This project studies the related areas of real and effective algebraic geometry, particularly enumerative problems, and combinatorial problems arising from enumerative algebraic geometry. A focus is the real number solutions to classical geometric problems arising from the Schubert calculus, work that is informed by substantial computer experimentation. These effective and computational techniques rely upon the combinatorics of Schubert varieties, Bruhat orders, and the cohomology rings of flag varieties, which Sottile also studies. An important part of this project is to write a book, "Applicable Algebraic Geometry" jointly with Jaochim Rosenthal and Alex Wang, on application-driven uses of algebraic geometry, providing a useful and motivated introduction to algebraic geometry for applied scientists.
Algebraic geometry, which is the study of solutions to systems of polynomial equations, is important for its potential applications-- polynomial equations are ubiquitous in mathematics and the applied sciences. Computational and real aspects of algebraic geometry are of particular interest, as applications often demand explicit, real-number answers to mathematical questions. Techniques to obtain such explicit answers often exploit special combinatorial structures of particular problems, and hence rely upon a good understanding of these structures. This project increases the applicability of algebraic geometry by studying computational and real aspects of algebraic geometry, by deepening the understanding of related combinatorial structures, and lastly by Sottile writing an application-driven algebraic geometry text for applied scientists with Rosenthal and Wang.
