Geometric Applications of Non-Abelian Hodge Theory

Pantev将使用来自阿贝尔和非阿贝尔霍奇理论的技术来解决具体的几何问题以及数学物理和弦理论中的问题。将研究四个问题。第一个是寻找一个线性代数对象的构造，它通常与射影变量X相关联，并计算X上所有局部系统及其Hodge结构的上同调，只要后者有意义。第二部分涉及曲线上的非阿贝尔霍奇猜想，并提供了动力局部系统轨迹的内在几何表征。第三部分讨论了补具有非残差有限基群的奇异平面曲线的构造。第四个项目提出了一种理解弦理论中增强规范对称的几何策略，并提出了一种分析F理论模空间的奇异分量的方法。理解这些问题对于统一代数几何和数学物理中的各种线性化过程是必不可少的。一方面，它将使我们更接近于理解代数变体的基本结构，另一方面，它将为最近发现的M弦理论和F弦理论的相互作用带来一个几何的视角。这是代数几何领域的研究。代数几何是现代数学中最古老的部分之一，但在过去的四分之一个世纪里，它已经有了革命性的发展。在它的起源中，它处理的图形可以用最简单的方程，即多项式，在平面上定义。如今，该领域不仅使用代数的方法，还使用分析和拓扑的方法，相反，它在这些领域以及物理学、理论计算机科学和机器人技术中也得到了应用。

Pantev 9800790 Pantev will use techniques from abelian and non-abelian Hodge theory to approach concrete geometric questions as well as problems in mathematical physics and string theory. Four problems will be studied. The first one is to look for a construction of a linear algebraic object that is canonically associated to a projective variety X and calculate the cohomology of all local systems on X together with their Hodge structures whenever the latter makes sense. The second concerns the non-abelian Hodge conjecture on curves and provides an intrinsic geometric characterization of the locus of motivic local systems. The third discusses a construction of a singular plane curve whose complement has a non-residually finite fundamental group. The fourth project suggests a geometric strategy for understanding the appearance of enhanced gauge symmetry in string theory and proposes a method for analyzing the exotic components of F theory moduli spaces. The understanding of these questions is essential for unifying various linearization procedures in algebraic geometry and mathematical physics. On one hand, it will bring us closer to understanding the basic structure of algebraic varieties, and on the other hand it will bring a geometric perspective to the interaction of the recently discovered M and F string theories. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one that has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.