Convex Bodies, Algebraic Geometry, and Symplectic Geometry

这个项目涉及代数、几何和组合学的交叉问题。 要探索的主要领域是代数和辛几何。代数几何是数学中最古老和最基本的分支之一。它有着广泛的应用，从编码和数据安全到高能物理和弦理论。主要的研究对象是代数簇，在几个变量的代数方程组的解决方案，经典的例子是椭圆，抛物线和双曲线。辛几何是经典力学的现代框架。它在现代物理学量子理论中也起着重要作用。主要研究对象是辛流形，它是物理学中相空间概念的抽象。辛流形的一个重要类别是代数簇。 通过这种联系，该研究项目探索了代数几何和辛几何之间的相互作用，目的是促进对这两个领域的理解。该项目的一个中心主题是将凸体（称为Newton-Okounkov体）与代数簇相关联，编码关于簇的几何信息。它给出了一个一般的框架，扩大范围的凸几何方法从环面品种一般品种。该项目将探索这一领域在辛几何和其他方向的应用。该计划的基本内容包括：（1）构造一般射影簇上的Hamilton环面作用;（2）对一类射影簇的几何量子化中的“极化无关原理”的一般方法;（3）建立与热带几何和计算代数的联系;（4）从Newton-Okunkov体理论的角度探讨熵的概念。在这些应用中，该项目预计将通过几何量子化对数学物理和量子力学做出贡献。该项目还将调查与热带几何，一个分支代数几何连接到凸优化的连接。

This project concerns questions at the intersection of algebra, geometry, and combinatorics. The main areas to be explored are algebraic and symplectic geometry. Algebraic geometry is one of the oldest and most fundamental branches of mathematics. It has a wide range of applications from coding and data security to high energy physics and string theory. The principal objects of study are algebraic varieties, the sets of solutions of systems of algebraic equations in several variables; classical examples are ellipses, parabolas, and hyperbolas. Symplectic geometry is the modern framework of classical mechanics. It also plays an important role in modern quantum theories in physics. The main objects of study are symplectic manifolds, which are abstractions of the notion of phase space from physics. An important class of symplectic manifolds are algebraic varieties. Through this connection, the research project explores interactions between algebraic geometry and symplectic geometry, with the goal of advancing understanding in both areas.A central theme of this project is to associate convex bodies, known as the Newton-Okounkov bodies, to algebraic varieties, encoding information about the geometry of the variety. It gives a general framework to extend the scope of convex geometry methods from toric varieties to general varieties. The project will explore applications of this area in symplectic geometry and other directions. Some fundamental topics to be addressed in the project are: (1) constructing Hamiltonian torus actions on general projective varieties; (2) a general approach to the "principle of independence of polarization" in geometric quantization for a large class of projective varieties; (3) establishing connections with tropical geometry and computational algebra; and (4) investigating the notion of entropy from the point of view of the theory of Newton-Okunkov bodies. Among the applications, the project is expected to contribute to mathematical physics and quantum mechanics, via geometric quantization. The project will also investigate connections with tropical geometry, a branch of algebraic geometry connected to convex optimization.