Operator Algebras and Topological Invariants

这个项目的指导前提是，广义上理解的代数拓扑学的思想和方法将继续在算子代数的研究中发挥关键作用。研究的基本主题是在算子代数的形变理论中设计新的技术和工具。从E-理论的发展可以看出，代数的变形思想与K-理论密切相关。特别地，人们可以从X上的连续函数代数的矩阵变形中读出局部紧连通空间X的K-同调。在可以被视为凝聚态物理学中这一性质的对应性质中，Kitaev最近提出了一种拓扑绝缘子的分类，该分类基于实K-同调，并且最终依赖于空间的矩阵变形的近似么正等价的分类。该项目的第一个目标是研究整合了本原谱和同调不变量的C*-代数的不变量。这自然地与形变和连续场理论联系在一起，并将涉及由不一定是Hausdorff的空间参数化为KK理论和E理论的研究。第二个目标是研究群C*-代数到矩阵代数的变形的存在性，这些变形具有检测群C*-代数的K-理论的性质。此外，首席研究员的目标是提供计算与这些变形相关的不变量的公式。第三个目标是研究简单C*-代数的不变量，这些不变量捕捉了K-理论或迹态没有检测到的性质。特别是，该项目将研究简单C*-代数的刚性性质。基本粒子的物理学导致了新的数学理论，其中数值函数被数字或矩阵的无限阵列所取代。矩阵可以相乘，但与数字乘法不同，因子的顺序是必不可少的，因此A乘以B并不总是等于B乘以A。矩阵乘法的这种称为非交换性的性质，对于解释量子力学中最初可能看起来完全违反直觉的非常复杂的现象至关重要(例如，海森伯格测不准原理)。从数字到矩阵的传递通常涉及一个变形过程，在这个过程中，空间结构被离散化，从而以数字不变量的形式捕捉曲面(甚至更复杂的对象)可能的扭曲。这个项目涉及这种变形的理论，并将经典几何与非对易几何联系起来，非对易几何提出了一种重建一般空间概念的理论，比传统几何更符合量子物理的原理。此外，首席研究员将研究的形变理论提供了与在某些特殊材料中观察到的物理现象相关的数学工具。这些材料在其内部起到绝缘体的作用，同时允许电荷在其表面移动，这是由于局域的外来金属状态。这种被称为“拓扑绝缘体”的引人注目的新材料已经对凝聚态物理产生了相当大的影响，并有望在磁电子学和量子计算中找到技术应用。磁电子学是一个新兴领域，使用自旋电流来代替电荷电流。

This project is guided by the premise that the ideas and the methods of algebraic topology, understood in a broad sense, will continue to play a key role in the study of operator algebras. The underlying theme of the research is to devise new techniques and tools in the deformation theory of operator algebras. The idea of deformations of algebras is intimately related to K-theory, as is revealed by the development of E-theory. In particular one can read the K-homology of a locally compact connected space X from matricial deformations of the algebra of continuous functions on X. In what may be regarded as a counterpart of this property in condensed-matter physics, Kitaev has proposed very recently a classification of topological insulators that is based on real K-homology and that relies ultimately on a classification up to approximate unitary equivalence of matricial deformations of spaces. A first objective of the project is to study invariants of C*-algebras that integrate the primitive spectrum along with homological invariants. This is naturally tied to the idea of deformation and the theory of continuous fields and will involve the study of KK-theory and E-theory parametrized by spaces that are not necessarily Hausdorff. A second objective is to investigate the existence of deformations of group C*-algebras into matrix algebras with the property that they detect the K-theory of the group C*-algebras. In addition, the principal investigator aims to provide formulas for computing invariants associated with these deformations. A third objective is to study invariants of simple C*-algebras that capture properties not detected by K-theory or tracial states. In particular, the project will investigate rigidity properties of simple C*-algebras.The physics of elementary particles has led to new mathematical theories in which numerical functions are replaced with infinite arrays of numbers or matrices. Matrices can be multiplied, but unlike numerical multiplication, the order of the factors is essential, so that A times B is not always equal to B times A. This property of matrix multiplication, called noncommutativity, is crucial for the interpretation of very complex phenomena in quantum mechanics that might initially appear completely counter-intuitive (e.g., Heisenberg's uncertainty principle). The passage from numbers to matrices often involves a process of deformation in which spatial structures are discretized in such a way that the possible twists of surfaces (or even more complicated objects) are captured in the form of numerical invariants. This project is concerned with the theory of such deformations and connects classical geometry with noncommutative geometry, a theory that proposes a reconstruction of the general concept of space that is better aligned with the principles of quantum physics than is traditional geometry. Moreover, the deformation theory that the principal investigator will study provides mathematical tools relevant to physical phenomena that are observed in certain special materials. These materials behave as insulators in their interiors, while permitting the movement of charges on their surfaces, due to exotic metallic states localized there. Such remarkable new materials, known as "topological insulators," have already had a considerable impact on condensed matter physics and are expected to find technological applications in magneto-electronics, an emerging field in which spin currents are used instead of charge currents, and in quantum computing.