Applications and Combinatorics in Algebraic Geometry

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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.