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

研究者们研究代数簇的基本群。 特别注意的是给予基本群体的判别位点的补充。 这些补集参数化了射影空间中的光滑超曲面。 它们是通过与超曲面的Hodge结构的变化相关联的周期映射和与超曲面自然相关联的簇来研究的。 在这项研究中，相应的单值表示也很重要。 Zenki研究了零维超曲面（投影线上的点）的情况，其中基本群是球面的辫子群，相应的表示包括置换和Burau表示。 研究人员正在高维超曲面的相应问题上取得进展。 特别是精确的结果正在获得的曲线和曲面的低程度。 作为一个应用，研究人员使用这些知识来构建具有有趣的基本群的投射和拟投射簇的例子。 代数几何是数学中最古老也是最活跃的领域之一。 最初它处理几何曲线在平面上定义的多项式方程。 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学许多分支的技术，它反过来影响这些分支和其他学科。 这项研究关注的是代数几何和拓扑之间的特定相互作用，这些相互作用不仅是数学家而且是研究费曼积分的物理学家开发的。 目前的研究也涉及辫子的研究。 自1925年埃米尔·阿廷（Emil Artin）引入辫子以来，辫子一直是数学的几个分支的核心，并且它们在数学和物理学中的影响力继续扩大。

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.