TCC course "Polytope Theory"
The TCC lecture on "Polytope Theory" will start October 10th 2022, will take place every Monday 10:00  12:00 on MS Teams, and will run for eight consecutive weeks.
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 posess 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 > 3), until today, where we understand that the richness of polytopal phenomena starts in dimension 4 and let to findings such as universality. The subject has shown proximity to algebraic geometry, representation theory, analysis and optimization.
Scope
In this course I aim to give an overview of this very diverse subject and to cover selected topics with focus towards research and open questions. I will try to cover the following:
 realization spaces and universality
 Gale duality and enumeration of combinatorial types
 face numbers, DehnSommerville equations and the Upper Bound Theorem
 reconstruction from the edgegraph
 spectral theory of polytopes, expansion and the Theorem of Izmestiev
 geometry and combinatorics of 3dimensional 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 (subgraph, 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 parts: linear/convex optimization, algebraic topology, real representation theory of finite groups.
Literature
 B. Grünbaum, "Convex Polytopes" (a bit older, but with all the essentials)
 G. Ziegler, "Lectures on Polytopes" (modern, more focus on combinatorial aspects in the later chapters)
 A. Brøndsted, "An Introduction to Convex Polytopes"
 I. Pak, "Lectures on Discrete and Polyhedral Geometry" (not specific to polytopes, but contains many neat proofs; freely available)
Course notes
Course notes will be uploaded shortly after each lecture.
 Lecture 1 (10/10/2022)
Introduction (overview, motivation, applications), definition of polytope, Vpolytopes, Hpolytopes, MinkowskiWeyl theorem  Lecture 2 (17/10/2022)
Polar duals, faces and facets, face lattice, vertex figures  Lecture 3 (24/10/2022)
3polytopes, edgegraphs, 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, DehnSommerville equations, upper bound theorem, gtheorem  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, pointline arrangements, Mnëv's universality theorem, universality of 4polytopes, nonrational polytopes

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