Algebraic Maps: Topology and Algebraic Cycles

代数图：拓扑和代数圈

• 批准号: 0202321
• 负责人: Mark Andrea de Cataldo
• 金额: $ 9.9万
• 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: 2002
• 资助国家: 美国
• 起止时间: 2002-06-15 至 2005-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0202321&HistoricalAwards=false
• 关键词: Algebraic; Maps; Topology; Cycles

The investigator and his collaborator work onthe topology of complex algebraic maps, with applicationsto algebraic cyclesand to non-algebraic maps.The study of the relations between the topologyof the domain and target of a mapis interesting and has broad applicationsin the fields of geometry and topology.The investigator works on:1) a topological and Hodge-theoretic proof of the so-called Decomposition Theorem for algebraic maps, 2) finding the topological obstructions to the validity of the theorem for other maps, 3) computing intersection forms arising from maps in significant cases, and 4) applications of the DecompositionTheorem to algebraic cycles.Several new methods are being introduced as a result of these studies.The term ``algebraic maps" above is a technical termfor 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 beautifuland 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 researchand proposes to study the deeper propertiesof the solutions to more complicated algebraic equations.As it has always been the case, pure and applied mathematicswill influence each other and new abstract ideas will fuel the progress of applications.

研究者和他的合作者研究复杂代数映射的拓扑，并应用于代数环和非代数映射。域拓扑和映射目标之间关系的研究很有趣，并且在几何和拓扑领域具有广泛的应用。研究者致力于：1）所谓的分解定理的拓扑和霍奇理论证明 代数映射，2）找到其他映射定理有效性的拓扑障碍，3）计算重要情况下映射产生的交集形式，以及4）分解定理在代数循环中的应用。这些研究的结果正在引入几种新方法。上面的术语“代数映射”是多项式方程的技术术语。 代数几何是专门研究他们的学科。它是一门古老的学科，植根于人类早期的成就，就像轮子、埃及人的椭圆花排列和阿基米德燃烧的抛物面镜一样。圆、椭圆和抛物线源于我们在高中学习的多项式方程。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 从行星的运动到叶子和花朵的形状，再到微观粒子的行为。该资助项目强烈倾向于纯研究，并提出研究更复杂的代数方程解的更深层次性质。一如既往，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。