Applications and Combinatorics in Algebraic Geometry

Algebraic Geometry is a deep and well-established field within pure mathematics that is increasingly finding applications outside of mathematics. These applications in turn are the source of new questions and challenges for the subject. Many applications flow from and contribute to the more combinatorial and computational parts of algebraic geometry, and this often involves real-number or positivity questions. The scientific development of this area devoted to applications of algebraic geometry is facilitated by the sociological development of administrative structures and meetings, and by the development of human resources through the training and education of younger researchers. One goal of this project is to deepen the dialog between algebraic geometry and its applications. This will be accomplished by supporting the research of Sottile in applications of algebraic geometry and in its application-friendly areas of combinatorial and computational algebraic geometry. It will be accomplished in a completely different way by supporting Sottile’s activities as an officer within SIAM and as an organizer of scientific meetings. Yet a third way to accomplish this goal will be through Sottile’s training and mentoring of graduate students, postdocs, and junior collaborators. The intellectual merits of this project include the development of applications of algebraic geometry and of combinatorial and combinatorial aspects of algebraic geometry. Specifically, Sottile will work to develop the theory and properties of orbitopes from the perspective of convex algebraic geometry, continue to investigate linear precision in geometric modeling, and apply the quantum Schubert calculus to linear systems theory. In combinatorial algebraic geometry, Sottile will work to clarify the foundations of tropical algebraic geometry, study equivariant cohomology of arithmetic toric varieties, and continue to investigate generalizations of the Shapiro conjecture for flag manifolds. The broader impacts of this project include Sottile’s training of graduate students, postdocs, and young researchers through his direct mentoring and web of collaboration (of 22 collaborators in active research related to this proposal, 9 are graduate students or postdocs). Other broader impacts include the wide dissemination of the research conducted by members of his research team at conferences and seminars where they will make presentations. A particularly important broader impact is the building of institutional infrastructure to support the applications of algebraic geometry—Sottile is the founding chair of the new SIAM Activity Group on Algebraic Geometry, and activities he carries out under this project will help to promote this activity group and make it relevant to the profession. This includes the organization of and Sottile’s attendance at some key conferences, his spending (but not organizing) the Winter/Spring term of 2011 at the Institut Mittag-Leffler for the program on applicable algebraic geometry, and the Winter term of 2013 at the MSRI program in Commutative Algebra (if that is approved).