Algebraic Cycles, Hodge Theory, and Arithmetic

代数圈、霍奇理论和算术

• 批准号: 1068974
• 负责人: Matthew Kerr
• 金额: $ 12.74万
• 依托单位: Washington University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1068974&HistoricalAwards=false
• 关键词: Algebraic; Cycles; Hodge; Theory; Arithmetic

The last two decades have seen a highly productive interaction between Hodge theory, symplectic geometry, and string theory in the mirror symmetry program. Recent discoveries suggest a new cross-fertilization between theoretical physics and algebraic K-theory, deepening the existing connection with topological K-theory. To make this new connection explicit, the P.I. proposes to construct families of cycles in low-degree algebraic K-groups of Calabi-Yau varieties, and to study their behavior under degeneration and homological mirror symmetry. This part of the project will have applications to the arithmetic of Gromov-Witten invariants. Underlying Hodge-theoretic problems which will be addressed include the classification of period subdomains parametrizing Calabi-Yau threefolds, and the determination of their Kato-Usui boundary components and automorphic cohomology. These results will be relevant not only to Hodge theory and physics but also for automorphic forms and the Langlands program. They will, in addition, involve the construction of new Calabi-Yau varieties.Hodge theory seeks to describe the influence of integrals and differential equations on the shape of an algebraic space. Famous conjectures including that of Hodge (which is one of the seven Clay Millenium Problems) predict that these computations, while non-algebraic a priori, are firmly governed by structures called algebraic groups and algebraic cycles. But these latter structures are far more ubiquitous, and useful, than the conjectures suggest; the P.I. will investigate their application to related problems in number theory and physics. For example, in the model of spacetime proposed by string theory (which purports to unify quantum mechanics and general relativity), 6 as-of-yet unobserved real dimensions are accounted for by Calabi-Yau spaces. These spaces are linked by dualities which preserve the physical theory while completely altering the mathematics. Incorporating algebraic cycles and their generalizations into these dualities will completely explain observed asymptotics of instanton numbers. This is part of the thrust of an interdisciplinary BIRS workshop on "Hodge Theory and String Duality" (Dec. 2011) co-organized by the P.I. Results from this project will be disseminated through such conferences, summer schools, journal articles and websites. The project consultants brought to Washington University by the grant will contribute to the research atmosphere, and there are specialized problems related to the project which will be suitable for training graduate students.

在过去的二十年里，霍奇理论、辛几何和弦理论在镜像对称程序中进行了高效的相互作用。 最近的发现表明理论物理学和代数 K 理论之间存在新的交叉融合，加深了与拓扑 K 理论的现有联系。 为了使这种新的联系变得明确，P.I.提出在 Calabi-Yau 簇的低次代数 K 群中构造周期族，并研究它们在退化和同调镜像对称下的行为。 该项目的这一部分将应用于 Gromov-Witten 不变量的算术。将解决的潜在霍奇理论问题包括参数化 Calabi-Yau 三重的周期子域的分类，以及它们的 Kato-Usui 边界分量和自守上同调的确定。 这些结果不仅与霍奇理论和物理学相关，而且与自同构形式和朗兰兹纲领相关。 此外，它们还将涉及新的 Calabi-Yau 簇的构造。霍奇理论试图描述积分和微分方程对代数空间形状的影响。 包括霍奇猜想（七个克莱千年问题之一）在内的著名猜想预测，这些计算虽然是非代数先验的，但却严格受到称为代数群和代数循环的结构的控制。 但后面这些结构比猜想所暗示的更加普遍和有用。 P.I.将研究它们在数论和物理学相关问题中的应用。 例如，在弦理论（旨在统一量子力学和广义相对论）提出的时空模型中，卡拉比-丘空间解释了 6 个尚未观测到的真实维度。 这些空间通过二元性联系在一起，既保留了物理理论，又完全改变了数学。 将代数循环及其推广纳入这些对偶性将完全解释观察到的瞬子数的渐近性。 这是 BIRS 跨学科研讨会“霍奇理论和弦对偶性”（2011 年 12 月）的一部分，该研讨会由 P.I. 共同组织。 该项目的成果将通过此类会议、暑期学校、期刊文章和网站传播。 通过赠款来到华盛顿大学的项目顾问将有助于营造研究氛围，并且存在与该项目相关的专门问题，适合培养研究生。