Applications and Combinatorics in Algebraic Geometry

代数几何是纯数学中一个深厚而成熟的领域，它越来越多地在数学之外找到应用。 许多应用程序流和有助于更多的计算和组合方面的代数几何，这往往涉及微妙的实数或积极的性质。 该项目将进一步发展的应用代数几何支持PI Sottile的工作中的应用代数几何和它的应用友好领域的真实的，组合和计算代数几何。这包括凸代数几何，复曲面品种的几何建模，量子舒伯特微积分的线性系统理论，热带几何，并继续调查的夏皮罗猜想。 它也将进一步增长的应用代数几何支持Sottile的活动作为一个官员在SIAM和组织者的科学会议，并通过支持Sottile的培训和指导研究生，博士后，和初级合作者。代数几何，这是关注的几何性质的解决方案，代数方程，正在产生新的工具，用于数学的应用。 这已经认识到社会的工业和应用数学通过他们的创造活动组代数几何。该提案将通过PI Sottile的研究、培训和组织活动的支持，支持进一步开发这些用于代数几何应用的新工具。

Algebraic geometry is a deep and well-established field within pure mathematics that is increasingly finding applications outside of mathematics. Many applications flow from and contribute to the more computational and combinatorial aspects of algebraic geometry, and this often involves subtle real-number or positivity properties. This project will further the development of applications of algebraic geometry by supporting PI Sottile's work in applications of algebraic geometry and its application-friendly realms of real, combinatorial, and computational algebraic geometry. This include convex algebraic geometry, toric varieties in geometric modeling, quantum Schubert calculus in linear systems theory, tropical geometry, and continued investigation of the Shapiro conjecture. It will also further the growth of applications of algebraic geometry by supporting Sottile's activities as an officer within SIAM and organizer of scientific meetings, and by supporting Sottile's training and mentoring of graduate students, postdocs, and junior collaborators.Algebraic geometry, which is concerned with geometric properties of solutions to algebraic equations, is giving rise to new tools for use in the applications of mathematics. This has been recognised by the Society for Industrial and Applied Mathematics through their creation of an activity group on algebraic geometry. This proposal will support the further development of these new tools for applications from algebraic geometry through the support of research, training, and organizational activities of PI Sottile.