Aspects of Sofic Entropy and Algebraic Actions

动力系统的数学理论本身涉及某些物理系统的状态（例如，房间内某种气体的含量，物种的数量，建筑物内的温度）。把“时间”本身变得更加抽象，用任何其他离散的对称性系统（比如说某个对象的对称性）来代替它，这是很有用的，也是很自然的（这样的对称性形成了所谓的离散群）。在一个非常混沌的物理系统中，人们无法有效地预测它的未来状态。这一事实产生了一个重要的量，即系统的熵，它衡量了系统的不可预测性。熵最初是直接在信息论的背景下定义的，但事实证明，如果人们从统计力学形式主义的角度来看待它，熵理论的范围就会扩大到包含一个非常大的类别，称为“sofic群”。通过这个新的透镜，熵成为物理系统有多少有限近似的度量。一个无穷过程如何被一个有限过程近似的一般问题是科学探索的一个基本问题，而sofic群类包括许多来自几何和数论的自然例子。该项目还揭示了与物理学的联系，这是其统计力学和信息理论起源所固有的。此外，该主题已被证明与数学的不同领域有着有趣的联系，包括：泛函分析，遍历理论，组合数学和算子代数。许多联系是主要研究者研究的直接结果。 该项目是正在进行的加强和理解这种联系的活动的一部分，围绕三个主要问题。第一个是了解轨道等价的后果，积极或完全积极的熵的行动，不顺从的群体。例如，主要研究者证明了具有完全正熵的作用是强遍历的，而正熵作用不是弱紧的。他试图证明具有竞争正熵的行为是完全遍历的，正如奇凡和约阿纳所定义的那样，这将进一步表明这些行为所表现出的伯努利行为。对这个目标的部分兴趣在于，它将不顺从群体作用的熵理论与顺从群体作用的熵研究形成了鲜明的对比，在顺从群体作用的熵研究中，不可能导出熵的任何轨道等价结果。完全正熵的这些结果适用于代数作用（即，作用于紧群的自同构），因为主要研究者已经可以证明与可逆卷积算子相关的代数作用具有完全正熵。第二个问题是有关调查的熵理论的代数行动的等价关系，这是联合工作与刘易斯鲍恩和迪伦艾雷。这一部分的项目的一个目的是推广的主要研究者的结果熵和Fuglede-Kadison决定因素从组的情况下，等价关系的情况下。最后，通过定义算子代数上作用的sofic熵，进一步加强了动力学与算子代数之间的联系。

The mathematical theory of dynamical systems concerns itself with states of certain physical systems (e.g., the amount of a certain gas inside a room, the population of a species, the temperature inside a building) as they evolve in time. It happens to be useful and natural to make "time" itself more abstract and replace it with any other discrete system of symmetries, say of some object (such symmetries form what is a called a discrete group). In the case of a very chaotic physical system, one cannot effectively predict its future state. This fact gives rise to an important quantity, the entropy of the system, that measures how unpredictable it is. Entropy was originally defined directly within an information theoretic context, but it turns out that, if one views it instead from the perspective of a statistical mechanics formalism, the scope of entropy theory is increased to encompass a very large class of groups known as "sofic groups." Seen through this new lens, entropy becomes a measurement of how many finitary approximations the physical system has. The general question of how well an infinitary process can be approximated by a finitary one is a fundamental question of scientific inquiry, and the class of sofic groups includes many natural examples coming from geometry and number theory. This project also sheds light on the connection to physics that is inherent in its statistical mechanical and information theoretic origins. Moreover, the subject has proven to have interesting links to diverse areas of mathematics, including the following: functional analysis, ergodic theory, combinatorics, and operator algebras. Many of the links are direct consequences of the principal investigator's research. The project is part of ongoing activity to strengthen and understand such connections.This project revolves around three main problems. The first is to understand the orbit equivalence consequences of positive or complete positive entropy for actions of nonamenable groups. For example, the principal investigator has shown that actions with complete positive entropy are strongly ergodic and that positive entropy actions are not weakly compact. He seeks to prove that actions with compete positive entropy are solidly ergodic, as defined by Chifan and Ioana, which will give further indication of the Bernoulli-like behavior that these actions exhibit. Part of the interest in this objective is that it places the theory of entropy for actions of nonamenable groups in stark contrast to the study of entropy for actions of amenable groups, where it is impossible to derive any orbit equivalence consequences of entropy. These consequences of complete positive entropy apply to algebraic actions (i.e., actions on compact groups by automorphisms), for the principal investigator can already demonstrate that algebraic actions related to invertible convolution operators have complete positive entropy. A second problem is concerned with investigating the entropy theory of algebraic actions of equivalence relations, which is joint work with Lewis Bowen and Dylan Airey. One aim of this part of the project is to generalize the principal investigator's results on entropy and Fuglede-Kadison determinants from the group case to the equivalence relation case. Lastly, the principal investigator will further the connections between dynamics and operator algebras by defining sofic entropy for actions on operator algebras.