The 2-D Peskin problem describes coupled motion of a 1-D closed elastic string and the ambient Stokes flow in the plane. Its global well-posedness was well-established when the initial string configuration is close to an equilibrium, which is an evenly-stretched circular configuration. In other words, initially, the shape of the string needs to be almost circular, and the string is almost evenly-stretched. In this talk, we present some recent progress on pursuing global solutions for a wider class of initial datum. We will show that certain geometric quantities of the string satisfy extremum principles and decay estimates. As a result, we can prove global well-posedness when the initial data satisfies a medium-size geometric condition on the string shape, while no assumption on the size of stretching is needed. This talk is based on a joint work with Dongyi Wei.