CAREER: Holomorphic Curves in Algebraic Geometry and Symplectic Topology

摘要：20世纪80年代，伪全纯曲线技术的发明彻底改变了辛拓扑，深刻地影响了代数几何，并与弦理论建立了惊人的联系。这个项目有三个不同的方向，由不同领域之间的相互作用所驱动。首先是对任意和一般复杂结构的变形曲线进行了详细的研究。它旨在基本理解伪全纯曲线的性质，包括Gromov-Witten不变量（这种曲线的计数）和它们行为的更多定性方面（例如存在性，唯一性等）。另一个方向的目标是将PI在弦理论预测中对五次三倍的1 gw不变量的证明技术应用于许多其他设置，测试额外的镜像预测并超越它们。第三个方向是进一步发展pi的局部超交方法及其在经典枚举几何中的应用。弦理论是一种物理模型，它通过振动弦来表示基本粒子，目的是统一自然界的四种基本力。当这样的弦在空间中移动时，它们会扫过黎曼曲面，也被称为全纯曲线。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可实验验证的预测。然而，它已经产生了大量的数学预测，并导致了辛拓扑和代数几何的基本发展，特别是在（伪）全纯曲线方面。这项提议旨在进一步从数学上测试弦理论，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。本提案中的部分项目将由研究生在PI的指导下进行。该奖项由几何分析、代数、数论和组合学项目联合资助。

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.