CAREER: Holomorphic Curves in Algebraic Geometry and Symplectic Topology

职业：代数几何和辛拓扑中的全纯曲线

• 批准号: 0846978
• 负责人: Aleksey Zinger
• 金额: $ 44.31万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-15 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0846978&HistoricalAwards=false
• 关键词: CAREER; Holomorphic; Curves; Algebraic; Geometry

AbstractAward: DMS-0846978Principal Investigator: Aleksey ZingerThe invention of pseudo-holomorphic curves techniques in the1980s revolutionized symplectic topology, profoundly impactedalgebraic geometry, and led to astounding connections with stringtheory. This project has three distinct directions, motivated bythis interplay between different fields. The primary one is adetailed study of deformations of such curves, for arbitrary aswell as generic complex structures. It aims at the fundamentalunderstanding of properties of pseudo-holomorphic curves,including Gromov-Witten invariants (counts of such curves) andmore qualititative aspects of their behavior (e.g. existence,uniruledness, etc.). The aim of another direction is to applytechniques employed in the PI's proof of the string theoryprediction for the genus 1 GW-invariants of a quintic threefoldin many other setting, testing additional mirror predictions andgoing beyond them. The third direction would further develop thePI's local excess intersection approach and its applications toclassical enumerative geometry.String theory is a physical model that represents elementaryparticles by vibrating strings with the aim of unifying the fourfundamental forces of nature. As such srings move in space, theysweep out Riemann surfaces, also called holomorphic curves. Whilestring theory is one of the main paradigms in physics today, ithas yet to make any experimentally testable predictions. However,it has generated plenty of mathematical predictions and led tofundamental developments in symplectic topology and algebraicgeometry, especially in relation to (pseudo-) holomorphiccurves. This proposal aims to further test string theorymathematically, while deepening the mathematical understanding ofsuch curves with an eye toward applications to more classicalproblems in geometry. Some of the projects in this proposal willpursued by graduate students under the PI's supervision.This award is jointly supported by the programs in Geometric Analysis and in Algebra, Number Theory, and Combinatorics.

AbstractAward：DMS-0846978首席研究员：Aleksey Zinger 20世纪80年代伪全纯曲线技术的发明彻底改变了辛拓扑，深刻影响了代数几何，并导致了与弦理论的惊人联系。 该项目有三个不同的方向，其动机是不同领域之间的相互作用。第一部分详细研究了任意复杂结构和一般复杂结构的曲线变形。它旨在对伪全纯曲线性质的基本理解，包括Gromov-Witten不变量（此类曲线的计数）和它们行为的更多定性方面（例如，存在性，单向性等）。 另一个方向的目的是applytechniques采用在PI的证明弦理论预测的亏格1 GW-不变量的五次threefolds在许多其他设置，测试额外的镜像预测和超越他们。第三个方向将进一步发展PI的局部过剩相交方法及其在经典枚举几何中的应用。弦理论是一种通过振动弦来表示基本粒子的物理模型，其目的是统一自然界的四种基本力。当这样的曲线在空间中运动时，它们会产生黎曼曲面，也称为全纯曲线。虽然弦理论是当今物理学的主要范式之一，但它还没有做出任何实验上可检验的预言。然而，它产生了大量的数学预言，并导致辛拓扑和代数几何的基本发展，特别是在（伪）全纯曲线。这个建议旨在进一步测试弦理论的数学，同时加深对这些曲线的数学理解，着眼于应用于更经典的几何问题。该项目中的一些项目将由PI监督下的研究生进行。该奖项由几何分析和代数，数论和组合学项目共同支持。