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Borsuk–Ulam Theorem and an Introduction to Topological Combinatorics

Fathi, Reza | 2024

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 57425 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Jafari, Amir
  7. Abstract:
  8. Topological combinatorics is an almost novel branch in mathematics whose dawn began to shine originally from Lovász studies at late 70's. The main purpose of the pursued subjects in this branch is to apply topological methods in order to conclude various combinatorial results based on the famous Borsuk-Ulam's theorem which itself is a very deep statement in algebraic topology and has many equivalent versions. For instance, one of these versions states that there is no continuous map f:S^n⟶S^(n-1) which behaves antipodally. In this thesis, we intend to take a considerably long journey in homology theory to pave the way toward a complete proof of this theorem and some of its generalizations. Along the way, we will be consecutively referring to some useful objects called simplicial complexes and on this account we will construct a bridge between two universes of continuous and discrete mathematics. Then later on, in the second half of this thesis, we want to show some combinatorial applications of these versions of Borsuk-Ulam theorem that one might categorize them into two general classes of calculating the chromatic number of Kneser hyper-graphs and partition problems. In this regard from the structural point of view, our approaches toward mentioned items will be to investigate them from the simplest levels to their most up-to-date forms
  9. Keywords:
  10. Borsuk–Ulam Theorem ; Kneser Hypergraph ; Partitions ; Topological Combinatorics