Classification of Real Structures and Actions on C*-algebras

我的研究涉及称为C*-代数的数学对象，以及它们可能被赋予的额外属性，称为群作用和实结构。C*-代数在物理学中量子力学现象的一些模型中出现，以及其他地方。在这些模型中，C*-代数描述了系统的可观测量，如动量，而群体行为描述了系统随着时间或参照系的其他变化的演变。C*-代数上的真实结构是许多自然产生的C*-代数所具有的额外对称性。把它们考虑进去，为研究C*-代数提供了额外的工具。它们最近也被用于研究拓扑绝缘体，物理上的绝缘体可以在表面导电，但不能在内部导电。我的研究计划是对C*-代数进行分类，这些代数具有实结构、群作用，或者既具有实结构又具有与之相容的群作用。针对的分类类型不是简单的列表。我们使用一套称为k理论的工具来构建描述所讨论对象的关键特征的标签。这就像收集了大量汽车的用户手册，而不仅仅是车牌号码列表。结果的目的有两个部分。第一部分被称为分类定理，它告诉我们，我们的手册区分了我们的对象，所以任何两个具有相同手册的对象本质上是相同的，就像两辆完全相同型号的汽车一样。第二部分称为不变定理的范围，告诉我们哪些手册对应于集合中的对象。以汽车为例，我们的手册库可能包含从未制造过的汽车的计划，或者没有出现在我们的产品中。一旦我们有了这样的分类，我们就可以通过阅读它们的使用手册来回答关于我们的对象的问题，而不必把它们拆开。

My research deals with mathematical objects called C*-algebras (read see-star), along with additional properties they may be endowed with, called group actions and real structures.C*-algebras arise, among other places, in some models of quantum mechanical phenomena in physics. In these models, the C*-algebra describes the observable quantities of the system, such as momentum, while group actions describe the evolution of the system through time or other changes of reference frame.A real structure on a C*-algebra is an additional symmetry that many naturally occurring C*-algebras possess. Taking them into account provides extra tools for studying the C*-algebra. They have also recently been used to study topological insulators, bodies in physics that may conduct electricity over their surfaces but not in their interiors.My research program looks to classify C*-algebras endowed with a real structure, a group action, or both a real structure and a group action that is compatible with it. The type of classification aimed at is not simply a list. We use a set of tools called K-theory to construct labels that describe the key features of the objects in question. This is like having a collection of owner's manuals for a lot full of cars, as opposed to just a list of license plate numbers. The results aimed at have two parts. The first part, called the classification theorem, tells us that our manuals distinguish between our objects, so that any two objects with the same manual are essentially the same, like two cars of exactly the same model. The second part, called the range of the invariant theorem, tells us which manuals correspond to objects in our collection. In our cars analogy, our library of manuals may contain plans for cars that were never built, or are not present in our lot.Once we have such a classification, we can answer questions about our objects by reading their manuals, rather than having to take them apart.