Non-Abelian Hodge Theory and Transcendence

Hodge理论涉及沿拓扑圈的代数形式的积分。对这些不变量的研究可以追溯到19世纪雅各比、阿贝尔和黎曼的工作；现代理论将代数、拓扑、复杂的分析和算术方面联系在一起，并有许多应用。辛普森在20世纪90年代的开创性工作发展了这一理论的非阿贝尔版本，其中基本群的表示空间扮演了拓扑圈的群的角色。由此产生的非阿贝尔霍奇理论同样触及了数学的许多领域，但它的许多方面仍然神秘。在这个项目中，PI将通过o-极小方法将经典Hodge理论和超越理论的最新进展推广到非阿贝尔环境。该项目将专门培养学生和职业生涯初期的数学家的参与。更详细地说，PI将应用o-极小技术来解决一些与代数簇上局部系统的几何有关的开放问题，以及它与复数分析、算术和超越理论的联系。这包括Riemann-Hilbert对应的超越理论，三代数子簇的分类，以及非交换Hodge轨迹的代数性和算术性。这些技术也将被应用于相关的几何问题，包括Shafarevich映射的构造，p进周期映射的超越理论，以及拉格朗日纤维的几何。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.