Algebraic geometry is a deep and well-established field within pure mathematics that is increasingly finding applications outside of mathematics. Many applications flow from and contribute to the more computational and combinatorial aspects of algebraic geometry, and this often involves subtle real-number or positivity properties. This project will further the development of applications of algebraic geometry by supporting PI Sottile's work in applications of algebraic geometry and its application-friendly realms of real, combinatorial, and computational algebraic geometry. This include convex algebraic geometry, toric varieties in geometric modeling, quantum Schubert calculus in linear systems theory, tropical geometry, and continued investigation of the Shapiro conjecture. It will also further the growth of applications of algebraic geometry by supporting Sottile's activities as an officer within SIAM and organizer of scientific meetings, and by supporting Sottile's training and mentoring of graduate students, postdocs, and junior collaborators.
Algebraic geometry, which is concerned with geometric properties of solutions to algebraic equations, is giving rise to new tools for use in the applications of mathematics. This has been recognised by the Society for Industrial and Applied Mathematics through their creation of an activity group on algebraic geometry. This proposal will support the further development of these new tools for applications from algebraic geometry through the support of research, training, and organizational activities of PI Sottile.
. This report covers the final year of the project. The Broader Impacts of this project include training of future scientists by PI Sottile, who advised two graduate students, a postdoctoral fellow, and two undergraduate students in this period. It also supported his work organizing and running the Texas A&M Math circle, a mathematics enrichment activity that serves 40 students each week. This also supported Sottile's travel to NIgeria in Summer 2014 which is part of a long term project to engage mathematicians and students in Africa's largest country. The Intellectual Merits of the work in the final year of the project include completed research projects of the PI and his collaborators that was reported on in nine preprints covering topics in combinatorial algebraic geometry and work that is useful for applications. We mention four. One, A Combinatorial proof that Schubert vs. Schur coefficients are nonnegative, written with Assaf and Bergeron, solved a 20-year old problem in the combinatorics of the Schubert calculus. Another, A primal-dual formulation for certifiable computations in the Schubert calculus, with Hein and Hauenstein, gives novel formulations of a class of geometric problems as solutions to systems of equations with the same numbers of equations as variables---heretofore, the only known formulations had far more equations than variables, which is numerically unstable, and does not allow solutions to be certified correct. A third, Higher convexity of coamoeba complements, with Nisse establishes a new and subtle property of these objects which form the set of angles appearing in soutions to systems of polynomial equations. The fourth, Cohomological consequences of the pattern map, was with Dr. H. Praise Adeyemo of Ibadan University in NIgeria. This was written during Sottile's trip to Nigeria, and is part of his project to help develop mathematicians in Nigeria.
