Insights into Mathematics Courses

The Fundamental Group - Algebraic Topology

Explore the fundamental group in algebraic topology, covering homotopic paths, loop multiplication, and equivalence classes. Understand key concepts for surfaces like disks and circles.

• YouTube
• 43 minutes
• Self-Paced
• Free Video

Knots and Surfaces I - Algebraic Topology - NJ Wildberger

Explore knot theory's origins, basic concepts, and key invariants in this introductory lecture on algebraic topology, covering Reidemeister moves, crossing numbers, and the Alexander-Conway polynomial.

• YouTube
• 52 minutes
• Self-Paced
• Free Video

The Two-Holed Torus and 3-Crosscaps Surface - Algebraic Topology - NJ Wildberger

Explore hyperbolic geometric structures of two-holed torus and 3-crosscaps surface through tessellations, hexagons, and the Beltrami Poincare model in this advanced algebraic topology lecture.

• YouTube
• 39 minutes
• Self-Paced
• Free Video

An Algebraic ZIP Proof of the Classification - Algebraic Topology - NJ Wildberger

Algebraic approach to classifying two-dimensional surfaces using spheres with holes, introducing a notation for manipulating edges between holes on spheres. Explores Conway's ZIP proof in a novel way.

• YouTube
• 42 minutes
• Self-Paced
• Free Video

Classification of Combinatorial Surfaces - Algebraic Topology - NJ Wildberger

Introduction to the classification of connected compact combinatorial surfaces in algebraic topology, covering key concepts like polygons, vertices, traversing, Euler number, and orientability.

• YouTube
• 51 minutes
• Self-Paced
• Free Video

More Applications of Winding Numbers - Algebraic Topology - NJ Wildberger

Explore winding numbers, degree of circle functions, and key theorems in algebraic topology. Learn about retractions, fixed points, and Borsuk's Lemma in this advanced mathematics lecture.

• YouTube
• 27 minutes
• Self-Paced
• Free Video

Non-Orientable Surfaces - The Mobius Band - Algebraic Topology

Explore non-orientable surfaces like the Möbius band, understanding their unique properties and applications in algebraic topology. Learn about crosscaps and key deformations.

• YouTube
• 42 minutes
• Self-Paced
• Free Video

The Klein Bottle and Projective Plane - Algebraic Topology - NJ Wildberger

Explore the Klein bottle, projective plane, and Platonic solids in this algebraic topology lecture. Learn about Euler's formula and its proof using triangulation and sphere flow.

• YouTube
• 40 minutes
• Self-Paced
• Free Video

Polyhedra and Euler's Formula - Algebraic Topology

Explore Platonic solids, Euler's formula, and spherical geometry in this engaging lecture on algebraic topology, featuring proofs and insights into fundamental mathematical concepts.

• YouTube
• 46 minutes
• Self-Paced
• Free Video

Two-Dimensional Objects - The Torus and Genus - Algebraic Topology

Explore surfaces like cylinders and tori, understanding their properties and connections to complex function theory. Learn about genus and how planes cover different surfaces.

• YouTube
• 50 minutes
• Self-Paced
• Free Video

Number Systems Throughout History - Elementary Mathematics Explained

Explore ancient number systems from Babylonian to Roman, understanding their development, unique features, and common elements. Gain insights into the foundations of modern mathematics.

• YouTube
• 31 minutes
• Self-Paced
• Free Video

The Cross Law in Universal Hyperbolic Geometry - Universal Hyperbolic Geometry - NJ Wildberger

Explores the Cross law in Universal Hyperbolic Geometry, proving it with a polynomial identity and demonstrating its application to equilateral triangles. Includes exercises and comparisons to Euclidean geometry.

• YouTube
• 35 minutes
• Self-Paced
• Free Video

The Spread Law in Universal Hyperbolic Geometry - Universal Hyperbolic Geometry

Explores the Spread law in universal hyperbolic geometry, including its definition, relation to Euclidean geometry, proof, and implications. Covers key concepts like quadrance, spread duality, and quadrea.

• YouTube
• 24 minutes
• Self-Paced
• Free Video

Visualizing Quadrance with Circles - Universal Hyperbolic Geometry - NJ Wildberger

Explore quadrance in universal hyperbolic geometry through circles, examining their appearance as ellipses, parabolas, or hyperbolas with different centers, enhancing understanding of this fundamental concept.

• YouTube
• 35 minutes
• Self-Paced
• Free Video

The Triple Quad Formula in Universal Hyperbolic Geometry

Explores the Triple quad formula in hyperbolic geometry, its proof, and related concepts. Compares with Euclidean geometry and discusses algebraic challenges, providing exercises for deeper understanding.

• YouTube
• 39 minutes
• Self-Paced
• Free Video