Instructors: Paul Arnaud Songhafouo Tsopméné and Donald Stanley
Course Description: A course intended to introduce students to Algebraic Topology, which is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homotopy equivalence (a weaker homeomorphism notion).
In this course, we will see that Algebraic Topology is not only useful for studying topological spaces. Among other things, it is used to solve efficiently many problems from other areas of mathematics such as (Differential) Geometry, Group Theory, Algebra,…
There are some nice applications (inside and outside mathematics) of the Algebraic Topology that we will mention. One has, for instance, an application to Data Analysis (for example, 3D image analysis). One can also use Algebraic Topology as a way of integrating local data about Sensor Networks into global information. For example, you may want to determine whether there are any holes in your sensor coverage.
Textbook: Algebraic Topology, by Allen Hatcher. This is free and available here.
Content: Topics include
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- A Brief Recall of the General Topology: Definition of a Topological Space, Subspaces,
Continuous Functions, Homeomorphisms, Compactness, Connectedness. - Homotopy Type: Definition and Examples, Contractible Spaces.
- The Fundamental Group: Paths, Homotopy of Paths, Loops and Definition of the Fundamental Group, The Fundamental Group of the Circle, Induced Homomorphism, Applications of the Fundamental Group, The Van Kampen Theorem.
- Homology: Simplicial Homology, Singular Homology, Homotopy Invariance, Exact Sequences, Applications of the Homology (Existence of continuous nonzero vector fields on the n-sphere, Which groups act freely on the n-sphere when n is even?).
- A Brief Recall of the General Topology: Definition of a Topological Space, Subspaces,
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