Topological Methods in Geometry (SS 2011)

Basic Information

Lecture time:	Wednesday 14-16
Lecture room:	023, Campus E 1.4
Tutorial:	Wednesday 16-17
Lecturers:	Saurabh Ray, Nabil H. Mustafa
Audience:	The course counts as computer science lecture (6 CP). The lecture will be given in English.
Schedule:	The first lecture will take place on Wenesday, April 13.
Exercises:	There will be weekly exercises.
Final Exam:	TBA
Re-Exam:	TBA
Course material:	Using the Borsuk-Ulam Theorem. Most of the material is also available here. Some parts of the book is available on this page.

Description

Topological methods are becoming increasingly useful in proving results in seemingly unrelated areas. There are many important results
in Discrete Geometry and Combinatorics for which only a topological proof is known. In this course we aim to give an introduction to the
use of topological methods in Discrete Geometry. We will also present some of the recent results in this area. Prior knowledge of
Topology or Discrete Geometry is not required.

Lectures

Date	Topic	Readings	Homework	Lecturer
Apr 13	Introduction - Intermediate value theorem, Sperner's lemma, Brouwer's Fixed Point theorem.	-	ex1.pdf	Saurabh
Apr 20	Basic concepts of Metric Spaces, Point Set Topology and Geometric Simplicial Complexes.	1.1,1.3	ex2.pdf	Saurabh
Apr 27	Abstract Simplicial Complexes, Simplicial Maps and Affine Extentions, Borsuk Ulam Theorem and its various equivalent forms.	1.4,1.5,2.1	ex3.pdf	Saurabh
May 04	Tucker's Lemma and its equivalence with Borsuk Ulam theorem, Combinatorial Proof of Tucker's Lemma.	2.3	ex4.pdf	Saurabh
May 11	Another proof of the Borsuk Ulam theorem.	2.4	ex5.pdf	Saurabh
May 18	Applications of Borsuk Ulam theorem - Ham Sandwich theorem, Lovasz-Kneser theorem, Dolnikov's theorem.	3.1,3.3,3.4	ex6.pdf	Saurabh
May 25	Deformation Retraction, Homotopy Equivalence, Joins of Topological spaces, Simplicial Complexes and Maps, Z_2 Spaces, Z_2 Maps, Z_2 Index.	1.2,4.1,4.2	ex7.pdf	Saurabh
June 1	Radon's theorem, geometric proof, re-statement as affine function from simplex, topological Radon's theorem, proof for d=1, inductive proof via Borsuk-Ulam [4], Z_2 spaces, Z_2 maps, Z_2 indices, transitivity, index of S^n, Z_2 action on product spaces, deleted products, proof of topological Radon's for d=1 using deleted products.	5.2, 5.3, 5.4 Notes	ex8.pdf	Nabil
June 8	Deleted Joins, Z_2 index of the deleted join of B^d, d-simplex, topological Radon's theorem via deleted joins.	5.5 Notes	ex9.pdf	Nabil
June 15	Z_2-simplicial complexes, Barycentric subdivision, order complexes of posets, Sarkaria's inequality, Proof of van Kampen-Flores theorem.	5.7 Notes	ex10.pdf	Nabil
June 22	Plan outline of Lovasz's original proof of Kneser conjecture, hypergraph of minimal nonfaces, upper-bounding \Delta(L0\L1) with the chromatic number of this hypergraph, Proof of Van-Kampen, non-planarity of K{3,3}, and another proof of Kneser conjecture.	5.8, 5.9 Notes	ex11.pdf	Nabil
June 29	Lecture cancelled, as Nabil is travelling.			Nabil
July 8	Bier Spheres, Proof of von Kampen-Flores theorem.	5.6		Weijia
July 13	Graph homomorphisms, Box complexes, lower-bound on chromatic number	5.9		Shay
July 20	k-connectedness, k-connectedness implies high Z_2 index.	4.3, 5.3		Vijay
July 29	Group actions on topological spaces, Z_p index, k-fold deleted joins, Topological Tverberg's theorem, Sarkaria's inequality of Z_p index, Kneser hypergraphs for k-fold joins, Colored Tverberg's theorem.	Chapter 6		Nabil

References

[1] Notes on compactness.
[2] Some Basic Geometry. See also Chapter 1 of these notes. and Chapter 2 of these notes.
[3] Video Tutorial on Measure Theory.
[4] Elementary deduction of topological Radon's theorem from Borsuk-Ulam.

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