Mathematical Sciences: K-Theory, C*-Algebras and Index Theory for Elliptic Operators

这个项目的核心数学概念是索引。在其最基本的表现形式中，索引与必须施加于微分方程的解上的额外条件的数量有关，以便明确地指定它。Atiyah和Singer的一个著名定理断言，在最简单的情况下，这种指标是由一个圈数给出的（计算平面上一条路径绕原点旋转的次数），更一般地说，是由一个类似的高维拓扑不变量给出的。Atiyah-Singer定理及其众多衍生定理都有这样的形式：左边（微分算子的索引）等于右边（你可以从算子的成分中计算出来的东西）。进步在于对右边到底是什么有了越来越复杂的理解。如今，这涉及到几何、拓扑和功能分析。以Kaminker教授和Ji教授等人为代表的最先进的观点是在适当的算子代数的k理论中计算指标。经典地，这将是交换的，一个在底层流形上的函数代数，但在额外结构的存在下，需要一个更复杂的代数。研究人员将寻求更好地理解代数及其K理论，并从K群的元素中提取新的微分算子不变量。

The central mathematical concept in this project is that of index. In its most basic manifestation, index has to do with the number of extra conditions that must be imposed on a solution to a differential equation in order to specify it unambiguously. A famous theorem of Atiyah and Singer asserts that this sort of index is given in the simplest case by a winding number (counting the number of times a path in the plane winds around the origin) and more generally by an analogous higher-dimensional topological invariant. The Atiyah-Singer Theorem and its numerous progeny all have the form: lefthand side (index of a differential operator) equals righthand side (something you can compute from the ingredients that go into the operator). Progress consists in achieving a more and more sophisticated understanding of what the righthand side really is. Nowadays, this involves geometry, topology, and functional analysis. The most advanced point of view, represented by Professors Kaminker and Ji among others, computes index in the K-theory of an appropriate algebra of operators. Classically, this would be commutative, an algebra of functions on the underlying manifold, but in the presence of additional structure a more complicated algebra is called for. The investigators will seek to understand the algebra and its K-theory better, and to extract new invariants for differential operators from elements of the K- groups.