KK-theory, quantum groups, and quaternions

这是一个纯数学项目。数学是关于联系和从我们对世界的现有信息中发展出可证明正确的推论。在历史上，它是高等教育本身的代名词。纯数学的进步慢慢地影响着应用科学；事实是，曾经深奥的纯数学概念现在出现在工程学和物理学中。对于学生或HQP来说，纯数学教授复杂主题的高级逻辑思维，并可能应用于复杂的科学问题。因此，这项研究计划的好处存在于今天的科学技术前沿，并可能影响明天的科学技术。该程序的基本主题是希尔伯特空间上算子的范闭自伴随代数，称为C*-代数。它们具有显著的特性，并连接不同的领域，如功能分析、代数、拓扑、几何和动力系统。也许在这个领域中最重要的单一努力是用k -理论数据（埃利奥特程序）对这些代数进行分类。现在正是研究将该项目拓展到新方向的时候。C*-代数可以看作是拓扑空间的非交换推广，如果我们寻找拓扑群的推广而不是拓扑空间，我们得到C*-代数量子群（qg），它可能在C*-代数上有协同作用。我们的主要目标是将Elliott规划推广到合适的qg和协同作用类。埃利奥特风格的qg分类与现有的代数分类非常不同，这种创新的方法很有可能被国际研究人员采用。HQP谁获得在拟议的程序的早期开始将有利于一个富有成效的研究生涯。上述程序的好处之一是，它允许将经典代数拓扑应用于qg。代数拓扑使用函子来证明某些事情不可能发生。对于C*-代数，存在一个Cuntz半群函子（CSF）。我们发现了许多qg的CSF的重要结构特性和它们的一些协同作用。这一发现可能使CSF成为研究量子群代数拓扑的有用工具。为了将经典代数拓扑扩展到量子动力学框架，需要进一步开发用于量子g及其协同作用的代数拓扑工具。例如，我们可以从CSF水平上的同构构造量子群同构。这种有趣的现象在经典代数拓扑中没有精确的对应，并且是一种明确的“量子”现象。我们与G.A. Elliott和C. Ivanescu共同研究了Villadsen代数的Cuntz半群。我们的目标是通过它们的Cuntz半群对Villadsen代数进行分类。这是一个令人兴奋的，及时的，更重要的是，可行的建议。

This is a project in pure mathematics. Mathematics is about connections and developing provably correct deductions from available information we have about the world. Historically, it was synonymous with higher education itself. Progress in pure mathematics slowly influences the applied sciences; it is a fact that once esoteric concepts in pure mathematics now appear in engineering and physics. For the student or the HQP, pure mathematics teaches advanced logical thinking about complex topics, with possible applications to complicated scientific problems. Thus the benefits of this research program exist at the frontiers of today's science and technology, and may affect tomorrow's science and technology. The basic topic of the program is norm-closed self-adjoint algebras of operators on Hilbert space, called C*-algebras. They have remarkable properties, and connect disparate areas such as functional analysis, algebra, topology, geometry, and dynamical systems. Perhaps the most important single endeavour in this field is to classify such algebras by K-theoretical data (the Elliott program). The time is right to investigate branching the program out to new directions. C*-algebras can be viewed as noncommutative generalizations of topological spaces, and if we look for generalizations of topological groups rather than topological spaces, we obtain C*-algebraic quantum groups (QGs), which may have co-actions on C*-algebras. Our main objective is to generalize the Elliott program to suitable classes of QGs and co-actions. An Elliott style classification of QGs works very differently from existing algebraic classifications, and there is strong potential that this innovative approach will be adopted by an international community of researchers. HQP who obtain an early start in the proposed program will be well positioned for a productive research career. One of the benefits of the above program is that it lets classical algebraic topology be applied to QGs. Algebraic topology uses functors to prove that certain things cannot happen. For C*-algebras, there exists a Cuntz semigroup functor (CSF). We found important structural properties of the CSF of many QGs and some of their co-actions. This discovery may make the CSF a useful tool for studying the algebraic topology of quantum groups. For extending classical algebraic topology to a quantum dynamical framework, it seems desirable to further develop algebraic topological tools for QG and their co-actions. For example, we can construct quantum group isomorphisms from isomorphisms at the CSF level. This interesting phenomenon has no precise counterpart in classical algebraic topology, and is an explicitly `quantum' phenomenon. Jointly with G.A. Elliott and C. Ivanescu, we are studying the Cuntz semigroup of Villadsen algebras. We aim to classify Villadsen algebras through their Cuntz semigroup. This is an exciting, timely, and more importantly, feasible proposal.