Course Descriptions

Spring 2023

Feng Luo

Subtitle:

Topics in discrete geometry and combinatorial topology  

Course Description:

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:

1. Koebe circle packing theorem of for planar graphs
2. Tutt embedding theorem of planar graphs
3. Brunn-Minkowski inequality
4. Minkowski theorem on convex polytopes 
5. Cauchy rigidity theorem for convex polytopes 
6. Bourgain’s embedding of finite metric spaces 
7. Johnson-Lindenstrass dimension reduction theorem
8. Johns’ ellipsoid theorem 
9. Hadwiger's theorem 
10. Duality of convex polytopes
11. Schramm’s rigidity theorem for infinite circle packing 
12. Discrete Schwarz lemma 
13. Discrete Laplacian and its applications 

The topics on combinatorial topology include a brief introduction to simplicial complexes and 

14. Sperner's lemma, Brouwer fixed point theorem and applications
15. Tucker's lemm,  Borsuk–Ulam theorem and applications
16. Chromatic and Tutt polynomials of graphs 
17. Jones polynomial of knots and an introduction topological quantum field theory (TQFT)
18. Turaev-Viro invariant of 3-manifolds and the volume conjecture  

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Prerequisites:

Real analysis, complex analysis and linear algebra.