Aspects of Sofic Entropy and Algebraic Actions

动力系统的数学理论关注的是某些物理系统的状态（例如，房间内某种气体的数量，物种的数量，建筑物内的温度）随着时间的推移而演变。把“时间”本身变得更加抽象，用任何其他离散的对称系统来代替它，比如某个物体（这种对称形成了所谓的离散群），这恰好是有用和自然的。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。熵最初是在信息论的背景下直接定义的，但事实证明，如果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括一个非常大的群，称为“sofic群”。从这个新的角度来看，熵变成了物理系统有多少有限近似值的度量。有限过程如何很好地近似无穷大过程的一般问题是科学研究的一个基本问题，而sofic群的类别包括许多来自几何和数论的自然例子。该项目还阐明了统计力学和信息论起源中固有的与物理学的联系。此外，这门学科已被证明与数学的各个领域有有趣的联系，包括：泛函分析、遍历理论、组合学和算子代数。许多联系是主要研究者研究的直接结果。该项目是加强和了解这种联系的持续活动的一部分。这个项目围绕着三个主要问题。首先是理解正或完全正熵对不可服从群的作用的轨道等价后果。例如，首席研究员已经证明，具有完全正熵的作用是强遍历的，而正熵的作用不是弱紧致的。正如Chifan和Ioana所定义的那样，他试图证明具有竞争性正熵的行为是完全遍历的，这将进一步表明这些行为所表现出的伯努利行为。对这一目标的部分兴趣在于，它将不可服从群作用的熵理论与可服从群作用的熵研究形成鲜明对比，在可服从群作用的熵研究中，不可能推导出熵的任何轨道等效结果。完全正熵的这些结果适用于代数作用（即，紧群上的自同构作用），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。第二个问题是研究等价关系的代数作用的熵理论，这是与Lewis Bowen和Dylan Airey共同完成的工作。这部分项目的目的之一是将主要研究者关于熵和Fuglede-Kadison行列式的结果从群情况推广到等价关系情况。最后，首席研究员将通过定义算子代数上的动作的sofic熵来进一步探讨动力学和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.