Joint invariant subspaces on the polydisc and a commutant lifting theorem on the unit ball

In this talk we will discuss two central problems in multivariable operator theory: (1) joint invariant subspaces of the tuple of shift operators on the Hardy space over the unit polydisc, and (2) a commutant lifting theorem of the tuple of shift operator on certain analytic reproducing kernel Hilbert spaces over the unit ball. If time permits, we will also talk about Nevanlinna-Pick interpolation theorem and a relevant factorization result for multipliers on reproducing kernel Hilbert spaces over the unit ball.