Plytopes are higher-dimensional analogues of polygons and polyhedra. They have attracted scientists' attention starting from Plato's time, and remain an important object of study in different branches of contemporary mathematics.

Some of the possible things to study in the project would be

• Platonic solids and other Regular polytopes (including higher-dimensional ones)
• Convex polytopes
• Discrete reflection groups and Kaleidoscopes
• Fullerenes (see here)
• Special polytopes like associahedra, permutohedra, zonotopes, cyclic polytopes, etc.

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

Prerequisites: Algebra II.

Corequisite: Geometry III. Topology may also be useful.

Resources:

There are many nice books and articles on combinatorics of polytopes. You may start with the following:

• Alexander Barvinok, Combinatorics of polytopes.
• V.M.Buchstaber, T.E.Panov, Toric Topology.
• Branko Grünbaum, Convex polytopes (library link).
• Günter Ziegler, Lectures on polytopes (library link).
• Günter Ziegler, Polytopes: Extremal Examples and Combinatorial Parameters. Slides.

More reading on polytopes can be found here.