Mathematical Sciences: Algebraic Topology of Algebraic Varieties and Conformal Field Theory

[214772]贝林森这个项目涉及两个主题。第一个是混合动机理论。这种推测性的理论旨在揭示算术几何中隐藏的拓扑模式。它有助于解释k理论和代数循环理论中的各种现象，并提供关于算术起源的特殊函数值的信息。在这个方向上已经得到了几个结果，例如，在与P. Deligne合著的一篇论文中，证明了D. Zagier关于多对数函数值的部分猜想。目前，混合动机范畴的实际构造问题（至少模格罗滕迪克的标准猜想）似乎是相当容易处理的。该项目的第二个主题是研究共形场论的几何。这一理论在过去十年中得到了迅速发展。这位研究者的第一个谦虚的目标是（与B. Feigin和B. Mazur一起）写下一个代数-几何设置的说明。接下来要做的事情（与V. Drinfeld和V. Ginzburg正在进行的工作）是一个理论，正如Drinfeld所预见的那样，将临界水平上的Kac-Moody代数表示与朗兰兹对应的几何版本联系起来。该项目的最终目标是将上述两个主题结合在一起，这仍然是一个诱人的希望。这个项目的两个主题，共形场理论和动机理论，有一些不同的起源。约十年前，共形场论首次出现在物理学中。人们在那里观察到，通常处于临界温度的物体（比如你杯子里正在融化的冰块）会突然获得比以前无限大的内部对称性；这种对称性支配着复杂的图案，使物体脱颖而出。在弦理论剧变之后，描述这一现象的数学结构很快成为理论物理学中最受欢迎的结构，这一发展也有助于将以前不相关但被深入研究的数学领域联系起来，如无限维群的表示理论、模空间的几何和朗兰兹的程序。这个项目的第一部分在这里。格罗滕迪克（Grothendieck）在60年代中期发现的动机理论，在某种意义上也有类似的味道。它始于一个古老的见解，即数论建议人们将整数视为某个复杂空间上的函数。这个空间，以其内部的对称性，取代了点的普通几何概念；在道德上，普通几何获得了一个额外的算术维度。动机理论研究这样一种几何。作为一种应用，它可以提供一系列关于经典算术函数值的不可预测的结果。* * *

9214772 Beilinson This project deals with two subjects. The first one is the theory of mixed motives. This, yet conjectural, theory is aimed to unfold the hidden topological pattern of arithmetic geometry. It helps to explain various phenomena in K-theory and the theory of algebraic cycles and to provide information about the values of the special functions of arithmetic origin. Several results in that direction have already been obtained, e.g., in a joint paper with P. Deligne, a proof of part of D. Zagier's conjecture on the values of the polylogarithm function. At the moment, the problem of the actual construction of the category of mixed motives (at least modulo Grothendieck's Standard Conjectures) seems to be quite tractable. The second subject of the project is the study of the geometry of conformal field theory. This theory has been developing rapidly during the last decade. The investigator's first modest aim was to write down (jointly with B. Feigin and B. Mazur) an account of an algebro-geometric setting. The next thing to pursue (work in progress with V. Drinfeld and V. Ginzburg) is a theory, as foreseen by Drinfeld, relating the representations of Kac-Moody algebras at the critical level with the geometric version of Langlands' correspondence. The ultimate aim of the project is to tie together the two subjects above, which remains a tantalizing hope. The two subjects of this project, conformal field theory and the theory of motives, have somewhat different origins. Conformal field theory came first to physics approximately ten years ago. It was observed there that often objects at the critical temperature (the melting ice cube in your glass) suddenly acquire infinitely greater inner symmetry than before; this symmetry governs the intricate pattern that singles out the object. The mathematical structures that described this soon became - after the string theory upheaval - the most popular structures of theoretical physics, a dev elopment that also helped to connect previously unrelated but intensively studied mathematical areas such as representation theory of infinite-dimensional groups, geometry of moduli spaces, and the Langlands' program. The first part of this project dwells here. The theory of motives, discovered by Grothendieck in the mid-60's, has, in a sense, a similar flavor. It starts with an old insight that number theory suggests that one consider the whole numbers as if they were functions on a certain complicated space. This space, with its inner symmetries, replaces the ordinary geometric notion of a point; morally, the ordinary geometry acquires an extra arithmetic dimension. The theory of motives studies such a geometry. As an application, it may provide a range of yet unpredictable results about the values of the classical arithmetic functions. ***