Seminar in Algebraic Topology
Seminar in algebraic topology
This seminar will give foundational background in topology and algebraic topology, with an eye to interesting applications. Participants will hold lectures on the basics of topology and their applications. As prerequisite, one should have successfully completed linear algebra 1 and 2 and analysis 1 and 2.
Instructors: Marc Levine and Daniel Harrer
Schedule: Mondays, 16-18 Uhr, WSC-N-U-4.05. The first meeting is on 17.10.2016.
Interested students who have a schedule conflict should send an email to Marc Levine.
Preliminary list of topics
What is a space?: topological spaces, open and closed subsets, continuous mappings, examples.
How does one construct spaces?: Subspaces, quotient spaces, product spaces, CW-complexes, compact spaces.
Simplicial spaces: Graphs and trees, witness complexes, cloud complexes, Vietoris-Rips complexes, phylogenetic trees, Cech-complexes, nerve complex.
Homology: Chain complexes, Euler characteristic, configuration spaces, tame topology, scissors equivalence, the Euler calculus.
Manifolds and topology: curvature and the Gauss-Bonnet theorem, vector fields, fixed point index and the Poincare-Hopf theorem.
Program
Lecture 1. Marc Levine. Basic notions of topology Lecture Notes
Lecture 2. Dario Antolini. Some basic constructions: union products, quotients, CW complex. An introduction to elementary homotopy theory. Lecture Notes
Lecture 3. Daniel Barth. Simplicial complexes and polytopes. Lecture Notes.
Lecture 4. Daniel Harrer. Some elementary homological algebra
Lecture 5. Marc Levine. Homology and cohomology of simplicial complexes and spaces. Lecture Notes
Lecture 6. ???.
Lecture 7. ???.