Non-Abelian Hodge Theory and Transcendence

霍奇理论涉及代数形式沿着拓扑圈的积分。 对这些不变量的研究可以追溯到世纪雅可比、阿贝尔和黎曼的工作;现代理论将代数簇的几何的代数、拓扑、复解析和算术方面联系在一起，并有许多应用。 开创性的工作辛普森在20世纪90年代开发了一个非阿贝尔版本的这一理论的空间表示的基本群体发挥的作用组的拓扑循环。 由此产生的非阿贝尔霍奇理论同样触及数学的许多领域，但它的许多方面仍然是神秘的。在这个项目中，PI将通过o-极小方法将经典霍奇理论和超越理论的最新进展扩展到非阿贝尔环境。该项目将特别面向促进学生和早期职业数学家的参与。更详细地说，PI将应用o-极小技术来解决一些与代数簇上的局部系统几何相关的开放性问题，以及它与复分析，算术和超越理论的联系。这包括超越理论的黎曼-希尔伯特对应，分类的三代数子簇，以及代数和算术的非阿贝尔霍奇轨迹。这些技术也将被带到承担相关的几何问题，包括建设的Shafarevich地图，超越理论的p-adic周期图，和几何的拉格朗日fibrations.This奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

Hodge theory is concerned with the integrals of algebraic forms along topological cycles. The study of these invariants traces its roots to the work of Jacobi, Abel, and Riemann in the nineteenth century; the modern theory ties together the algebraic, topological, complex analytic, and arithmetic facets of the geometry of an algebraic variety, and has many applications. Pioneering work of Simpson in the 1990s developed a non-abelian version of this theory where the space of representations of the fundamental group plays the role of the group of topological cycles. The resulting non-abelian Hodge theory touches equally many fields of mathematics, but many aspects of it remain mysterious. In this project, the PI will extend recent progress in classical Hodge theory and transcendence theory via o-minimal methods to the non-abelian setting. The project will specifically be geared towards fostering the involvement of students and early-career mathematicians.In more detail, the PI will apply o-minimal techniques to address a number of open questions related to the geometry of local systems on algebraic varieties, and its connection to complex analysis, arithmetic, and transcendence theory. This includes the transcendence theory of the Riemann—Hilbert correspondence, the classification of tri-algebraic subvarieties, as well as the algebraicity and arithmeticity of non-abelian Hodge loci. These techniques will also be brought to bear on related geometric questions, including the construction of Shafarevich maps, transcendence theory of p-adic period maps, and the geometry of Lagrangian fibrations.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.