This project supports research in the field of mathematics, focusing on a problem known as the d-bar-Neumann problem. This problem is central to understanding how shapes and structures behave in several complex variables, an area of math with important connections to physics and engineering. The project will develop new tools and methods to solve long-standing questions and will also strengthen research and education at the University of Texas Rio Grande Valley (UTRGV), a Hispanic-Serving Institution. Through workshops, courses, and community engagement, the work will support the training of the next generation of researchers. The research will explore the relationship between the d-bar-Neumann problem and the geometry of complex domains. One of the main goals is to find geometric conditions that ensure solutions to the d-bar-Neumann problem are smooth up to the boundary, which has been a major open question in the field. The investigator will use advanced mathematical tools from homological algebra and microlocal analysis to study this question. The work builds on recent progress made by the principal investigator and will seek to clarify the role of boundary geometry in solving this problem. The results are expected to improve understanding in complex analysis and geometry and may lead to progress on other major problems involving regularity and geometric structure. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation u