Srivatsav K Elayavalli
University of California, San Diego
January 7, 2025
Large N-limits of matrices and applications to operator algebras.: The study of matrix algebras is of fundamental importance in non-commutative analysis, especially in disciplines such as operator algebras. In recent decades, there has been a significant effort to uncover the analytic and probabilistic behavior of large N-limits of matrices. Voiculescu initiated his free probability theory with a crucial insight on the limiting joint spectral distribution of pairs of independent random Gaussian ensembles. This limiting distribution actually is concretely seen inside the von Neumann algebra associated to the free group on two generators. Since this result, there have been several deep contributions in this line of research, including theories of entropy, strong convergence, etc, with powerful applications to the study of operator algebras. In this talk I will describe my recent contributions to this area studying the so-called ultraproduct of matrix algebras, with applications to the internal structure of the free group von Neumann algebras, continuous model theory, and new considerations related to the famous Connes embedding problem which has been recently resolved using quantum complexity theory.
University of California, San Diego
January 7, 2025
Large N-limits of matrices and applications to operator algebras.: The study of matrix algebras is of fundamental importance in non-commutative analysis, especially in disciplines such as operator algebras. In recent decades, there has been a significant effort to uncover the analytic and probabilistic behavior of large N-limits of matrices. Voiculescu initiated his free probability theory with a crucial insight on the limiting joint spectral distribution of pairs of independent random Gaussian ensembles. This limiting distribution actually is concretely seen inside the von Neumann algebra associated to the free group on two generators. Since this result, there have been several deep contributions in this line of research, including theories of entropy, strong convergence, etc, with powerful applications to the study of operator algebras. In this talk I will describe my recent contributions to this area studying the so-called ultraproduct of matrix algebras, with applications to the internal structure of the free group von Neumann algebras, continuous model theory, and new considerations related to the famous Connes embedding problem which has been recently resolved using quantum complexity theory.