Dynamical Systems Seminar

Recently, Fisher and Melnick classified \(SL(n,\mathbb R)\)-actions on \(n\)-dimensional manifolds for \(n \ge 3\). In this talk, we generalize this result by classifying smooth or real-analytic \(SL(n,\mathbb R)\)-actions on \(m\)-dimensional manifolds for \(3 \le n \le m \le 2n-3\). This work is motivated by the Zimmer program and is central to it, as Lie group actions restrict to their lattice actions. This classification relies on the linearization of \(SL(n, \mathbb R)\)-actions when there is a global fixed point. The analytic case was proved by Guillemin-Sternberg and Kushinirenko. We discuss the smooth case which is ongoing joint work with Insung Park.