Algebraic Topology (GE450)

Accademic year: 2019/2020 - First Semester
Lecturer: Giulio Codogni

 

Il corso è momentaneamente sospeso. La prossima lezione sarà martedì 19 novembre.

Schedule

Office hours: Tuesday, 11:00-13:00, office 302

Bibliography

  1. A. Hatcher: Algebraic Topology
  2. Edelsbrunner, Harer,  Computational Topology
  3. M. J. Greenberg, J. R. Harper, Algebraic Topology, A first course
  4. R. Bott, L.W. Tu, Differential forms in algebraic topology
  5. Lecture notes by E. Arbarello and R. Salvati Manni, Chapter 5
  6. MacLane, Homology
  7. J. Milnor, Morse Theory,
  8. G. Carlsson, Topology and Data,
  9. F. Chazal, B. Michele, An introduction to Topological Data Analysis: fundamental and practival aspects for data scientist, arxiv
  10. Fast and Accurate Tumor Segmentation of Histology Images using Persistent Homology and Deep Convolutional Features, arxiv

This course is partially inspired by the course "Topics in Geometry - Computational Topology", by G. Székelyhidi

Prelimary plan of course:

Omology of topological spaces, with focus on simplicial complexes and manifolds. Homotopy invariance, Mayer-Vietoris, excision, Kunneth formula. Examples and applications. Vietoris-Rips and Chech complexes associated to a data set, persistent homology,  application to topologicla data analysis.

According to time and to interests of the class, we will also cover some of the following topics.

CW-complexes, Morse functions, persistent homology of manifolds associated to a Morse function, spectral sequences associated to persistent homology. Leray-Serre spectral sequences. Application to topological data analysis.

Cohomology, Poincaré duality, differential forms, De Rahm cohomology.

Language: English or Italian, to be decided together with the students.

Exam: Homeworks, presentation of at least one execrcise at the blackboard, and final presentaton in the form of a seminar.

Homeworks Exercises are taken from the Lecture notes by E. Arbarello and R. Salvati Manni, Chapter 5

Possible topics for the final presentation:

Lecture Schedule:

  1. 27/9 Introduction
  2. 1/10 Complexes of modules. Definition of the homology of topological spaces.
  3. 3/10 Categories and functuors. Homology of a point. Zero degree homology of a path-connected space
  4. 4/10 Direct sum of complexes; homology and decomposition in path-connected components. Five lemma
  5. 8/10 Homotopies of morphisms of complexes; homotopy invariance of the homology of topological spaces. Exercises.
  6. 10/10 Snake lemma
  7. 11/10 Relative homology of a pair and the associated long exact sequence
  8. 15/10 Relative homology and homology of the quotient: statement and applications. Homology of the spheres.
  9. 17/10 More about the homology of the sphere.Satement of the barycentric subdivision. Mayer-Vietoris exact sequence. Hopf fibration.
  10. 18/10 Homology of complex projective spaces. Exercises.
  11. 22/10 Excission. Proof of the theorem of the homology of the quotient. Exercises.
  12. 24/10 Symplicial complexes and symplicial homology.
  13. 25/10  NO CLASS
  14. 29/10 Naturality of the snake lemma. Isomorphism of symplicial and singular cohomology. Homology of the real projective plane