While funded on this grant, the PI worked in the Stanford University Department of Mathematics under the mentorship of Professor Ravi Vakil. During this time, the PI completed several research projects related to matrices of polynomials and he launched a project (now completed) about algebraic geometry over finite fields. Matrices of polynomials are at the heart of the study of free resolutions over the polynomial ring. In linear algebra, one studies matrices where the entries are numbers, and this theory has been extremely well developed from both a theoretical and computational perspective. By contrast, much less is known about the analogous questions for matrices of polynomials. Yet this an area of broad interest due to its connections with algebraic geometry, algebraic combinatorics, and computational algebra. One recent breakthrough in this area was the development of Boij-Soederberg theory, which provides a powerful structural tool for studying the behavior of free resolutions. The theory began in conjectures of Boij and Soederberg, but it was essentially launched by Eisenbud-Schreyer who proved these conjectures. Much of the PI's work during this fellowship helped extend and refine Boij-Soederberg theory. For instance, the PI wrote a paper with Berkesch, Kummini, and Sam that provides an analogue of the theory for regular local rings. In addition, the PI expanded on the central construction of pure resolutions, in a separate paper with Berkesch, Kummini, and Sam. In a different direction, the PI launched a research project on algebraic geometry over finite fields. This involved joint work with Melanie Matchett Wood. Finite fields are important for computational questions, as computational algebra algorithms generally run much faster over finite fields than they would over the rational numbers. The PI began work on a project that attempted to understand an analogue of a famous theorem from classical algebraic geometry, the Bertini Theorem, in the context of algebraic geometry over a finite field. The foundational result in this area is due to Poonen, and the PI and Matchett Wood began work on expanding Poonen's result to new situations. Beyond research, the PI engaged in a number of projects related to education, mentoring, and building scientific communitites. These include a consulting project on K-12 education with the Instructional Research Group; mentoring of an undergraduate through the Association for Women in Mathematics Mentoring Program; and co-organizing a 150 person conference, the Western Algebraic Geometry Seminar, which was held at Stanford University in Spring 20111.
