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

C*动力学系统最初是用来模拟量子力学系统的。可观测的量用C*-代数的元素表示，在C*-代数上给出了时间演化（一组自同构）。除了时间进化之外，其他群体行为（例如对称）也可以通过这一思想的概括来建模。动态或几何信息可以包含在称为交叉积代数的代数结构中，它已被证明在许多抽象情况下与其原始物理动机无关。本研究旨在发展分析C*-动力系统的对称性和平衡态的方法，并将这些方法用于研究数论、动力系统和几何群论等数学领域的问题。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提供新的思路，并可以产生相关的问题和新的例子，以检验现有的知识。其中一种现象是具有自发对称性破缺的相变，在这种现象中，系统可以在低温下以多种不对称构型处于平衡状态，而在高温下只有一种平衡状态。这方面的日常例子是水冻结成冰，铁磁体在室温下自发磁化，通过加热失去。

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.