Asymptotic Hodge Theory, Fibered Motives, and Algebraic Cycles

这个项目的重点是多项式方程及其解集，这已经被代数几何学者专门和深入地研究过。深层次的猜想，包括Bloch、Beilinson和Hodge的猜想，试图将解析对象（如积分和微分方程）的行为与这些解集的潜在代数结构和拓扑形状联系起来，这些分析对象是先验的非代数的。即使这些猜想在总体上仍然难以解决，个别情况下的解决方案都继续证明它们的有效性，并产生新的代数结构（例如“循环”和“动机”），这些结构有助于解决数学和其他科学中看似遥远的领域的问题，例如，在数论和物理学的界面（如弦论和量子场论）。最近的技术创新，基于在家庭中考虑多项式方程，已经开始提供这些猜想的新案例。它们的进一步发展和应用是本项目的主题，其结果将通过会议、暑期学校、期刊文章和网站传播。这项资助为华盛顿大学带来的项目顾问将有助于改善其研究氛围，项目中包含的专门问题将为项目负责人的研究生提供培训。霍奇理论不变量，如周期和调节器映射，提供了现代几何中代数和超越世界之间的基本接口。该项目的目标是更好地理解这些不变量的渐近性质，并将结果应用于算术几何、物理和代数几何中当前感兴趣的密切交织的问题。具体来说，PI计划：(I)展示Fano变量的Apery常数作为高级正态函数的极限（因此是周期），并构建与动机Gamma函数相关的动机，以验证Beilinson和Green-Griffiths-Kerr猜想的具体实例；（II）计算与一组双环图相关的费曼幅值，并将量子曲线的谱与正态函数的零点和极限联系起来（从而确认马里诺猜想的两个结果）；（III）利用奇点普遍变形的混合Hodge理论解释模中几何边界分量的振动，并利用李氏理论方法研究周期映射的Hodge理论紧化的局部和全局方面。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The focus of this project is on polynomial equations and their solution sets, which have been studied specifically and intensively by algebraic geometers. Deep conjectures, including those of Bloch, Beilinson and Hodge, seek to relate the behavior of analytic objects (like integrals and differential equations), which are a priori non-algebraic, to the underlying algebraic structure and topological shape of such solution sets. Even as these conjectures remain intractable in general, solutions in individual cases both continue to bear out their validity and produce new algebraic structures (for example "cycles" and "motives") which facilitate the solutions of problems in apparently remote areas of mathematics and other sciences, for example, at the interface of number theory and physics (such as string theory and quantum field theory). Recent technical innovations, based on considering polynomial equations in families, have begun to provide access to new cases of these conjectures. Their further development and application is the subject of this project, whose results will be disseminated through conferences, summer schools, journal articles and websites. The project consultants brought to Washington University by the grant will contribute to its research atmosphere, and specialized problems embedded in the project will provide training for the PI's graduate students.Hodge-theoretic invariants such as period and regulator maps provide the basic interface between the algebraic and transcendental worlds in modern geometry. The goal of this project is to better understand the asymptotic properties of these invariants, and apply the results to closely intertwined problems of current interest in arithmetic geometry, physics, and algebraic geometry. Specifically, the PI plans to: (I) exhibit the Apery constants of Fano varieties as limits of higher normal functions (hence periods), and construct motives related to motivic Gamma functions to verify specific instances of conjectures of Beilinson and Green-Griffiths-Kerr; (II) compute the Feynman amplitudes associated to a family of two-loop graphs, and relate the spectra of quantum curves to zeroes and limits of normal functions (thereby confirming two consequences of a conjecture of Marino); and (III) use the mixed Hodge theory of miniversal deformations of singularities to interpret fibrations of geometric boundary components in moduli, and use Lie-theoretic methods to study local and global aspects of Hodge-theoretic compactifications of period maps.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.