Topics in discrete geometry and combinatorial topology
This course covers many topics in discrete geometry and combinatorial topology.
In the discrete geometry part, we begin with some of the basics on convexity and decompositions (separation theorem, Caratheodory, Helly, Minkowski’s lattice theorems and Voronoi cells). More advanced topics are:
Koebe circle packing theorem of for planar graphs
Tutt embedding theorem of planar graphs
Brunn-Minkowski inequality
Minkowski theorem on convex polytopes
Cauchy rigidity theorem for convex polytopes
Bourgain’s embedding of finite metric spaces
Johnson-Lindenstrass dimension reduction theorem
Johns’ ellipsoid theorem
Hadwiger's theorem
Duality of convex polytopes
Schramm’s rigidity theorem for infinite circle packing
Discrete Schwarz lemma
Discrete Laplacian and its applications
The topics on combinatorial topology include a brief introduction to simplicial complexes and
Sperner's lemma, Brouwer fixed point theorem and applications
Tucker's lemm, Borsuk–Ulam theorem and applications
Chromatic and Tutt polynomials of graphs
Jones polynomial of knots and an introduction topological quantum field theory (TQFT)
Turaev-Viro invariant of 3-manifolds and the volume conjecture
None
Real analysis, complex analysis and linear algebra.