  # Algebraic Topology

## Video Lectures

Displaying all 27 video lectures.
 Lecture 1 Play Video Introduction to Algebraic TopologyIn this video lecture, Prof. N.J. Wildberger introduces some of the topics of the course and three problems. Lecture 2 Play Video One-dimensional Objects 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. Lecture 3 Play Video Homeomorphism and the Group Structure on a Circle 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. Lecture 4 Play Video Two-dimensional Surfaces: The Sphere 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. Lecture 5 Play Video More on the Sphere 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. Lecture 6 Play Video Two-dimensional Objects: The Torus and Genus 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. Lecture 7 Play Video Non-orientable Surfaces: The Mobius Band 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. Lecture 8 Play Video The Klein Bottle and Projective Plane 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. Lecture 9 Play Video Polyhedra and Euler's Formula 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. Lecture 10 Play Video Applications of Euler's Formula and GraphsIn this video lecture, Prof. N.J. Wildberger talks about the applications of Euler's formula and graphs.  We use Euler's formula to show that there are at most 5 Platonic, or regular, solids. We discuss other types of polyhedra, including deltahedra (made of equilateral triangles) and Schafli's generalizations to higher dimensions. In particular in 4 dimensions there is the 120-cell, the 600-cell and the 24-cell. Finally we state a version of Euler's formula valid for planar graphs. Lecture 11 Play Video More on Graphs and Euler's Formula In this video lecture, Prof. N.J. Wildberger discusses applications of Euler's formula to various planar situations, in particular to planar graphs, including complete and complete bipartite graphs, the Five neighbours theorem, the Six colouring theorem, and to Pick's formula, which lets us compute the area of an integral polygonal figure by counting lattice points inside and on the boundary. Lecture 12 Play Video Rational Curvature, Winding and Turning 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. Lecture 13 Play Video Duality for Polygons and the Fundamental Theorem of AlgebraIn this video lecture, Prof. N.J. Wildberger talks about duality for polygons and the fundamental theorem of algebra.  We define the dual of a polygon in the plane with respect to a fixed origin and unit circle. This duality is related to the notion of the dual of a cone.Then we give a purely rational formulation of the Fundamental Theorem of Algebra, and a proof which keeps track of the winding number of the image of concentric circles about the origin. Lecture 14 Play Video More Applications of Winding NumbersIn this video lecture, Prof. N.J. Wildberger talks about more applications of winding numbers.  We define the degree of a function from the circle to the circle, and use that to show that there is no retraction from the disk to the circle, the Brouwer fixed point theorem, and a Lemma of Borsuk. Lecture 15 Play Video The Ham Sandwich Theorem and the ContinuumIn this video lecture, Prof. N.J. Wildberger talks about the Borsuk Ulam theorem: a continuous map from the sphere to the plane takes equal values for some pair of antipodal points. This is then used to prove the Ham Sandwich theorem (you can slice a sandwich with three parts (bread, ham, bread) with a straight planar cut s so that each slice is cut cut in two. Also, we give an application to the continuum: the plane is different (not homeomorphic) 3 dimensional space. Lecture 16 Play Video Rational Curvature of a PolytopeIn this video lecture, Prof. N.J. Wildberger talks about rational curvature of a polytope. We use our new normalization of angle called turn-angle, or "tangle" to define the curvature of a polygon P at a vertex A. This number is obtained by studying the opposite cone at the vertex A, whose faces are perpendicular to the edges of P meeting at A. A classical theorem of Harriot on spherical triangles is important. Lecture 17 Play Video Rational Curvature of Polytopes and the Euler NumberIn this video lecture, Prof. N.J. Wildberger talks about rational curvature of polytopes and the Euler number. We show that the total curvature of a polyhedron is equal to its Euler number. This only works with the rational formulation of curvature, using an analog of the turn angle suitable for the 2 dimensional sphere. Lecture 18 Play Video Classification of Combinatorial Surfaces (Part I)In this video lecture, Prof. N.J. Wildberger introduces the central theorem in Algebraic Topology: the classification of two dimensional combinatorial surfaces. We use cut and paste operations to reduce any combinatorial surface into a standard form, and also introduce an algebraic expression to encode this standard form. Lecture 19 Play Video Classification of Combinatorial Surfaces (Part II)In this video lecture, Prof. N.J. Wildberger presents the traditional proof of the most important theorem in Algebraic Topology: the classification of (two-dimensional) surfaces using a reduction to a normal or standard form. The main idea is to carefully cut and paste the polygons forming the surface in a particular way, creating either a sphere, or a number of crosscaps or handles. Lecture 20 Play Video An Algebraic ZIP Proof 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. Lecture 21 Play Video The Geometry of Surfaces 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. Lecture 22 Play Video The Two-holed Torus and 3-Crosscaps Surface In this video lecture, Prof. N.J. Wildberger Lecture 22 Play Video The Two-holed Torus and 3-Crosscaps Surface In this video lecture, Prof. N.J. Wildberger Lecture 23 Play Video Knots and Surfaces (Part I) In this video lecture, Prof. N.J. Wildberger talks about knots and surfaces. Lecture 24 Play Video Knots and Surfaces (Part II) 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. Lecture 25 Play Video The Fundamental Group 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. Lecture 26 Play Video More on the Fundamental Group 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.