Mathematical Sciences: Motives and Motivic Cohomology

利希滕鲍姆继续着他在动机和动机上同源这两个密切相关的领域的研究。在给出了代数闭域上的混合动机的定义后，他将这个定义推广到任意基方案上的动机。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。根据他关于动机的动机上同调的定义，研究者寻找一个证据，证明这个定义满足贝林森和他自己提出的大部分(如果不是全部)“公理”。特别是，库默公理和“有限维”公理似乎是可以理解的。最后，在已经为每个变种定义了混合动机范畴中的同调动机之后，研究者研究了对应于上同调、Borel-Moore同调和具有紧支撑的上同调的可能的动机定义，并在曲线的情况下显式地计算了这些动机。调查人员还研究了庞加莱和亚历山大的二元性动机。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25年里，它已经取得了革命性的成就。在它的起源，它处理的是平面上的曲线，可以由多项式方程式定义，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，而且使用分析和拓扑学的方法，反过来，在这些领域以及物理、理论计算机科学和机器人学中也找到了应用。

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.