Algebraic Cycles, Hodge Theory, and Arithmetic

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卓有成效的相互作用。最近的发现表明理论物理与代数k理论之间存在新的交叉受精，深化了与拓扑k理论的现有联系。为了明确这一新的联系，P.I.提出构造Calabi-Yau变种的低次代数k群的环族，并研究它们在退化和同调镜像对称下的行为。项目的这一部分将应用于Gromov-Witten不变量的算法。在此基础上，研究了周期子域Calabi-Yau三倍参数化的分类，确定了周期子域的Kato-Usui边界分量和自同构上同调。这些结果不仅与霍奇理论和物理学有关，而且与自同构形式和朗兰兹纲领有关。此外，它们还将涉及培育新的卡拉比-丘品种。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。著名的猜想包括霍奇的猜想（这是七个克莱千禧年问题之一）预言，这些计算虽然是非代数的先验，但却被称为代数群和代数循环的结构牢牢地控制着。但后一种结构远比这些猜想所暗示的更为普遍和有用；私家侦探将研究它们在数论和物理相关问题中的应用。例如，在弦理论（意在统一量子力学和广义相对论）提出的时空模型中，6个迄今未观测到的实维由卡拉比-丘空间来解释。这些空间由对偶联系在一起，这保留了物理理论，同时完全改变了数学。将代数循环及其推广到这些对偶中，将完全解释瞬时数的渐近性。这是由P.I.共同组织的跨学科BIRS研讨会“Hodge理论和弦对偶性”（2011年12月）的一部分，该项目的结果将通过此类会议、暑期学校、期刊文章和网站传播。该资助为华盛顿大学带来的项目顾问将有助于研究氛围，并且与项目相关的专业问题将适合培养研究生。

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.