C*-algebras: structure, symmetries, and equilibrium states

C*-动力学系统最初被引入来模拟量子力学系统。可观测量由C*-代数的元素表示，在C*-代数上给出时间演化（一组自同构）。除了时间演化，其他群体行为（例如对称性）也可以通过这个想法的推广来建模。动力学或几何信息可以包含在称为交叉积代数的代数结构中，该代数结构已被证明在超出其原始物理动机的许多抽象情况下都是相关的。拟议的研究旨在开发方法来分析C*-动力系统的对称性和平衡态，并使用这些方法研究从数论，动力系统和几何群论等数学领域产生的问题。有证据表明，受物理系统启发的结构和技术可以为长期存在的问题带来新的启示，并可以产生相关的问题和新的例子来测试现有的知识。其中一种现象是具有自发破缺的相变，其中系统在低温下可以处于各种不对称构型的平衡，而在高温下只有一个平衡态。这方面的日常例子是水冻结成冰和铁磁体在室温下的自发磁化，其通过加热而失去。

C*-dynamical systems were initially introduced to model quantum mechanical systems. The observable magnitudes are represented by elements of a C*-algebra, on which a time evolution (a group of automorphisms) is given. In addition to time evolutions, other group actions (e.g. symmetries) can also be modelled via a generalization of this idea. The dynamical or geometric information can be subsumed in an algebraic construction, called the crossed product algebra, which has proven relevant in a number of abstract situations beyond its original physical motivation. The proposed research aims to develop methods to analyze symmetries and equilibrium states of C*-dynamical systems, and to use these methods in the study of questions arising from various areas of mathematics like number theory, dynamical systems, and geometric group theory. There is evidence that constructions and techniques inspired by physical systems can shed new light onto long-standing problems and can generate relevant questions and new examples on which to test the existing knowledge. One such phenomenon is that of phase transition with spontaneous symmetry-breaking, in which a system can be in equilibrium in a variety of asymmetric configurations at low temperature, while having only one equilibrium state at high temperature. Everyday examples of this are the freezing of water into ice and the spontaneous magnetization of a ferromagnet at room temperature which is lost through heating.