Lecture time:  Wednesday 1416 

Lecture room:  023, Campus E 1.4 
Tutorial:  Wednesday 1617 
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 
ReExam:  TBA 
Course material: 
Using
the
BorsukUlam
Theorem. Most of the material is also available here.
Some parts of the book is available on this page. 
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, LovaszKneser 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, restatement as
affine function from simplex, topological Radon's theorem, proof for d=1, inductive proof via BorsukUlam [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, dsimplex, topological Radon's theorem via deleted joins. 
5.5 Notes 
ex9.pdf 
Nabil 
June 15 
Z_2simplicial complexes, Barycentric subdivision, order complexes of posets, Sarkaria's inequality, Proof of van KampenFlores theorem. 
5.7 Notes 
ex10.pdf 
Nabil 
June 22 
Plan outline of Lovasz's original proof of Kneser conjecture, hypergraph of minimal nonfaces, upperbounding \Delta(L0\L1) with the chromatic number of this hypergraph, Proof of VanKampen, nonplanarity 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 KampenFlores theorem.  5.6 
Weijia 

July 13 
Graph homomorphisms, Box complexes, lowerbound on chromatic number  5.9 
Shay 

July 20 
kconnectedness, kconnectedness implies high Z_2 index.  4.3, 5.3 
Vijay 

July 29 
Group actions on topological spaces, Z_p index, kfold deleted joins, Topological Tverberg's theorem, Sarkaria's inequality of Z_p index, Kneser hypergraphs for kfold joins, Colored Tverberg's theorem. 
Chapter 6 
Nabil 