Mathematical Sciences: Menodromy Kernels, Discriminant Loci, and Fundamental Groups of Algebraic Varieties

研究人员研究代数簇的基本群。对判别基因的基本互补群给予了特别关注。这些补将射影空间中的光滑超曲面参数化。通过与超曲面的Hodge结构的变化相关的周期映射和与超曲面自然相关的簇来研究它们。相应的单字表征在本研究中也很重要。Zariski研究了零维超曲面(射影直线上的点)的情况，其中基本群是球面的辫群，相应的表示包括置换表示和Burau表示。研究人员在高维超曲面的相应问题上取得了进展。对于低次曲线和曲面，得到了特别精确的结果。作为一个应用，研究人员利用这一知识来构造具有有趣的基本群的投射和拟投射簇的例子。这项研究是在代数几何领域进行的，代数几何是数学中最古老的领域之一，也是目前最活跃的领域之一。最初，它处理由多项式方程定义的平面上的曲线几何。现在，它在许多维度上处理类似的问题，使用数学和理论物理的许多分支的技术，并影响这些分支和其他学科。这项研究关注代数几何和拓扑之间的特定相互作用，这不仅是由数学家发展的，也是由研究费曼积分的物理学家发展的。本研究还涉及到对辫子的研究。自1925年埃米尔·阿丁引入数学以来，辫子一直是数学的几个分支的中心，它们在数学和物理学中的影响继续扩大。

The investigators study fundamental groups of algebraic varieties. Particular attention is given to the fundamental groups of complements of discriminant loci. These complements parametrize smooth hypersurfaces in projective space. They are studied by means of the period mappings associated to the variation of Hodge structure of the hypersurfaces and of varieties naturally associated to the hypersurfaces. The corresponding monodromy representations are also important in this study. Zariski studied the case of zero-dimensional hypersurfaces (points on the projective line), where the fundamental group is the braid group of the sphere and the corresponding representations include the permutation and Burau representations. The investigators are making progress on corresponding questions for higher-dimensional hypersurfaces. Particularly precise results are being obtained for curves and surfaces of low degree. As one application, the investigators use this knowledge to construct examples of projective and quasi-projective varieties with interesting fundamental groups. This research is in the field of algebraic geometry, which is one of the oldest fields in mathematics as well as one of the most active at present. Originally it treated the geometry of curves in the plane defined by polynomial equations. Now it treats similar questions in many dimensions, uses techniques from many branches of mathematics and theoretical physics, and it turn it influences these branches, and other subjects. This research is concerned with specific interactions between algebraic geometry and topology that have been developed not only by mathematicians but also by physicists studying Feynman integrals. The present research also concerns the study of braids. Braids have been central to several branches of mathematics since their mathematical introduction by Emil Artin in 1925, and their influence continues to expand in mathematics and physics.