Applications and Combinatorics in Algebraic Geometry

代数几何是纯数学中一个深厚而久负盛名的领域，它越来越多地在数学之外找到应用。许多应用程序来自于代数几何中计算性和组合性更强的方面，这通常涉及到微妙的实数或正性性质。该项目将通过支持Pi Sottile在代数几何及其实数、组合和计算代数几何的应用友好领域的工作，进一步推动代数几何应用的发展。这包括凸代数几何、几何建模中的环面变化、线性系统理论中的量子舒伯特微积分、热带几何以及对夏皮罗猜想的继续研究。它还将通过支持索蒂尔作为SIAM官员和科学会议组织者的活动，以及通过支持索蒂尔对研究生、博士后和初级合作者的培训和指导，进一步促进代数几何应用的增长。代数几何关注代数方程解的几何性质，正在为数学应用提供新的工具。工业和应用数学学会通过创建一个代数几何活动小组认识到了这一点。这项提议将通过对Pi Sottile的研究、培训和组织活动的支持，支持这些新工具的进一步开发，用于代数几何的应用。

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.