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

Tuesday, 9:0011:00, room 009

Thursday, 9:0011:00, room 009

Friday, 11:0013:00, room 009
Office hours: Tuesday, 11:0013:00, office 302
Bibliography

A. Hatcher: Algebraic Topology

Edelsbrunner, Harer, Computational Topology

M. J. Greenberg, J. R. Harper, Algebraic Topology, A first course

R. Bott, L.W. Tu, Differential forms in algebraic topology

Lecture notes by E. Arbarello and R. Salvati Manni, Chapter 5

MacLane, Homology

J. Milnor, Morse Theory,

G. Carlsson, Topology and Data,

F. Chazal, B. Michele, An introduction to Topological Data Analysis: fundamental and practival aspects for data scientist, arxiv

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, MayerVietoris, excision, Kunneth formula. Examples and applications. VietorisRips 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.
CWcomplexes, Morse functions, persistent homology of manifolds associated to a Morse function, spectral sequences associated to persistent homology. LeraySerre 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

Section 1, page 7, exercises from1 to 5 ; Tuesday 8th October

Section 2 page 16, exercises 1 to 3 ; Tuesday 15th October

Section 3 page 2829, from 1 to 15 except 7 ; Tuesday 29th October
Possible topics for the final presentation:

Spectral sequences and homology of fiber bundles. For instance, you can focus on Serre spectral sequence
J.P. Serre, Homologie Singulière des esapces fibres
Hatcher, Spectral sequences in algebraic topology

Group cohomology.
Kenneth S. Brown, Cohomology of groups, (The definition of group homology is in chapter 2)
J.P. Serre, Galois Cohomolgy (for the first read, you can try to read directly chapter 2, replacing profinite with finite)
S. Maclane, Homology, Chapter 4
A.I. Kostrikin, I. R. Shafarevich (Eds.), Algebra V, Homological Algebra, Chapter 2, especially 2.7

Morse theory
J. Milnor, Morse Theory

Topological Data Analysis
see for instance these two papers arxiv and arxiv
Lecture Schedule:

27/9 Introduction

1/10 Complexes of modules. Definition of the homology of topological spaces.

3/10 Categories and functuors. Homology of a point. Zero degree homology of a pathconnected space

4/10 Direct sum of complexes; homology and decomposition in pathconnected components. Five lemma

8/10 Homotopies of morphisms of complexes; homotopy invariance of the homology of topological spaces. Exercises.

10/10 Snake lemma

11/10 Relative homology of a pair and the associated long exact sequence

15/10 Relative homology and homology of the quotient: statement and applications. Homology of the spheres.

17/10 More about the homology of the sphere.Satement of the barycentric subdivision. MayerVietoris exact sequence. Hopf fibration.

18/10 Homology of complex projective spaces. Exercises.

22/10 Excission. Proof of the theorem of the homology of the quotient. Exercises.

24/10 Symplicial complexes and symplicial homology.

25/10 NO CLASS

29/10 Naturality of the snake lemma. Isomorphism of symplicial and singular cohomology. Homology of the real projective plane