The Algebraic Topology: A Beginner's Course (UNSW). Taught by Professor N. J. Wildberger, this course provides an introduction to algebraic topology, with emphasis on visualization, geometric intuition and simplified computations.
999 years, 10 months
36
Algebraic topology is one of the most dynamic and exciting areas of 20th century mathematics, with its roots in the work of Riemann, Klein and Poincare in the latter half of the 19th century. This course introduces a wide range of novel objects: the sphere, torus, projective plane, knots, Klein bottle, the circle, polytopes, curves in a way that disregards many of the unessential features, and only retains the essence of the shapes of spaces. And it also has some novel features, including Conway's ZIP proof of the classification of surfaces, a rational form of turn angles and curvature, an emphasis on the importance of the rational line as the model of the continuum, and a healthy desire to keep things simple and physical.
Course Currilcum
- Lecture 01 – Introduction to Algebraic Topology Unlimited
- Lecture 02 – One-dimensional Objects Unlimited
- Lecture 03 – Homeomorphism and the Group Structure on a Circle Unlimited
- Lecture 04 – Two-dimensional Surfaces: the Sphere Unlimited
- Lecture 05 – More on the Sphere Unlimited
- Lecture 06 – Two-dimensional Objects: the Torus and Genus Unlimited
- Lecture 07 – Non-orientable Surfaces: the Mobius Band Unlimited
- Lecture 08 – The Klein Bottle and Projective Plane Unlimited
- Lecture 09 – Polyhedra and Euler’s Formula Unlimited
- Lecture 10 – Applications of Euler’s Formula and Graphs Unlimited
- Lecture 11 – More on Graphs and Euler’s Formula Unlimited
- Lecture 12 – Rational Curvature, Winding and Turning Unlimited
- Lecture 13 – Duality for Polygons and the Fundamental Theorem of Algebra Unlimited
- Lecture 14 – More Applications of Winding Numbers Unlimited
- Lecture 15 – The Ham Sandwich Theorem and the Continuum Unlimited
- Lecture 16 – Rational Curvature of a Polytope Unlimited
- Lecture 17 – Rational Curvature of Polytopes and the Euler Number Unlimited
- Lecture 18 – Classification of Combinatorial Surfaces I Unlimited
- Lecture 19 – Classification of Combinatorial Surfaces II Unlimited
- Lecture 20 – An Algebraic ZIP Proof Unlimited
- Lecture 21 – The Geometry of Surfaces Unlimited
- Lecture 22 – The Two-holed Torus and 3-crosscaps Surface Unlimited
- Lecture 23 – Knots and Surfaces I Unlimited
- Lecture 24 – Knots and Surfaces II Unlimited
- Lecture 25 – The Fundamental Group Unlimited
- Lecture 26 – More on the Fundamental Group Unlimited
- Lecture 27 – Covering Spaces Unlimited
- Lecture 28 – Covering Spaces and 2-oriented Graphs Unlimited
- Lecture 29 – Covering Spaces and Fundamental Groups Unlimited
- Lecture 30 – Universal Covering Spaces Unlimited
- Lecture 31 – An Introduction to Homology Unlimited
- Lecture 32 – An Introduction to Homology (cont.) Unlimited
- Lecture 33 – Simplices and Simplicial Complexes Unlimited
- Lecture 34 – Computing Homology Groups Unlimited
- Lecture 35 – More Homology Computations Unlimited
- Lecture 36 – Delta Complexes, Betti Numbers and Torsion Unlimited
