Abstract: Symplectomorphism groups are one of the classical infinite-dimensional Lie groups that have been studied. Arnold's paper in 1966, where he used methods of infinite-dimensional Lie theory to study the hydrodynamics of a perfect incompressible fluid, has motivated intensive research in infinite-dimensional Lie theory. He showed that the geodesics on the group of volume preserving diffeomorphisms are essentially solutions of the Euler's equations. In a fundamental paper in 1970 Marsden and Ebin studied some infinite-dimensional groups in more details which included symplectomorphism groups. In this talk we study a special class of symplectomorphism groups that resemble compact Lie groups in a particular way. We see there is a similar notion of the so-called maximal tori in the symplectomorphism groups of toric manifolds. As a consequence we see there is an analogue of the Schur-Horn-Kostant convexity theorem in this infinite-dimensional setting. It also should be mentioned that these results are a generalization of results that were obtained by Bao-Ratiu 1997, Bloch-Flaschka-Ratiu 1993 and El-hadrami 1996 for special cases of toric manifolds.

Abstract: The goal of this talk is to introduce some of the most basic notions in the field of topology. We focus on the concept of a surface or 2-dimensional manifold. A surface is a mathematical abstraction of the familiar concept of a surface made of paper - like the surface of a sphere, the Mobius strip, and so on. We will spend time constructing these surfaces and then I will demonstrate the tools an algebraic topologist would use to classify all possible surfaces.