The invariant subspace problem (ISP) is a long-standing open problem that asks whether there is a nontrivial closed subspace that is invariant for a given bounded linear operator acting on some Hilbert space. This talk will briefly review some successes, failures, and potential future paths of the ISP. We will provide numerous examples of invariant subspaces for certain operators and highlight the breadth of the topic. We will point out some baby versions of the ISP that have remained open for a long time.

The design of this talk assumes a broad audience familiar with basic complex analysis and linear operators. We will also briefly review these concepts during the talk.