Algebraic Maps: Hodge Theory and Algebraic Cycles

代数图：霍奇理论和代数圈

• 批准号: 0501020
• 负责人: Mark Andrea de Cataldo
• 金额: $ 10.5万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-01 至 2008-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0501020&HistoricalAwards=false
• 关键词: Algebraic; Maps; Hodge; Theory; Cycles

The investigator works on the topology of complex algebraic maps, with applications to algebraic cycles. The study of the relations between the topology of the domain and target of a map is interesting and has broad applications in the fields of geometry and topology. This project aims at1) Furthering the study and the understanding of the Hodgeand Chow-theoretic properties of complex algebraic maps and at2) Explicitly studying these properties in significant cases.The term ``algebraic maps" above is a technical term for polynomial equations. Algebraic geometry is the discipline devoted to their study. It is an ancient subject rooted in the early achievements of humanity, like the wheel, the Egyptians' elliptical flowers arrangements and Archimedes' burning parabolic mirrors. Circles, ellipses and parabolas arise from the polynomial equations we study in high school. They are both beautiful and ubiquitous in nature as they describe many natural phenomena, from the motion of planets to the shape of leaves and flowers, to the behavior of microscopic particles. The funded project is strongly inclined towards pure research and proposes to study the deeper properties of the solutions to more complicated algebraic equations. As it has always been the case, pure and applied mathematics will influence each other and new abstract ideas will fuel the progress of applications.

调查工作的拓扑结构复杂的代数地图，与应用代数循环。定义域的拓扑与映射目标的拓扑之间的关系是一个有趣的研究课题，在几何学和拓扑学领域有着广泛的应用。本项目的目标是：1）进一步研究和理解复代数映射的HodgeandChow理论性质; 2）在重要情况下探索这些性质。代数几何是专门研究它们的学科。这是一个古老的主题，植根于人类早期的成就，就像车轮，埃及人的椭圆形插花和阿基米德的燃烧抛物面镜。圆、椭圆和抛物线都是从我们在高中学习的多项式方程中产生的。它们在自然界中既美丽又无处不在，因为它们描述了许多自然现象，从行星的运动到树叶和花朵的形状，再到微观粒子的行为。该资助项目强烈倾向于纯研究，并建议研究更复杂的代数方程的解的更深层次的性质。正如以往一样，纯数学和应用数学将相互影响，新的抽象思想将推动应用的进步。