Motives of algebraic varieties

代数几何，研究由多项式方程定义的形状（代数变体），是现代数学最古老的领域之一，其根源可以追溯到古希腊对二次曲线的研究。然而，它也是数学研究中最活跃的领域之一，每年产生数百篇科学论文，涉及各种各样的主题，与许多其他领域（如数论和高能物理）有关。动机是由法国数学家Alexander Grothendieck（现代代数几何的创始人之一）在20世纪60年代末作为“大统一上同调理论”引入的，该理论汇集了用于研究代数几何中物体形状的几种不同理论。从那时起，动机就成为代数几何中不可或缺的工具。它们与代数循环的研究密切相关，代数循环是理论中最困难但最迷人的课题之一。我计划在动机和代数循环领域的四个主要项目上工作：(1)射影齐次变量的动机分解，(2)周期和费曼振幅，(3)混合霍奇结构的变化，(4)代数几何中的Steenrod运算。第一个项目对代数有直接的应用。第二个项目的目标是研究物理学家德克·克莱默（Dirk Kreimer）在高能物理中计算基本粒子之间相互作用时发现的动机和费曼振幅之间的深层关系。第三和第四个项目的灵感来自于代数几何中最重要的未解决问题——霍奇猜想（Hodge conjecture），该猜想提出了一个形状何时可以用多项式方程来定义。

Algebraic geometry, the study of shapes (algebraic varieties) defined by polynomial equations, is one of the oldest areas of modern mathematics with roots reaching back to the ancient Greek study of conic sections.  It is, however, also one of the most active areas of research in mathematics producing hundreds of scientific papers a year on a diverse array of topics with connections to many other areas such as number theory and high energy physics. Motives were introduced by the French mathematician Alexander Grothendieck, one of the founders of modern algebraic geometry, in the late 1960s as a "grand unified cohomology theory" bringing together several different theories used to study the shapes of objects in algebraic geometry. Since then, motives have become an indispensable tool in algebraic geometry.  They are intimately related to the study of algebraic cycles, one of the most difficult but fascinating subjects in the theory. I plan to work on four main projects in the area of motives and algebraic cycles: (1) motivic decompositions of projective homogeneous varieties, (2) periods and Feynman amplitudes, (3) variations of mixed Hodge structure and (4) Steenrod operations in algebraic geometry. The first project has direct applications to algebra.  The goal of the second is to study a deep relationship discovered by the physicist Dirk Kreimer between motives and the Feynman amplitudes which arise in calculations of interactions between elementary particles in high energy physics.  The third and fourth projects are loosely motivated by the circle of ideas surrounding the most important unsolved question in algebraic geometry, the Hodge conjecture, which asks when a shape can be defined by polynomial equations.