The study of infinite-dimensional Lie groups is pioneered by Arnold in a celebrated paper. Ebin and Marsden wrote Arnold's idea in a rigorous language. The theory of infinite-dimensional Lie groups is somehow unsatisfactory, although it has been an active area of research in the past 40 years. An analog of a maximal torus has been studied previously by Bao-Raiu for the volume preserving diffeomorphisms of finite cylinder which motivated a later work by El Hadrami on complex projective spaces. Later on, Bloch-Flaschka-Ratiu motivated by Bao-Ratiu's result obtained an analog of Schur-Horn-Kostant convexity theorem. In this talk we will see that there is an analog of maximal torus in the symplectomorphism group of toric manifolds. We also study the existence of an analog of Schur-Horn-Kostant convexity theorem.

When a compact torus (S^1)^m acts on a connected manifold M of dimension n, the dimension of each orbit should be greater than or equal to 2m-n. In case there is an orbit of dimension 2m-n, we can say that the action of T^m on M is ``maximal" in some sense. In this talk, we describe compact connected complex manifolds with ``maximal" torus actions.