This project aims to determine the relationship between galaxies and the hot gas surrounding them. The portion of the intergalactic medium (IGM) that is photo-ionized and heated to temperatures of order 10,000 K outweighs the baryonic (i.e., ordinary) matter inside galaxies by roughly a factor of 4. This gas can be studied in relatively nearby galaxies by observing the hydrogen Lyman-alpha absorption line in the spectra of more distant active galactic nuclei (AGN) near the same line of sight. The Principal Investigator and collaborators have shown that about half of the Lyman-alpha absorbers at redshifts less than 0.017 are located within about 400 kiloparsecs of a moderate-sized or larger galaxy. The team will continue and expand upon this work, in four parts: (1) improve the statistics by a factor of order 10 using archival AGN spectra obtained with the Cosmic Origins Spectrograph on the Hubble Space Telescope; (2) identify all bright galaxies near all observed AGN lines of sight, using public archival data and new observations with ground-based telescopes; (3) improve the quality of the existing estimates of galaxy distances, luminosities, classifications, and orientations, using published data and new data when appropriate; and (4) perform an in-depth analysis of the physical conditions in individual absorbers. The goal is to determine the properties of the absorbers as a function of distance from the galaxy, separately for galaxies of different types, sizes, luminosities and environments. This project will train a graduate student to make use of space-based ultraviolet and ground-based optical and radio observational facilities, some involving international collaborations. Some parts of the project will be made appropriate for research projects for undergraduate students, including those participating in the formal Research Experience for Undergraduates (REU) program at the performing institution. An exhibit illustrating the existence of matter outside galaxies will be created for the Space Place museum at the University of Wisconsin.