Mathematical Sciences: Non-Commutative Differential Geometry

在技术层面上，Wodzicki教授的项目主要涉及计算代数的循环同调，这些代数来自微分几何或代数几何或泛函分析。微分算子的代数，特别是光滑解析变元上的微分算子代数，在这一研究中起着重要的作用，但它还有一个重要的方面是处理C*-代数（希尔伯特空间上的某些算子代数）及其k理论和循环同调。这里所设想的数学研究是微积分在流形、粗糙曲面及其高维类似物上的深远发展。微分算子来自于微积分，它只能看到流形的小块，但却包含了流形整体形状和构象的信息。与Wodzicki教授的工作相关的一种策略是从微分算子中提取这类信息，包括将算子组合成代数结构，然后使用代数机制。该机制的进一步发展，其中两个重要且密切相关的部分被称为k理论和循环同源，及其在阐明几何和拓扑结构中的应用构成了该奖项支持的研究议程。

At a technical level, Professor Wodzicki's project is largely concerned with computing the cyclic homology of algebras that come, variously, from differential or algebraic geometry or from functional analysis. Algebras of differential operators, especially on smooth analytic varieties, play a major role in this research, but there is also an important aspect of it that deals with C*-algebras (certain algebras of operators on Hilbert space) and their K-theory and cyclic homology. The mathematical research here envisioned is a far-reaching outgrowth of calculus on manifolds, roughly surfaces and their higher-dimensional analogues. Differential operators, which come from calculus, see the manifold only in small pieces, but nevertheless contain information about its overall shape and conformation. One strategy, relevant to the work of Professor Wodzicki, for extracting this sort of information from differential operators involves assembling the operators into algebraic structures and then using algebraic machinery. The further development of this machinery, two important and intimately related parts of which are called K-theory and cyclic homology, and its use in elucidating geometric and topological structure constitute the agenda of the research supported by this award.