Project IV 2014-2015

Curves on surfaces

Anna Felikson

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;
laminations, train tracks, ideal triangulations;
skein relations, cluster algebras...

How far would you get (and WHERE would you get) depends only on your curiosity and persistence!

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Prerequisites: Topology III and Algebra II would be useful, but not required.

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Resources:

Many of the basic notions for the project are introduced in

"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"
S.K.Lando, A.K.Zvonkin, "Graphs on surfaces"

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

There are also lecture notes for the graduate course "Curves on Surfaces" by Dylan Thurston ---- (not always an easy reading!)

email: Anna Felikson

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