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The Algebraic Topology: A Beginner's Course

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. 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.

Introduction


Lecture 01 - Introduction to Algebraic Topology
Lecture 02 - One-dimensional Objects
Lecture 03 - Homeomorphism and the Group Structure on a Circle
Lecture 04 - Two-dimensional Surfaces: the Sphere
Lecture 05 - More on the Sphere
Lecture 06 - Two-dimensional Objects: the Torus and Genus
Lecture 07 - Non-orientable Surfaces: the Mobius Band
Lecture 08 - The Klein Bottle and Projective Plane
Lecture 09 - Polyhedra and Euler's Formula
Lecture 10 - Applications of Euler's Formula and Graphs
Lecture 11 - More on Graphs and Euler's Formula
Lecture 12 - Rational Curvature, Winding and Turning
Lecture 13 - Duality for Polygons and the Fundamental Theorem of Algebra
Lecture 14 - More Applications of Winding Numbers
Lecture 15 - The Ham Sandwich Theorem and the Continuum
Lecture 16 - Rational Curvature of a Polytope
Lecture 17 - Rational Curvature of Polytopes and the Euler Number
Lecture 18 - Classification of Combinatorial Surfaces I
Lecture 19 - Classification of Combinatorial Surfaces II
Lecture 20 - An Algebraic ZIP Proof
Lecture 21 - The Geometry of Surfaces
Lecture 22 - The Two-holed Torus and 3-crosscaps Surface
Lecture 23 - Knots and Surfaces I
Lecture 24 - Knots and Surfaces II
Lecture 25 - The Fundamental Group
Lecture 26 - More on the Fundamental Group
Lecture 27 - Covering Spaces
Lecture 28 - Covering Spaces and 2-oriented Graphs
Lecture 29 - Covering Spaces and Fundamental Groups
Lecture 30 - Universal Covering Spaces
Lecture 31 - An Introduction to Homology
Lecture 32 - An Introduction to Homology (cont.)
Lecture 33 - Simplices and Simplicial Complexes
Lecture 34 - Computing Homology Groups
Lecture 35 - More Homology Computations
Lecture 36 - Delta Complexes, Betti Numbers and Torsion