This award is funded in part under the American Rescue Plan Act of 2021 (Public Law 117-2). The project serves the national interest, as stated by NSF's mission, by promoting the progress of science, and laying foundations that will help advance the national prosperity and welfare; In particular, the award will enable research on compact binary systems in general relativity and modified gravity by combining theory and multi-messenger observations with gravitational waves. The work will advance our understanding of nuclear physics at densities and temperatures that arise in the deep interiors of colliding neutron stars, which cannot be probed by experiments on Earth. New tools will be developed for understanding gravity through a synergy of gravitational wave observations by LIGO, and the anticipated electromagnetic signals that will be detected by ground and space-based telescopes observing from the radio to the hard gamma-ray bands. The group will develop new codes for modeling binary neutron stars and binary black hole-neutron stars with the aid of supercomputers. Such theoretical work is crucial for the successful interpretation of and the extraction of fundamental physics from existing and future observations of compact binary systems. The award will support the training, education and further career development of students in STEM fields, including minorities and women. The award will enable the development of the Computation and Mathematics Program for incoming Arizona Science Students (COMPASS) initiative, which will bridge and smooth the high school-to-college transition for students, and facilitate the success and retention of underrepresented minorities in Physics, Astronomy and other scientific fields. The specific projects have a two-fold goal: a) advance our understanding of strong field, relativistic gravity, nuclear physics and theoretical astrophysics; b) facilitate the detection and physical interpretation of gravitational waves and their possible electromagnetic counterparts. A substantial part of the effort will focus on the investigation of the inspiral and merger of compact binaries (binary neutron stars, binary black hole-neutron stars) and their gravitational wave and electromagnetic signatures. Methods will be developed for utilizing these observable signatures to extract information about fundamental physics, in particular about the nature of gravity and nuclear physics above the nuclear saturation density. More specifically, codes that run on supercomputers will be developed to solve the equations of a scalar-tensor theory of gravity with future application to binary black hole-neutron stars in order to develop techniques that test whether there are any scalar-field induced deviations from Einstein gravity that can be triggered by either charge (dark, electric or magnetic) or by magnetic fields, and to test the equivalence principle in this regime for the first time. Using existing codes, methods will be developed for probing the unknown thermal part of the equation of state at nuclear densities by using the gravitational wave and electromagnetic signatures from binary neutron star mergers. The proposed studies are extremely timely in light of the recent detection of gravitational waves by the LIGO and Virgo collaborations not only as generated by binary black holes, but also as generated by compact binaries with at least one neutron star. The project will benefit the interpretation and detection of gravitational waves and their electromagnetic counterparts not only by current and upcoming gravitational wave detectors and telescopes, but also by future ones. The proposed research will be accomplished via large-scale supercomputer simulations accompanied by analytic modeling. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.