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

研究者们研究代数变体的基本群。特别注意的是鉴别位点补的基本群。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结构的变化和与超曲面自然相关的变异相关的周期映射来研究的。相应的单一性表示在本研究中也很重要。Zariski研究了零维超曲面（射影线上的点）的情况，其中基本群是球的编织群，相应的表示包括置换和布洛表示。研究者们正在高维超曲面的相应问题上取得进展。对于低次曲线和曲面，得到了特别精确的结果。作为一种应用，研究者使用这些知识来构造具有有趣基本群的射影和拟射影变体的例子。代数几何是数学中最古老也是目前最活跃的研究领域之一。最初它是用多项式方程来处理平面上曲线的几何形状。现在，它在许多维度上处理类似的问题，使用了许多数学和理论物理分支的技术，它反过来影响了这些分支，以及其他学科。本研究关注的是代数几何和拓扑学之间的特定相互作用，这些相互作用不仅由数学家开发，而且由研究费曼积分的物理学家开发。目前的研究也涉及到对辫子的研究。自从1925年Emil Artin将辫子引入数学以来，它一直是几个数学分支的核心，而且它在数学和物理学中的影响还在继续扩大。

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.