A projective structure on a Riemann surface is determined by a holomorphic quadratic differential via the Schwarzian differential equation. The monodromy of this equation, or equivalently the holonomy of the projective structure, defines a representation of the fundamental group of the surface to $PSL(2,\mathbb{C})$, determined up to conjugation. In this talk, we shall describe some recent work concerning the space of projective structures on a punctured Riemann surface, corresponding to meromorphic quadratic differentials. In particular, we shall describe a geometric parametrization of that space which is an analogue of Thurston’s grafting theorem for closed surfaces, and discuss results about the monodromy map to the moduli space of framed $PSL(2,\mathbb{C})$-representations, which was first defined by Allegretti-Bridgeland. This represents work in several papers, some written in collaboration with Gianluca Faraco, Spandan Ghosh and Mahan Mj.

Venue: TBA
Time: TBA

TBA

Venue: Online (TBA)
Time: TBA

We show that a homotopy equivalence between two non-compact orientable surfaces is homotopic to a homeomorphism if and only if it preserves the Goldman bracket, provided our surfaces are neither the plane nor the punctured plane.

Venue: 112, APC Ray LHC, IISER Kolkata
Time: 2:30-3:30 pm
Slides

The notion of topological rigidity and Haken manifolds were introduced and several related interesting results were stated. Also, a proof sketch of Nielsen's theorem on topological rigidity of compact surfaces with boundary was presented.

Venue: 211, APC Ray LHC, IISER Kolkata
Time: 3-4 pm
Slides