The principal investigator will investigate problems in dynamics and geometry in the framework of operator algebras. Sample topics include: 1. Study the relation between entropy of principal algebraic actions of countable groups and Fuglede-Kadison determinant; 2. Develop an operator algebraic approach to entropy of actions of sofic groups; 3. Investigate combinatorial independence in measurable and topological dynamics; 4. Develop a convex analysis approach to the study of noncommutative Choquet boundary; 5. Study quantum isometry of compact quantum metric spaces.

In order to get a new mathematical tool to study quantum mechanics, von Neumann introduced operator algebras. The theory of operator algebras has grown into a huge exciting area of modern mathematics, and is related to many other areas of physics and mathematics. Ergodic theory and dynamics arose from studying the long-term behavior of complicated processes. This project will deepen and broaden connections between geometry, dynamics, and operator algebras, and has application in physics. The proposed study of entropy of algebraic actions and actions of sofic groups will enhance our understanding of complicated symmetries. The study of the new interrelation between combinatorics and dynamics has application to the local theory of Banach spaces. The development of noncommutative Choquet boundary has application in operator theory. The study of metric geometry will provide a concrete mathematical foundation for certain statements in the theoretical high-energy physics literature.

Project Report

This project discovers and investigates new connection between operator algebras and dynamical systems. There are two main threads, intertwining with each other. The first is the development of sofic entropy theory. Together with David Kerr, the principal investigator extended Lewis Bowen's sofic measure entropy to full generality, including both measure entropy and topological entropy. This was done first using operator algberaic method, and then later using dynamical system method. These invariants were shown to extend the classical ones in amenable group case. The relation between positive entropy and combinatorial independence was also extended from amenable group case to sofic group case. The principal investigator also extended Gromov and Lindenstrauss-Weiss' mean dimension theory from amenable group case to sofic group case. The second thread is the study of algebraic actions, using operator algebraic methods. Together with Andreas Thom, the principal investigator found that for amenable groups the entropy for algeraic actions corresponds to the L2-torsion in L2-invariants theory. This includes Deninger's conjecture about relation between entropy and determinants as a special case, and proves Lueck's conjecture about vanishing L2-torsion as a consequence. Deninger's conjecture was also established for some residually finite group case by the principal investigator with David Kerr and Lewis Bowen. Together with Bingbing Liang, the principal investigator found that mean dimension for algebraic actions corresponds to von Neumann-Lueck rank in L2-invariants theory, first in the amenable group case, and later in the general sofic group case. The principal investigator, together with Nhan-Phu Chung, also studied expansive algebraic ations using operator algebraic method and combinatorial independence, and established relation between entropy properties and homoclinic points for a large class of groups. Together with Jesse Peterson and Klaus Schmidt, the principal investigator also used operator algebraic methods to study ergodicity of principal algebraic actions. In order to get a new mathematical tool to study quantum mechanics, von Neumann introduced operator algebras. The theory of operator algebras has grown into a huge exciting area of modern mathematics, and is related to many other areas of physics and mathematics. Ergodic theory and dynamics arose from studying the long-term behavior of complicated processes. This project discovers new connections between operator algebras and dynamics, which have yields applications to both fields. Entropy is a numerical invariant describing how complicated or chaotic a system is, and is one of the most important invariants in dynamical systems. For a long time people thought that entropy could be defined only for amenable symmetries. The discoveries in this project has helped the sofic entropy theory to get fully founded and become a fast-developing field in the last several years. The algebraic actions provide many interesting examples, including ones from physics and number theory. Earlier work on algebraic actions concentrate on algebraic actions of abelian groups, where the tools of commutative algebra is avaible. This project uses operator algebraic methods to study algebraic actions instead, enabling us to undertand many algebraic actions of nonabelian groups.

Agency
National Science Foundation (NSF)
Institute
Division of Mathematical Sciences (DMS)
Type
Standard Grant (Standard)
Application #
1001625
Program Officer
Bruce P. Palka
Project Start
Project End
Budget Start
2010-09-01
Budget End
2014-08-31
Support Year
Fiscal Year
2010
Total Cost
$119,764
Indirect Cost
Name
Suny at Buffalo
Department
Type
DUNS #
City
Buffalo
State
NY
Country
United States
Zip Code
14228