The language of the lecture will be Czech or English depending on the audience. (If there is an attendee who does not understand Czech, the lecture will be in English).
Lecture | Contents | Literature |
---|---|---|
Lecture 1, Feb 22 | Motivation: Kneser graphs, Colorful Tverberg theorem. Basic notions from topology: Topological space (examples: standard topology on $\mathbb R^d$, Sierpinski space); open and closed sets; subspace; Hausdorff spaces; continuous maps; homeomorphism; the closure, the boundary and the interior of a set in a topological space; compact topological spaces. Examples: $\mathbb R$ is not compact, $[0,1]$ is compact (with a proof). A closed subset of a compact space is compact. A compact subset of a Hausdorff space is closed. | [KMS, Chap. 6.1, 6.2 and 6.3] (But these chapters contain more than what have been said during the lecture). |
Lecture 2, Feb 29 | Compact sets, continued: An image of a compact set is compact. Basis of a topology. Product of topological spaces. Tychonoff's theorem (without proof). Characterization of compact subsets of $\mathbb R^d$. A continuous map attains a minimum on a compact set. Connected topological space and a component of connectedness. Homotopy equivalence (started): Derfomation rectract; homotopy of maps; nullhomotopic maps. | Compactness [KMS, Chap. 6.3]; connectedness [KMS, Chap. 6.2] or wikipedia (for components); homotopy equivalence [M, Chap. 1.2] or [KMS, Chap. 6.4]. |
Lecture 3, Mar 7 | Homotopy equivalence of topological spaces. (Remark on equivalent characterization using deformation retracts.) Simplicial complexes: Geometric simplicial complexes. Polyhedron (underlying space) of a geometric simplicial complex; dimension. Triangulation of a topological space. Examples how to get $S^{n-1}$ and torus. Abstract simplicial complexes. Relation between the abstract and the geometric simplicial complexes. A simplicial map. | Literature: [M, Chap. 1.2--1.5] (or [KMS, Chap 6.4 and 6.7] but explained slightly differently) |
Lecture 4, Mar 14 | Simplicial map induces a continuous map on polyhedra. Proposition on properties of the induced map (without proof). Barycentric subdivision. Borsuk-Ulam theorem: six equivalent formulations. Proof of the equivialence of the formulations (so far (BU1a) $\Leftrightarrow$ (BU1b) $\Leftrightarrow$ (BU2a) and (BU2a) $\Rightarrow$ (BU2b)). | Literature: [M, Chap, 1.5--2.1] (only parts of the chapters). |
Lecture 5, Mar 21 | Finishing the proof of the equivalence from last time. One more equivalent formulation (LS-co), proof will be a homework. Applications: Kneser graphs. Kneser conjecture/Lovász theorem. Greene's proof of the theorem. Coloring hypergraphs and colorability defect. Doľnikov's theorem. (Proof will appear on the next lecture.) | Literature: [M, Chap 2.1, 3.3, 3.4] |
Lecture 6, Mar 28 | Proof of Doľnikov's theorem. Schrijver's graphs. Schrijver's theorem (on chromatic number of Schrijver's graphs). Moment curve and its basic properties. Gale's lemma. Proof of Schrijver's theorem via Gale's lemma and then a proof of Gale's lemma. | Literature: [M, Chap 3.4, 3.5, 1.6 (for the moment curve)] |
Lecture 7, Apr 4 | Ham sandwich theorem for measures. Explanation of the statement including a definition of finite Borel measure. Proof (modulo Dominated convergence theorem; few details skipped here and there). Ham sandwich theorem for point sets. (Few details in the proof skipped again.) | Literature: [M, Chap 3.1] |
Lecture 8, Apr 11 | Ham sandwich theorem for point sets in general position. Akiyama-Alon theorem on partitioning into rainbow $d$-tuples. Necklace theorem. Hobby-Rice theorem. (Sketched a proof via ham-sandwich; more complete proof directly from Borsuk-Ulam.) Remarks on equipartition theorems. Tucker's lemma. | Literature: [M, Chap 3.1, 3.2, 2.3] |
Lecture 9, Apr 14 | An equivalent version of Tucker's lemma (via simplicial maps). Equivalence of the Borsuk-Ulam theorem and Tucker's lemma. A proof of Tucker's lemma via `happy simplices'. The proof works only for special triangulations but this is enough to deduce the Borsuk-Ulam theorem which in turn implies Tucker's lemma in full generality. | Literature: [M, Chap 2.3] |