I’ll present a metric construction over compact surfaces called Abelian

Differentials or Flat Structures (flat surfaces). The goal of this

talk will be to present my results regarding tight bounds on the

number of invariant components on a flat surface.

Let S be a compact surface with a complex structure. A flat structure on S

is an atlas of charts defined on S minus a finite set of points, such that

the transition functions are translations. This results in a locally

Euclidean surface with a directional field (vector field) globally

defined.

I’ll also define the moduli space of flat structures and the problem -

finding the best upper bounds in every stratum on the number of invariant

components (invariant to the vertical flow one can define on such

surfaces).

I’ll present the results, and, time allows, will prove some of them. the

talk and proofs will be geometric in nature.