1. Elementary topology: topological spaces, homeomorphisms, connectivity, homotopy, homotopy equivalence. 2. Simplicial complexes: polyhedrons, abstract simplicial complex, Czech complex, Vietoris-Rips complex. 3. Simplicial homology: chain complex, boundary operator, homology. Invariance, Euler characteristics. Exact sequences. Betti numbers. 4. Algorithm of computation of Betti numbers. Important cases: algorithms of computation of zero Betti number; algorithm of computation of Betti numbers of homology of 2D and 3D simplicial complexes. 5.Morse theory, discrete Morse theory, Morse-Smale complex. 6. Persistent topology: filtration, persistent homology groups, persistent diagram, barcodes. Algorithm of computation of persistent diagrams. 7. Stability of persistent doagrams. 8. Various applications of computational topology.
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The aim of the course is to introduce students to the modern field, located at the intersection of topology and computer science. In many applied problems (in geophysics, chemistry, medicine, biology, etc.) there is a need for a qualitative analysis related to the global behavior of the objects included in the system. Topology methods turn out to be an indispensable tool. At the same time, the problem of creating computationally efficient algorithms for calculating topological characteristics becomes very important and this is one of the main objectives of course.
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