Lecture – Polytope Theory

I taught this TCC course on “Polytope Theory” starting from October 10th 2022 for 8 weeks. Due to the nature of TCC courses (being streamed to several universities) this was an online course taught via MS Teams.

Introduction

Polytope Theory is the study of (convex) polytopes, the generalization of polygons (2D) and polyhedra (3D) to general dimension. Besides their geometric nature as convex sets, polytopes possess a rich combinatorial structure, making them exceptionally accessible by combinatorial techniques. The study of polytopes reaches from the antiquity (starting from the Platonic solids), over the 19th/20th century (understanding polytopes in 3D and initializing the study of polytopes in dimension \(\ge 3\)), until today, where we understand that the richness of polytopal phenomena starts in dimension 4 and which led to findings such as universality. The subject has shown proximity to algebraic geometry, representation theory, analysis, optimization and many more.

Scope

In this course I aimed to give an overview of this very diverse subject and cover selected topics with focus towards research and open questions. I tried to cover the following, though not everything made it into the final course:

  • realization spaces and universality
  • Gale duality and enumeration of combinatorial types
  • face numbers, Dehn-Sommerville equations and the Upper Bound Theorem
  • reconstruction from the edge graph
  • spectral theory of polytopes, expansion and Izmestiev’s Theorem
  • geometry and combinatorics of 3-dimensional polytopes
  • inscribability and related geometric constraints
  • symmetry properties of polytopes
  • zonotopes
  • the polytope algebra

Prerequisites

Besides an elementary geometric understanding, the prerequisites are minimal:

  • linear algebra
  • elementary graph theory: basic definitions (sub-graph, vertex degree, bipartite graph), connectivity, handshaking lemma, planar graphs, etc
  • basic convex geometry: convex sets and cones, hyperplanes, some central theorems (Carathéodory’s theorem, hyperplane separation theorem) though we will give ample reminders for these
  • basic combinatorics: mostly some counting coefficients
  • basics of partially ordered sets, lattices, etc

Not strictly necessary, but a background in any of the following will provide motivation and can help the understanding at some points: linear/convex optimization, algebraic topology, real representation theory of finite groups.

Literature

Course notes

These course notes were created during each lecture. I am aware that there is the occasional typo, but overall they are correct.

  • Lecture 1 (10/10/2022)
    Introduction (overview, motivation, applications), definition of polytope, V-polytopes, H-polytopes, Minkowski-Weyl theorem
  • Lecture 2 (17/10/2022)
    Polar duals, faces and facets, face lattice, vertex figures
  • Lecture 3 (24/10/2022)
    3-polytopes, edge-graphs, Steintz’ theorem, Balinski’s theorem, simple/simplicial polytopes, neighborly polytopes, cyclic polytopes, Gale’s evenness criterion, Kalai’s simple way to tell a simple polytope from its graph
  • Lecture 4 (31/10/2022)
    Counting faces, Euler’s polyhedral formula + Euler Poincaré identity, Dehn-Sommerville equations, upper bound theorem, g-theorem
  • Lecture 5 (07/11/2022)
    Polytopal complexes, shellability, line/linear shelling, Schlegel diagrams, abstract polytopal/simplicial complexes
  • Lecture 6 (14/11/2022)
    Gale duality (linear/affine), spherical Gale diagrams, classifying small polytopes (d+1, d+2, d+3 vertices)
  •  Lecture 7 (21/11/2022) + GeoGebra files: addition, multiplication, squaring, golden ratio Realization spaces, centered realization space, (affine) Gale diagrams, point-line arrangements, Mnëv’s universality theorem, universality of 4-polytopes, non-rational polytopes

  • Lecture 8 (28/11/2022)
    Selection of research directions in polytope theory