FRG: Collaborative Research: Noncommutative dimension theories

FRG：协作研究：非交换维度理论

基本信息

批准号：
1564401
负责人：
Nathanial Brown
金额：
$ 52.72万
依托单位：
Pennsylvania State Univ University Park
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2016
资助国家：
美国
起止时间：
2016-06-01 至 2020-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1564401&HistoricalAwards=false
关键词：
FRG Collaborative Research Noncommutative dimension

项目摘要

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.

近似定义了我们的世界。例如，此屏幕上的字母具有光滑的曲线和弯曲。但是放大，您看到的只是正方形：像素。像素越小（即近似值越大），曲线看起来更顺畅。添加三种颜色，突然我们可以近似彩虹。从某种意义上说，科学是关于完善和改善现实的近似值。以牛顿物理学为例。它在中等尺度上效果很好，但是当事情太大或太小时会分解。爱因斯坦的相对性和量子力学在这些量表上的工作效果要好得多。这些理论需要复杂的数学。运算符代数作为量子力学的框架出现。多年来，许多古典理论都扩展到了这种非交通性背景：几何，拓扑，概率等。这个重点的研究小组项目解决了操作员代数及其在其他数学领域中的类比中的几个杰出问题。具体而言，PI应在核维度和渐近维度之间推动类比，这两个概念是通过近似来定义的，并包含大量示例，以解决kseatient的问题，例如通用系数定理和Baum-Connes和Baum-Connes和Farrell-Jones的想法。