Combinatorial and Real Algebraic Geometry

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Algebraic geometry, which is the mathematical study of solutions to systems of polynomial equations, is a core mathematical discipline noted for its theoretical depth and its interactions with other areas of mathematics. Algebraic geometry is also a tool for applications, since physical objects can be described by polynomial equations, and relations between concepts in science and engineering may be modeled by polynomials. Whatever their source, once polynomials enter the picture, the theoretical base, trove of classical examples, and modern computational tools of algebraic geometry may be brought to bear on the problem at hand. This research project aims to strengthen the role of algebraic geometry as a tool for applications, in two ways. First, a major focus will be on developing the interface between algebraic geometry and applications. Second is work on combinatorial aspects of algebraic geometry, for this develops tools and ideas to better understand objects in algebraic geometry with special structure, and these highly structured objects are those that appear most commonly in the interactions between algebraic geometry and other fields, both within mathematics and in the applied sciences. This project will also involve training of students and postdocs, the investigator's outreach activities at the local level through a mathematics circle, and his continued interaction with mathematics in Nigeria. Oftentimes, objects from algebraic geometry that arise in other parts of mathematics or science have strong combinatorial structures (e.g., toric varieties or Grassmannians) or else the application demands real solutions. Consequently, the research in areas of combinatorial and real algebraic geometry in this project will serve both to advance our basic understanding of these topics and to help build a foundation for applications. This project involves research in three topics within algebraic geometry: real toric varieties, tropical geometry, and Schubert calculus. In each of these areas the investigator will work with collaborators and students on projects ranging from developing foundations to understanding key examples to work inspired by problems from applications. Of particular note are the goals of developing a rich and robust theory of irrational toric varieties, understanding the geometry and topology of complements of tropical objects, establishing positivity in type C Schubert calculus via a useful theory of shifted dual equivalence, and understanding Galois groups in the Schubert calculus. This award is jointly funded by the Algebra and Number Theory and Combinatorics programs.