Equivariant and combinatorial techniques in algebraic and symplectic geometry

代数和辛几何中的等变和组合技术

• 批准号: 326749-2012
• 负责人: Harada, Megumi
• 金额: $ 1.82万
• 依托单位: McMaster University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2014
• 资助国家: 加拿大
• 起止时间: 2014-01-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=548889
• 关键词: Equivariant; combinatorial; techniques; algebraic; symplectic

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.

应用数学和纯数学中的许多问题都涉及到方程组的解。代数几何正是对代数方程组解的空间的研究，因此是数学的一个核心领域。例如，费马大定理的证明给出了与费马方程的非平凡解相关的代数几何对象（椭圆曲线）的性质。此外，理论物理中的镜像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、图像和信号处理方面也有应用。另一方面，辛几何是经典物理学的数学表述。辛几何中的对称理论和守恒定律与表征理论有联系，表征理论可以被认为是量子物理学的数学框架，以及现代物理学的其他领域，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概括，如三角形、梯形和平行四边形。多面体的凸几何在优化理论等许多研究领域具有重要意义。这三个数学的核心领域在很多方面都是相关的。申请人建议详细研究这些丰富联系的几个实例：牛顿-奥昆科夫体、环面变异体及其堆栈类似物，以及Goresky-Kottwitz-MacPherson理论。申请人的研究计划有许多方面，这些方面构成了对未来科学家的有效培训计划。这个研究项目的长期效益是双重的：首先，这项研究的结果将揭示许多新的组合技术来分析重要空间的等变几何，这些技术在许多现实世界的应用中出现（例如镜像对称，密码学，流体力学，优化理论），其次，该提案的培训方面将培养本科生，研究生和研究生水平的高素质人才。在几何学的重要领域具有竞争力的研究和技术技能。