CAREER: Computation, Combinatorics, and Reality in Algebraic Geometry, with Applications

Texas A&M Research Foundation, College Station TX

Investigators

Abstract

Sottile will work on projects ranging from research in algebraic geometry and algebraic combinatorics, to strengthening links between algebraic geometry and applied science, to educational outreach. His work in algebraic geometry will focus on concrete aspects of flag manifolds and their Schubert varieties, involving a range of problems from the real Schubert calculus, from extensions of quantum Schubert calculus, from combinatorial problems in Schubert calculus, and from the geometry of Schubert varieties. This will deepen our understanding of computation, combinatorics, and real-number phenomena in algebraic geometry. Sottile will study structural properties of certain combinatorial Hopf algebras, work that has applications to polytopes, to the Schubert calculus, and to certain combinatorial schemes in physics. He is actively engaged in applications of computational algebraic geometry to other fields,particularly discrete geometry and the applied sciences. A unifying theme in this scientific work is studying concrete mathematical objects using computer experimentation to inform theoretical work. While algebraic geometry is concerned with abstract questions about solutions to equations, its value to other disciplines is largely through concrete objects and computational tools. A major activity of this project is to help disseminate the modern computational tools from algebraic geometry to other fields, including the applied sciences, through research collaboration, organizing conferences, and co-authoring a textbook on algebraic geometry for applied scientists. Sottile will also organize an outreach program for talented high school students in Western Massachusetts with the particular goal of serving the needs of students in nearby urban areas, many of whom come from groups traditionally underrepresented in science. The intention of this work: the research, the technology transfer, the service and the outreach, is to provide a firm foundation for a lifetime of integrated research and educational activities for Sottile.