Time-varying primary magnetic fields generated outside Earth by the magnetospheric ring current induce electrical currents in Earth's interior, which give rise to secondary magnetic fields with a complementary geometry. Geomagnetic depth sounding involves the analysis of magnetic field data to compute frequency-dependent response functions which yield information about the electrical conductivity of Earth's interior. I explore methods and results of forward-modeling global electromagnetic induction under a variety of assumptions about Earth conductivity and the spatial structure of the primary field. I begin by developing computational tools to perform time- and frequency-domain simulations of global induction in models with arbitrary conductivity and primary field structure using FlexPDE, a general-purpose software package that employs the finite-element method to solve partial differential equations. The method is shown to produce solutions with better than 1% accuracy when the simulated fields and response functions are compared to analytic solutions for a variety of problems in electromagnetic induction, and to qualitatively reproduce fields and response functions measured by satellites and observatories. The technique is employed in combination with analytic methods to explore the effect on the response of Earth models to primary fields with asymmetric structure. Standard methods of producing response functions from scalar and vector magnetic data are compared, and scalar methods are found to generate responses with significantly greater spatial bias for models with non-zonal fields. I develop the mathematical formulation for including Earth-rotation in the forward models, and use it to calculate frequency-dependent estimates of the amount of non-zonal structure required to produce previously reported local-time bias in empirical satellite response functions. Because it is difficult to validate solutions to induction problems that lack analytic solutions, we participate in an ongoing project with other researchers who simulate the global induction problem with different methods. We compare the synthetic fields calculated with the FlexPDE method to those calculated with the integral equation method and with the time-domain spectral method for a variety of conductivity models

Galaxies evolve under the influence of gas flows between their interstellar medium and their surrounding gaseous halos known as the circumgalactic medium (CGM). The CGM is a major reservoir of galactic baryons and metals, and plays a key role in the long cycles of accretion, feedback, and recycling of gas that drive star formation. In order to fully understand the physical processes at work within galaxies, it is therefore essential to have a firm understanding of the composition, structure, kinematics, thermodynamics, and evolution of the CGM. In this white paper we outline connections between the CGM and galactic star formation histories, internal kinematics, chemical evolution, quenching, satellite evolution, dark matter halo occupation, and the reionization of the larger-scale intergalactic medium in light of the advances that will be made on these topics in the 2020s. We argue that, in the next decade, fundamental progress on all of these major issues depends critically on improved empirical characterization and theoretical understanding of the CGM. In particular, we discuss how future advances in spatially-resolved CGM observations at high spectral resolution, broader characterization of the CGM across galaxy mass and redshift, and expected breakthroughs in cosmological hydrodynamic simulations will help resolve these major problems in galaxy evolution.