Real and Combinatorial Algebraic Geometry

这个项目研究实数和有效代数几何的相关领域，特别是计数问题，以及由计数代数几何产生的组合问题。一个焦点是舒伯特微积分产生的经典几何问题的实数解，这项工作是由大量的计算机实验提供的。这些有效的计算技术依赖于舒伯特簇、Bruhat序和旗帜簇的上同调环的组合学，索蒂尔也对此进行了研究。这个项目的一个重要部分是与Jaochim Rosenthal和Alex Wang共同撰写一本关于代数几何的应用驱动用途的书《应用代数几何》，为应用科学家提供了一本有用且有动力的代数几何入门。代数几何研究多项式方程组的解，对其潜在的应用非常重要--多项式方程在数学和应用科学中普遍存在。代数几何的计算和实数方面特别有趣，因为应用程序通常需要明确的实数答案来回答数学问题。获得这种明确答案的技术利用了特定问题的特殊组合结构，因此依赖于对这些结构的良好理解。这个项目通过研究代数几何的计算和真实方面，通过加深对相关组合结构的理解，以及通过Sottile与Rosenthal和Wang为应用科学家编写一本由应用程序驱动的代数几何教材，增加了代数几何的适用性。

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.