Operator Algebras and Applications

算子代数及其应用

• 批准号: RGPIN-2017-06719
• 负责人: Elliott, George
• 金额: $ 2.7万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=686986
• 关键词: Operator; Algebras; Applications

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.)

拟议研究的目的是证实一个惊人的现象。我发现，一大类相当复杂的数学对象（几乎所有自然产生的希尔伯特空间中的算子的规范闭代数，包括那些对数学物理感兴趣的代数，以及考虑复杂系统的其他数学分支，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——现在通常称为艾略特不变量。这是出乎意料的，但证据是压倒性的。这一发现具有深远的意义，无论是对所涉及的对象类别（可调节的C*代数类别），还是对这些对象出现的相关数学和物理领域。（例如，它与动力系统轨道的研究有关。）这个分类图的证据确实很充分，但仍需要做很多工作。现在全面评价这一发现可能还为时过早。早期的分类方案，被公认为是重要的——实际上，我的分类方案只应该放在它们旁边进行概念比较！——，是林奈对生物物种的分类，门捷列夫对化学元素的分类，以及基本粒子物理学中亚原子粒子的分类。（最近，林奈的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，以至于不可能用一个简单的标签来标记它们，以使两个本质上相同的对象的标签相同。人们只能用其他更简单的对象来标记这些对象，如果两个对象本质上相同，那么标记的对象也本质上相同（“同构”）。我在四十年前为一类重要的对象（AF C*-代数——一种特殊的可服从C*-代数）做了这个。15年后，我才意识到有可能采取进一步的措施。从那以后的25年里，许多人都为艾略特计划（用于可调节C*-代数的分类）做出了贡献。***最近已经取得了很大的进展（关于简单的、统一的、特别是表现良好的案例），但发展仍在滚雪球。接下来的五年对艾略特项目来说将是激动人心的。（非简单、非单一和表现不佳的情况——确切地说，是最后一种情况——揭示了诱人的挑战。）