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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).
Research articles:
Lecture | Contents | Literature |
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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 18 | 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] |
Lecture 10, Apr 25 | Joins: Join of (abstract) simplicial complexes. Examples. Associativity (up to isomorphism), join of multiple copies of the same complex. (Example for 2-point discrete sets.) Quotient space (example). Topological join. Proposition on geometric join (some steps in the proof only sketched). Together with a homework this implies $|K * L| \cong |K|*|L|$ where $K$ and $L$ are simplicial complexes. | Literature: [M, Chap 4.2] |
Lecture 11, May 2 | Join of maps. Embeddability: Embedding. Generalizations of $K_5$ and $K_{3,3}$. The van Kampen--Flores theorem. Partial proof (only for a generalization of $K_{3,3}$) via a certain map that does not identify antipodal points and via the Borsuk-Ulam theorem. Motivation for homology. | Literature: [KMS, Chap 6.8] but the proof of the van Kampen--Flores theorem is different. (No Gauss maps were used at the lecture.) |
Lecture 12, May 9 | Homology: Oriented $k$-simplex. Group of $k$-chains (everything done over integers). Boundary operator and a lemma that it is well defined. Applying the boundary operator twice yields a trivial homomorphism. Groups of $k$-cycles and $k$-boundaries and $k$th (simplicial) homology group. Example: computation of the homology of a hollow triangle. Homology of a simplex (sketched). Homology does not depend on triangulation and is a homotopy invariant. (Only stated without proofs). A remark on singular homology. | Literature: [Mu, Chap 1.5]---this one is the closest to the lecture contents. Other sources are [H, Chap 2.1] but the homology is treated in more geenral setting for $\Delta$-complexes or [KMS, Chap 6.10] but here the homology is treated with $\mathbb{Z}_2$-coefficients and the upgrade to $\mathbb{Z}$ is only sketched. |
Lecture 13, May 16 | Using homology for showing that $\mathbb{R}^m$ and $\mathbb{R}^n$ are not homeomorphic if $m \neq n$ (modulo unproved results). The map $f_\#$. Lemma: It commutes with the boundary operator. (Proof is a homework). Corollary: $f_\#$ induces a homomorphism $f_*$ on homology. Theorem: homology is a functor. (A proof is straightforward, sketched.) Remarks on the induced maps in singular homology. Homological degree of a map: definition; theorem on properties of the degree (partially proved); computing the degree via preimages in $f_\#$. | Literature: [Mu, Chap 1.12]--again the closest one but only a part of the chapter is relevant. Other sources [H, Chap 2.1] (treatment for singular homology) or [KMS, Chap 6.10] again with $\mathbb{Z}_2$-coefficients. For the homological degree, the closest one is: [H, Chap 2.2]. |
Lecture 14, May 23 | The colorful Tverberg theorem: statement. Blagojević-Matschke-Ziegler theorem (optimal value of $t$ is $r$ if $r+1$ is prime). BMZ-collections and a theorem which implies an existence of the rainbow Tverberg paritition of BMZ-collections if $r$ is prime. (This theorem implies the Blagojević-Matschke-Ziegler theorem.) Sketch of the main ideas of the latter theorem (using the degree). Sketch of the definition of the fundamental group. | Literature: For colored Tverberg theorem: [BMZ] or [MTW] (But both are journal papers and contain much more than what was on the lecture an in a different language.) For fundametal group: [H, Chap 1.1] or [KMS, Chap 6.9]. |