calculated by summing all the IMFs in all embedded star clusters which formed throughout the galaxy in 10 Myr time epochs. The GalIMF code recalculates the galaxy-wide IMF at each time step because the integrated galaxy- wide IMF depends on the galactic star formation rate and metallicity. The resulting galaxy-wide IMF and metal abundance evolve with time. Using this code, we examine the chemical evolution of early-type galaxies (ETGs) from dwarf to the most massive. We find that the introduction of the non-canonical IMF affects the best estimation of the galaxy properties such as their mass, star formation history, and star formation efficiency. Moreover, we are able to provide an independent estimation on the stellar formation timescale of galaxies, the...