Motives of algebraic varieties

代数几何，研究由多项式方程定义的形状（代数簇），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圆锥曲线的研究。 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，涉及与数论和高能物理等许多其他领域有关的各种主题。现代代数几何的创始人之一、法国数学家亚历山大·格罗滕迪克（英语：Alexander Grothendieck）在1960年代后期提出了一种“大统一上同调理论”，将几种用于研究代数几何中物体形状的不同理论结合在一起。从那时起，动机已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学科之一。我计划在动机和代数圈领域开展四个主要项目：（1）投射齐次簇的动机分解，（2）周期和费曼振幅，（3）混合霍奇结构的变化和（4）代数几何中的Steenrod运算。第一个项目有直接的应用代数。 第二个目标是研究物理学家Dirk Kreimer发现的动机和费曼振幅之间的深刻关系，这些振幅是在高能物理中基本粒子之间相互作用的计算中出现的。 第三和第四个项目的动机是围绕代数几何中最重要的未解决问题的思想圈，霍奇猜想，它询问何时可以用多项式方程定义形状。

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.