Geometric Applications of Non-Abelian Hodge Theory

潘特夫9800790将使用阿贝尔和非阿贝尔霍奇理论的技巧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。第一种方法是寻找一个与射影簇X典型相关的线性代数对象的结构，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。第二种是关于曲线上的非阿贝尔Hodge猜想，并给出了Motivic局部系轨迹的内在几何刻画。第三章讨论了补具有非剩余有限基本群的奇异平面曲线的构造。第四个项目提出了一种几何策略来理解弦理论中增强规范对称性的出现，并提出了一种分析F理论模空间奇异分量的方法。对这些问题的理解对于统一代数几何和数学物理中的各种线性化过程是至关重要的。一方面，它将使我们更接近于了解代数簇的基本结构，另一方面，它将为最近发现的M和F弦理论的相互作用带来几何视角。这是在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25年里，它已经取得了革命性的成就。在它的起源中，它处理的是可以在平面上用最简单的方程定义的图形，即多项式。如今，该领域不仅使用代数的方法，而且使用分析和拓扑学的方法，反过来，在这些领域以及物理、理论计算机科学和机器人中也找到了应用。

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.