The investigator will try to answer questions about algebras of bounded operators on Hilbert space; in particular, these questions involve finite von Neumann algebras, which are algebras of operators on Hilbert space that are closed under taking adjoints and strong--operator limits, and on which there exist tracial linear functionals. The main question the investigator proposes to work on is known as Connes'' embedding problem, which asks (in one of many equivalent formulations): can all operators in a von Neumann algebra that has a bounded trace be approximated by matrices over the complex numbers? This question is fundamental and its resolution would have profound ramifications on our understanding of operator algebras and operator spaces. One important goal is to seek a group lacking certain analogous approximation properties (i.e., a non-sofic group), both as it''s own goal and as a guide for particular von Neumann algebras to investigate vis-a-vis Connes'' embedding problem. In other directions, the investigator proposes to use the "practical Schubert calculus" to learn more about finite von Neumann algebras, and to study certain natural questions in finite von Neumann algebras, such as the single commutator question and the Schur-Horn question, using diverse methods. Group theory is the study of fundamental symmetries that are found in nearly all branches of mathematics and natural sciences. On the other hand, von Neumann algebras were invented in the 1930''s and 1940''s by Murray and von Neumann; they consist of operators on infinite dimensional space and are related to the formalism of quantum mechanics, which motivated their invention. In modern mathematics, the study of these von Neumann algebras has important connections to several other areas of mathematics, such as group theory and dynamical systems. The investigator will try to answer related, fundamental problems in group theory and von Neumann algebras, namely, "can certain infinite objects always be approximated by finite ones?" Answers to these questions would have broad impact; they are related to many other open problems about groups and, respectively, operators on infinite dimensional space. The investigator will also study certain other natural questions about von Neumann algebras. The research effort will furthermore support the educational efforts of the principal investigator: most directly in teaching graduate students the prerequisites to understanding current research problems and then to guide them to vital and important research problems and but also in educating undergraduate students in a way that shows the utility and beauty of mathematics and teaches them essential mathematical skills and methods of mathematical (and logical) reasoning.
