Operator Algebras and Applications
算子代数及其应用
基本信息
- 批准号:RGPIN-2017-06719
- 负责人:
- 金额:$ 2.7万
- 依托单位:
- 依托单位国家:加拿大
- 项目类别:Discovery Grants Program - Individual
- 财政年份:2020
- 资助国家:加拿大
- 起止时间:2020-01-01 至 2021-12-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
The objective of the proposed research is to confirm a striking phenomenon. I have discovered that a large class of quite complicated mathematical objects (virtually all naturally arising norm-closed algebras of operators in Hilbert space, including those of interest in mathematical physics, and in other branches of mathematics in which complex systems are considered, such as the theory of dynamical systems and the theory of foliations) can apparently be described in terms of very simple data---often now called the Elliott invariant.
This was unexpected, but the evidence is overwhelming. The discovery has far-reaching implications, both for the class of objects involved (the class of amenable C*-algebras), and, it appears likely, for the related areas of mathematics and physics in which these objects appear. (For instance, it is pertinent to the study of the orbits of dynamical systems.)
The evidence for this classification picture is indeed strong, but much work is still needed.
It may be too early to evaluate this discovery fully. Earlier classification schemes, well recognized to be important---indeed, mine should only be placed beside them for conceptual comparison!---, are the Linnaeus classification of biological species, the Mendeleev classification of chemical elements, and the classification of subatomic particles in elementary particle physics. (More recently, the Linnaeus classification has virtually given way to that of Watson and Crick!) A mathematical analogue, perhaps, is the classification of finite simple groups. (While the justification of this takes several thousand pages, that much has already been written concerning my conjecture.)
My classification scheme (based necessarily on the mathematical notion of a functor, owing to the complexity of the objects considered) has quite new features. The objects mentioned are so complicated that it is not possible to label them with a simple label, in such a way that the label is the same for two objects that are essentially the same. One can only label the objects with other, simpler, objects, in such a way that if two objects are essentially the same, then also the labelling objects are essentially the same ("isomorphic").
I did this forty years ago for an important class of objects (AF C*-algebras---a special kind of amenable C*-algebra). It was fifteen years before I realized that further steps were possible. In the twenty-five years since then, many people have contributed to what has become known as the Elliott program (for the classification of amenable C*-algebras).
Great progress has taken place recently (concerning the simple, unital, especially well behaved case), but development continues to snowball. The next five years will be exciting for the Elliott program. (The non-simple, the non-unital, and the not especially well behaved cases---the last in a quite precise sense---have revealed tantalizing challenges.)
这项研究的目的是证实一个惊人的现象。我发现有一大类相当复杂的数学对象(实际上,希尔伯特空间中所有自然产生的算子的范数闭代数,包括数学物理中感兴趣的代数,以及考虑复杂系统的其他数学分支,例如动力系统理论和叶理理论)显然可以用非常简单的数据来描述-现在常被称为艾略特不变量
这是出乎意料的,但证据是压倒性的。这一发现有着深远的意义,不仅对所涉及的对象(顺从C*-代数类),而且很可能对这些对象出现的数学和物理学的相关领域产生影响。(For例如,它与研究动力系统的轨道有关。)
这一分类图的证据确实很有力,但仍需做大量工作。
对这一发现进行全面评价可能还为时过早。早期的分类方案,被公认为是重要的-事实上,我的分类方案只应该放在它们旁边进行概念上的比较!是生物物种的林奈分类,化学元素的门捷列夫分类,以及基本粒子物理学中的亚原子粒子分类。(More最近,林奈分类法实际上已经让位于沃森和克里克的分类法!)一个数学上的类比,也许是有限单群的分类。(虽然证明这一点需要几千页,但关于我的猜想已经写了很多。
我的分类方案(由于所考虑对象的复杂性,必须基于函子的数学概念)具有相当新的特征。所提到的对象是如此复杂,以至于不可能用简单的标签来标记它们,以这种方式,标签对于本质上相同的两个对象是相同的。人们只能用其他更简单的对象来标记这些对象,以这样的方式,如果两个对象本质上相同,那么标记对象也本质上相同(“同构”)。
40年前,我对一类重要的对象(AF C*-代数--一种特殊的顺从C*-代数)做了这件事。15年后,我才意识到进一步的步骤是可能的。在此后的25年里,许多人为后来被称为埃利奥特计划(用于对顺从C*-代数进行分类)的计划做出了贡献。
最近已经取得了很大的进展(关于简单的,单一的,特别是表现良好的情况),但发展继续滚雪球。未来五年将是令人兴奋的埃利奥特计划。(The非简单的、非单位的和不是特别好表现的情况--最后一个在相当精确的意义上--揭示了诱人的挑战。
项目成果
期刊论文数量(0)
专著数量(0)
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会议论文数量(0)
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Elliott, George其他文献
Elliott, George的其他文献
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{{ truncateString('Elliott, George', 18)}}的其他基金
Operator Algebras and Applications
算子代数及其应用
- 批准号:
RGPIN-2017-06719 - 财政年份:2021
- 资助金额:
$ 2.7万 - 项目类别:
Discovery Grants Program - Individual
Operator Algebras and Applications
算子代数及其应用
- 批准号:
RGPIN-2017-06719 - 财政年份:2019
- 资助金额:
$ 2.7万 - 项目类别:
Discovery Grants Program - Individual
Operator Algebras and Applications
算子代数及其应用
- 批准号:
RGPIN-2017-06719 - 财政年份:2018
- 资助金额:
$ 2.7万 - 项目类别:
Discovery Grants Program - Individual
Operator Algebras and Applications
算子代数及其应用
- 批准号:
RGPIN-2017-06719 - 财政年份:2017
- 资助金额:
$ 2.7万 - 项目类别:
Discovery Grants Program - Individual
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