Project IV 2015-2016

Curves on surfaces

Anna Felikson and Pavel Tumarkin

Description:

Curves on surfaces are elementary objects encoding a lot of topological and geometric properties of surfaces.

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...

Anna Felikson will supervise in Michaelmas and Pavel Tumarkin in Epiphany.

----

Prerequisites: Algebra II and either Topology III or Geometry III

----

Resources:

Many of the basic notions for the project are introduced in the course

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

Some of the more advanced topics are described in the following books:

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"

Other topics are studied in numerous expository and research papers, you may start from ones listed here.

email: Anna Felikson, Pavel Tumarkin

----