Algebraic Cycles, Hodge Theory, and Arithmetic

在过去的二十年里，霍奇理论、辛几何和弦论在镜像对称计划中进行了卓有成效的互动。 最近的发现表明理论物理和代数K理论之间的新的交叉施肥，深化了与拓扑K理论的现有联系。 为了使这种新的连接明确，PI。提出在Calabi-Yau簇的低次代数K-群中构造圈族，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。基本的霍奇理论的问题，将得到解决，包括分类的周期子域parametrizing卡拉比-丘三倍，并确定其加藤-Kazi边界组件和自守上同调。 这些结果不仅与霍奇理论和物理学有关，而且与自守形式和朗兰兹纲领有关。 他们将，此外，涉及建设新的卡-丘品种。霍奇理论试图描述的影响，积分和微分方程的形状的代数空间。 包括霍奇（Hodge）在内的著名理论（这是七个粘土千禧年问题之一）预测，这些计算虽然不是代数的先验，但被称为代数群和代数圈的结构所牢牢控制。 但是，这些后一种结构远比插图所暗示的更普遍，更有用; 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-丘空间解释了6个尚未观测到的真实的维度。 这些空间通过对偶性联系在一起，这些对偶性保留了物理理论，同时完全改变了数学。 将代数循环及其推广到这些对偶中将完全解释所观察到的瞬子数的渐近性。 这是一个跨学科的BIRS研讨会“霍奇理论与弦对偶”（2011年12月）的一部分，由P.I. 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。 赠款带来的项目顾问到华盛顿大学将有助于研究气氛，有专门的问题，有关的项目将适合培养研究生。

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.