Below is a list of possibly useful material for the summer related to surfaces and closed curves on surfaces.

A very basic introduction to topology of curves and surfaces

An introduction to hyperbolic geometry

A basic introduction to the Goldman bracket

Relations between word length, hyperbolic length and self-intersection numbers of curves on surfaces

Rachel Zhang’s paper clearly explains (among other things) the concept of linked pairs.

•Here is a link that implements the algorithm to compute the Goldman bracket of two free homotopy classes of closed curves on a surface with boundary. (It “kind of” works for closed surfaces, although the final answer is not in its most reduced form)

•This article, by Tony Phillips, explains ideas about curvature. You can use it to compute the curvature of crocheted hyperbolic planes.

•Mostly surfaces by Richard Schwartz. You can also find here for an earlier draft of the book, provided by the author. (Advised reading is sections 1.1 to 1.5, 2.1 to 2.7, the whole chapters 3, 4 and 5. If you have extra time, Chapter 10, 6 and 7, in that order).

•Beginning topology, Sue E. Godman.

• A primer on mapping class groups by Benson Farb and Dan Margalit

•Topology of surfaces by Christine Kinsey, the first five chapters,

•Low-dimensional geometry, from Euclidean surfaces to hyperbolic knots, by Francis Bonahon, the first five chapters. (this book is probably for more mature readers than Kinsey’s).

Anderson's book "Hyperbolic geometry”,

Mumford-Series-Wright's "Indra's Pearls" .

Ratcliffe's book "Hyperbolic manifolds" is another text on hyperbolic geometry too.