FRG: Collaborative Research: Noncommutative dimension theories

近似定义了我们的世界。例如，此屏幕上的字母具有平滑的曲线和弯曲。但放大后，你所看到的都是正方形：像素。像素越小(即，近似越精细)，曲线看起来就越平滑。再加上三种颜色，我们突然就能看到一道彩虹。从某种意义上说，科学就是改进和改进对现实的近似。以牛顿物理学为例。它在中等规模下效果很好，但当东西太大或太小时就会崩溃。爱因斯坦的相对论和量子力学在这些尺度上的效果要好得多。而这些理论需要复杂的数学知识。算符代数作为量子力学的框架应运而生。多年来，许多经典理论被扩展到这种非对易的背景下：几何学、拓扑学、概率论等等。这个重点研究小组的项目解决了算子代数及其在其他数学领域的类似物中的几个悬而未决的问题。PI将带头开展一项国际努力，以利用算子代数与其他领域如动力学、测度论、粗几何和K-理论之间的最新联系。具体地说，PI应推动核维度和渐近维度之间的类比，这两个概念通过近似定义，并包含大量实例，以解决K理论问题，如宇宙系数定理、Baum-Connes和Farrell-Jones猜想。

Approximation defines our world. For example, the letters on this screen have smooth curves and bends. But zoom in and all you see are squares: pixels. The smaller the pixels (i.e., the finer the approximation), the smoother a curve looks. Add three colors and suddenly we can approximate a rainbow. In a sense, science is all about refining and improving approximations to reality. Take Newtonian physics, for example. It works great at medium scales, but breaks down when things are too big or too small. Einstein's relativity and quantum mechanics work much better at those scales. And these theories require sophisticated mathematics. Operator algebras arose as a framework for quantum mechanics. Over the years many classical theories were extended to this noncommutative context: geometry, topology, probability and more. This focused research group project addresses several outstanding questions in operator algebras and their analogies in other areas of mathematics.The PIs will spearhead an international effort to capitalize on recent connections between operator algebras and other areas such as dynamics, measure theory, coarse geometry and K-theory. Specifically, the PIs shall push analogies between nuclear dimension and asymptotic dimension, two notions defined via approximation and encompassing a huge swath of examples, to address K-theoretic questions such as the Universal Coefficient Theorem and the Baum-Connes and Farrell-Jones conjectures.