Hodge Theory, Dualities and Non-Commutative Geometry

这是代数几何领域的一项研究，代数几何是研究多项式方程组解的经典学科。目前的项目解决了四个问题，在复杂几何、弦理论和量子物理之间提供了令人兴奋的接口。第一个问题旨在建立一个计算与高同伦相关的复射影变不变量的Hodge理论技术工具箱。第二章探讨了复杂几何中经典沙法里维奇均匀化猜想的局限性。第三个问题在由K3曲面构成的gerby Calabi-Yaumanifold上提出了一种显式的束结构，并研究了它对场非交换变形物理概念的影响。项目四探讨了Calabi-Yau流形的极值跃迁退化族、可积系统的极限行为和非交换代数几何之间的关系。理解这些问题对于统一代数几何、辛拓扑、理论和数学物理中的各种线性化过程是必不可少的。该项目为理解代数品种的基本结构奠定了基础，以一种适合于广泛应用的实用方法。所提出的工作将直接与分类理论、组合群论、可积系统理论、弦理论、量子引力和宇宙学等深层次问题相关。该项目提出了具体的跨学科应用到矩阵量子力学，弦二象性和粒子物理模型构建。

This is a research in the field of algebraic geometry - a classicalsubject studying the solutions to systems of polynomial equations. Thecurrent project addresses four problems providing exciting interfacesbetween complex geometry and string theory and quantum physics. Thefirst problem aims to build a toolbox of Hodge theoretic techniques forcomputing invariants of complex projective varieties related to higherhomotopy. The second explores the limits of the classical Shafarevichuniformization conjecture in complex geometry. The third problemproposes an explicit construction of sheaves on a gerby Calabi-Yaumanifold which is fibered by K3 surfaces, and studies its impact on thephysical notion of non-commutative deformation of fields. The fourthproject probes the relationship between the extremal transitions indegenerating families of Calabi-Yau manifolds, the limiting behavior ofintegrable systems, and non-commutative algebraic geometry.The understanding of these questions is essential for unifying variouslinearization procedures in algebraic geometry, symplectic topology,theoretical and mathematical physics. The project sets the stage forunderstanding the basic structure of algebraic varieties in a waysuitable for pragmatic use in a broad spectrum of applications. Thework proposed will be immediately relevant to deep questions incategory theory, combinatorial group theory, the theory of integrablesystems, string theory, quantum gravity and cosmology. The projectproposes concrete interdisciplinary applications to matrix quantummechanics, string dualities and particle physics model building.