FRG: Collaborative Research: Noncommutative dimension theories

近似定义了我们的世界。例如，这个屏幕上的字母有平滑的曲线和弯曲。但是放大后你看到的都是正方形：像素。像素越小（即，近似越精细），曲线看起来越平滑。把三种颜色加在一起，我们就能突然看到彩虹。从某种意义上说，科学就是提炼和改进接近现实的东西。以牛顿物理学为例。它在中等规模下工作得很好，但当事情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工作得更好。这些理论需要复杂的数学。算子代数作为量子力学的框架而出现。多年来，许多经典理论被扩展到这个非交换的背景下：几何，拓扑，概率和更多。 这个重点研究小组项目解决了算子代数及其在其他数学领域的类比中的几个突出问题。PI将带头国际努力，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。具体来说，PI将推动核维数和渐近维数之间的类比，这两个概念通过近似定义，并包含大量的例子，以解决K理论问题，如泛系数定理和鲍姆-康纳斯和法雷尔-琼斯定理。

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.