Project III, 2020-2021

Associahedron

Anna Felikson

Description:

An associahedron is an (n-2)-dimensional convex polytope in which each vertex corresponds to a way of correctly inserting opening and closing parentheses in a word of n letters and the edges correspond to single application of the associativity rule. The number of its vertices C_n is an element of the sequence of Catalan numbers, one of the most frequently used sequences of natural numbers appearing as answer to numerous enumerative problems, completely different from the first glance. An associahedron and its generalisations are also central objects in modern theory of cluster algebras connecting numerous fields in mathematics and theoretical physics.

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Prerequisites: Algebra II, Geometry III or Topology III

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Resources:

The wikipedia page for Catalan numbers contain many examples of problems where these numbers arise. You could start your reading from "Catalan numbers" by Tom Davis. The following elementary book includes Catalan numbers in a series of other types of distinguished number sequences:

Conway and Guy (1996) The Book of Numbers. New York: Copernicus.

You can find the connections to combinatorics of polytopes, triangulated surfaces, reflection groups and many other domains in mathematics in

The wikipedia page for Associahedron;
G. Ziegler, Lecture on the Associahedron;
S. Fomin, N. Reading, Root systems and generalized assosiahedra.
Editors: F. Müller-Hoissen, J. Pallo and J. Stasheff, Associahedra, Tamari Lattices, and Related Structures.

You can find some more references here.

Finally, you can go for Catalan numbers page by Igor Pak, where you can find unlimited quantity of selected and sorted by topics information on Catalan numbers, associahedra and related topics.

email: Anna Felikson

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