Equivariant and combinatorial techniques in algebraic and symplectic geometry

应用数学和纯数学中的许多问题都涉及到方程组的求解。代数几何是研究代数方程组的解空间的学科，因此是数学的核心领域。例如，费马大定理的证明给出了与费马方程的非平凡解有关的代数几何对象(椭圆曲线)的性质。此外，理论物理中的镜面对称理论在根本上使用了代数几何的语言。代数几何在量子计算、数学生物学、密码学以及图像和信号处理中也有应用。另一方面，辛几何是经典物理的数学公式。辛几何中的对称论和守恒定律与表象理论有联系，表象理论可以被认为是量子物理的数学框架，以及现代物理的其他领域，如流体力学。组合几何和凸几何包括对多面体的研究，多面体是平面几何中图形的推广，如三角形、梯形和平行四边形。多面体的凸几何在最优化理论等许多研究领域都具有重要意义。数学的这三个核心领域在许多方面是相关的。申请人建议详细研究这些丰富联系的几个例子：牛顿-奥孔科夫天体、环面变种及其堆叠类似物，以及戈雷斯基-柯特维茨-麦克弗森理论。申请者的研究计划有许多方面，这构成了对未来科学家的有效培训计划。这项研究计划的长期好处有两方面：第一，这项研究的结果将揭示许多新的组合技术，用于分析许多现实世界中出现的重要空间的等变几何(例如，镜面对称、密码学、流体力学、最优化理论)，第二，这项建议的培训方面将培养出高素质的本科生、研究生和研究生水平的人才，他们在重要的几何领域拥有具有竞争力的研究和技术技能。

Many problems in both applied and pure Mathematics involve the solution of a system of equations. Algebraic Geometry is precisely the study of the space of solutions to systems of algebraic equations, and is therefore a core area of Mathematics. For instance, the proof of Fermat's Last Theorem gives properties of an algebraic- geometric object (an elliptic curve) associated to a non-trivial solution of Fermat's equation. Also, the theory of Mirror Symmetry in theoretical physics uses the language of algebraic geometry in a fundamental way. Algebraic geometry also has applications in quantum computing, mathematical biology, cryptography, and image and signal processing. Symplectic geometry, on the other hand, is the mathematical formulation of classical physics. The theory of symmetries and conservation laws within symplectic geometry has connections with representation theory, which can be thought of as the mathematical framework of quantum physics, and other areas of modern physics such as fluid mechanics. Combinatorial and convex geometry includes the study of polytopes, which are generalizations of the figures in plane geometry such as triangles, trapezoids, and parallelograms. The convex geometry of polytopes is important in many research areas, such as optimization theory. These three core areas of mathematics are related in many ways. The applicant proposes to study in detail several instances of these rich connections: Newton-Okounkov bodies, toric varieties and their stack analogues, and Goresky-Kottwitz-MacPherson theory. There are many aspects of the applicant's research program which form an effective training program for future scientists. The long-term benefits of this research program are two-fold: first, the results of this research will bring to light many new combinatorial techniques for analyzing the equivariant geometry of important spaces which arise in many real-world applications (e.g. Mirror Symmetry, cryptography, fluid mechanics, optimization theory), and secondly, the training aspects of this proposal will produce highly qualified personnel at the undergraduate, graduate, and postgraduate level, who possess competitive research and technical skills in important areas of geometry.