Description

How do the curves intersect? How many (different) closed curves can you place in a surface without intersection? Which groups of transformations are acting on the curves? How many lengths of the curves you need to know to recover a precise metric structure of the surface?

Studying the curves from topological, combinatorial or geometrical point of view one can reach the following (simple, advanced or very advanced) areas:

• fundamental group of surface, intersection numbers, invariants of curves;
• pants decompositions, curve complex;
• Dehn twists, Mapping Class Group, Teichmüller space;
• Heegaard splittings of 3-manifolds, invariants of 3-manifolds;
• closed geodesics on hyperbolic surfaces;
• billiards;
• laminations, train tracks, ideal triangulations;
• skein relations, cluster algebras...

Prerequisites

Resources

• "Introduction to Modern Topology and Geometry" by Anatole Katok, Alexey Sossinsky ---- (in particular, Chapter 5)

• R.C.Penner, J.L.Harer, "Combinatorics of train tracks", Chapter 1
• R.C.Penner, "Decorated Teichmüller theory"
• B.Farb, D.Margalit, "A primer on mapping class groups"