Real and Combinatorial Algebraic Geometry

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This project studies the related areas of real and effective algebraicgeometry, particularly enumerative problems, and combinatorial problemsarising from enumerative algebraic geometry. A focus is the real numbersolutions to classical geometric problems arising from the Schubertcalculus, work that is informed by substantial computer experimentation.These effective and computational techniques rely upon the combinatorics ofSchubert varieties, Bruhat orders, and the cohomology rings of flagvarieties, which Sottile also studies. An important part of thisproject is to write a book, "Applicable Algebraic Geometry" jointly withJaochim Rosenthal and Alex Wang, on application-driven uses of algebraicgeometry, providing a useful and motivated introduction to algebraicgeometry for applied scientists.Algebraic geometry, which is the study of solutions to systems ofpolynomial 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 tomathematical questions. Techniques to obtain such explicit answersoften 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 anapplication-driven algebraic geometry text for applied scientists with Rosenthal and Wang.