Cluster algebras were introduced by Fomin and Zelevinsky in 2002 and got a growing wave of interest due to numerous connections to various fields of mathematics and mathematical physics.

The project will be centred around various geometric aspects of cluster algebras and may include investigation of

• oriented graphs and their mutations;
• triangulated surfaces, flips of arcs;
• hyperbolic surfaces and their Teichmuller spaces;
• combinatorics of snake graphs and their perfect matchings;
• root systems, associahedra, finite and infinite reflection groups;
• Ptolemy relations (similar to the one in Ptolemy theorem);
• skein relations (similar to ones for knots)
• frieze patterns;
• laminations on surfaces;
• intersection numbers of curves on surfaces.

Prerequisites: Algebra II, Geometry III or Topology III.

Resources:

You can get an idea about cluster algebras and some of connected topics from the following texts:

• Lauren Williams, Cluster algebras: an introduction.
• Robert Marsh, Lecture notes on cluster algebras.
• Philipp Lampe, Cluster algebras.

You can also take a look at the pictures in the following lecture notes:

• Sergey Fomin and Nathan Reading, Root systems and generalised associahedra.
• Dylan Thurston, The Geometry and Algebra of Curves on Surfaces, notes by Qiaochu Yuan.

For more information on cluster algebras see Cluster Algebras Portal by Segey Fomin.