Mathematical Sciences: Motives and Motivic Cohomology

研究者继续在密切相关的动机和动机上同源领域进行研究。在给出代数闭域上混合动机的定义后，他将这个定义扩展到任意基格式上的动机。然后，他研究了X相对于Y的动机、Y相对于Z的动机和X相对于Z的动机之间的关系。利用他在动机方面的动机上同调的定义，研究者寻找一个证明，证明这个定义满足Beilinson和他自己提出的大部分（如果不是全部的话）“公理”。特别是，Kummer公理和“有限维数”公理似乎是可以理解的。最后，研究者已经在混合动机的范畴中为每一个变量定义了一个同调动机，研究了对应于上同调、Borel-Moore同调和紧支撑上同调的动机的可能定义，并明确地计算了曲线情况下的这些动机。研究者还研究了庞加莱和亚历山大二元性的动机。这一建议构成了对代数几何基本结构的基础研究。代数几何是现代数学中最古老的部分之一，但在过去的四分之一个世纪里，它已经有了革命性的发展。在它的起源中，它处理平面上可以用多项式方程定义的曲线，推广了希腊人所知的经典圆锥曲线。如今，该领域不仅使用代数的方法，还使用分析和拓扑的方法，相反，它在这些领域以及物理学、理论计算机科学和机器人技术中也得到了应用。

Lichtenbaum The investigator continues his researcn in the closely related areas of motives and motivic cohomology. Having given a definition of mixed m otives over an algebraically closed field, he extends this definition to motives over an arbitrary base scheme. He then studies the relationship between the motive of X relative to Y, the motive of Y relative to Z, and the motive of X relative to Z. Using his definition of motivic cohomology in terms of motives, the investigator searches for a proof that this definition satisfies most, if not all, of the "axioms" proposed by Beilinson and himself. In particular, the Kummer axiom and the "finite dimensionality" axiom seem to be accessible. Finally, having already defined for each variety a homology motive in the category of mixed motives,the investigator studies possible definitions of motives corresponding to cohomology, Borel-Moore homology, and cohomology with compact supports, and explicitly computes these motives in the case of curves. The investigator also studies Poincare and Alexander duality for motives. This proposal constitutes fundamental research into the basic structures of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated curves in the plane that could be defined by polynomial equations, generalizing the classic conic sections known to the Greeks. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as physics, theoretical computer science, and robotics.