Algebraic Topology

In this video lecture, Prof. N.J. Wildberger introduces the two basic one-dimensional objects: the line and circle. The latter has quite a few different manifestations: as a usual Euclidean circle, as the projective line of one-dimensional subspaces of a two-dimensional space, as a polygon, or as a space of orbits of a translation group on the line. 

In this video lecture, Prof. N.J. Wildberger gives the basic definition of homeomorphism between two topological spaces, and explains why the line and circle are not homeomorphic.

Then, we introduce the group structure on a circle, or in fact a general conic, in a novel way, following Lemmermeyer and as explained by S. Shirali. This gives a gentle intro to the definition of a group. It also uses Pascal's theorem in an interesting way, so we give some background on projective geometry.

After the plane, the two-dimensional sphere is the most important surface, and in this video lecture we give a number of ways in which it appears. As a Euclidean sphere, we relate it to stereographic projection and the inversive plane.

In this video lecture, Prof. N.J. Wildberger continues the discussion of the sphere, relating inversive geometry on the plane to the more fundamental inversive geometry of the sphere, introducing the Riemann sphere model of the complex plane with a point at infinity. Then he discusses the sphere as the projective line over the (rational!) complex numbers.

In this video lecture, Prof. N.J. Wildberger introduces some surfaces: the cylinder, the torus or doughnut, and the n-holed torus. We define the genus of a surface in terms of maximal number of disjoint curves that do not disconnect it. We discuss how the plane covers the cylinder and the torus, and the associated group of translations.

In this video lecture, Prof. N.J. Wildberger talks about Non-orientable Surfaces: The Mobius Band. 

A surface is non-orientable if there is no consistent notion of right handed versus left handed on it. The simplest example is the Mobius band, a twisted strip with one side, and one edge. An important deformation gives what we call a crosscap.

In this video lecture, Prof. N.J. Wildberger talks about the Klein bottle and projective plane. 

The Klein bottle and the projective plane are the basic non-orientable surfaces. The Klein bottle, obtained by gluing together two Mobius bands, is similar in some ways to the torus, and is something of a curiosity. The projective plane, obtained by gluing a disk to a Mobius band, is one of the most fundamental of all mathematical objects. Of all the surfaces, it most closely resembles the sphere.

In this video lecture, Prof. N.J. Wildberger investigates the five Platonic solids: tetrahedron, cube, octohedron, icosahedron and dodecahedron. Euler's formula relates the number of vertices, edges and faces. We give a proof using a triangulation argument and the flow down a sphere.

In this video lecture, Prof. N.J. Wildberger introduces an important re-scaling of curvature, using the natural geometric unit rather than radians or degrees. We call this the turn-angle, or tangle, and use it to describe polygons, convex and otherwise. We also introduce winding numbers and the turning number of a planar curve.

In this video lecture, Prof. N.J. Wildberger gives a description of a variant to the proof of the Classification theorem for two dimensional combinatorial surfaces, due to John Conway and called the ZIP proof. Our approach to this is somewhat algebraic. We think about spheres with holes that are then zipped together rather than polygonal pieces which are glued together.

In this video lecture, Prof. N.J. Wildberger talks about the geometry of surfaces. 

This lecture relates the two dimensional surfaces we have just classified with the three classical geometries- Euclidean, spherical and hyperbolic. Our approach to these geometries is non-standard (the usual formulations are in fact deeply flawed) and we concentrate on isometries, avoiding distance and angle formulations. In particular we introduce hyperbolic geometry via inversions in circles---the Beltrami Poincare disk model.

In this video lecture, Prof. N.J. Wildberger 

In this video lecture, Prof. N.J. Wildberger 

In this video lecture, Prof. N.J. Wildberger talks about knots and surfaces. 

In this video lecture, Prof. N.J. Wildberger talks about knots and surfaces. 

In the 1930's H. Siefert showed that any knot can be viewed as the boundary of an orientable surface with boundary, and gave a relatively simple procedure for explicitly constructing such `Seifert surfaces'. We show the algorithm, exhibit it for the trefoil and the square knot, and then discuss Euler numbers for surfaces with boundaries.

In this video lecture, Prof. N.J. Wildberger introduces the fundamental group of a surface. We begin by discussing when two paths on a surface are homotopic, then defining multiplication of paths, and then multiplication of equivalence classes or types of loops based at a fixed point of the surface.The fundamental groups of the disk and circle are described.

In this video lecture, Prof. N.J. Wildberger continues the discussion of the fundamental group of a space. We show that the homotopy classes of closed loops from a fixed point on a space actually form a group. And the important cases of the torus and the projective plane are studied in some detail.