**Unit 1: The Zariski Topology** |

Lecture 01 - What is Algebraic Geometry? |

Lecture 02 - The Zariski Topology and Affine Space |

Lecture 03 - Going Back and Forth between Subsets and Ideals |

**Unit 2: Irreducibility in the Zariski Topology** |

Lecture 04 - Irreducibility in the Zariski Topology |

Lecture 05 - Irreducible Closed Subsets Correspond to Ideals whose Radicals are Prime |

**Unit 3: Noetherianness in the Zariski Topology** |

Lecture 06 - Understanding the Zariski Topology on the Affine Line; The Noetherian Property in Topology and in Algebra |

Lecture 07 - Basic Algebraic Geometry: Varieties, Morphisms, Local Rings, Function Fields and Nonsingularity |

**Unit 4: Dimension and Rings of Polynomial Functions** |

Lecture 08 - Topological Dimension, Krull Dimension and Heights of Prime Ideals |

Lecture 09 - The Ring of Polynomial Functions on an Affine Variety |

Lecture 10 - Geometric Hypersurfaces are Precisely Algebraic Hypersurfaces |

**Unit 5: The Affine Coordinate Ring of an Affine Variety** |

Lecture 11 - Why should We Study Affine Coordinate Rings of Functions on Affine Varieties? |

Lecture 12 - Capturing an Affine Variety Topologically from the Maximal Spectrum of its Ring of Functions |

**Unit 6: Open Sets in the Zariski Topology and Functions on Such Sets** |

Lecture 13 - Analyzing Open Sets and Basic Open Sets for the Zariski Topology |

Lecture 14 - The Ring of Functions on a Basic Open Set in the Zariski Topology |

**Unit 7: Regular Functions in Affine Geometry** |

Lecture 15 - Quasi-Compactness in the Zariski Topology; Regularity of a Function at a Point of an Affine Variety |

Lecture 16 - What is a Global Regular Function on a Quasi-Affine Variety? |

**Unit 8: Morphisms in Affine Geometry** |

Lecture 17 - Characterizing Affine Varieties; Defining Morphisms between Affine or Quasi-Affine Varieties |

Lecture 18 - Translating Morphisms into Affines as k-Algebra Maps and the Grand Hilbert Nullstellensatz |

Lecture 19 - Morphisms into an Affine Correspond to k-Algebra Homomorphisms from its Coordinate Ring of Functions |

Lecture 20 - The Coordinate Ring of an Affine Variety Determines the Affine Variety and is Intrinsic to It |

Lecture 21 - Automorphisms of Affine Spaces and of Polynomial Rings - The Jacobian Conjecture; The Punctured Plane is Not Affine |

**Unit 9: The Zariski Topology on Projective Space and Projective Varieties** |

Lecture 22 - The Various Avatars of Projective n-Space |

Lecture 23 - Gluing (n+1) Copies of Affine n-Space to Produce Projective n-Space in Topology, Manifold Theory and Algebraic Geometry; The Key to the Definition of a Homogeneous Ideal |

**Unit 10: Graded Rings, Homogeneous Ideals and Homogeneous Localization** |

Lecture 24 - Translating Projective Geometry into Graded Rings and Homogeneous Ideals |

Lecture 25 - Expanding the Category of Varieties to Include Projective and Quasi-Projective Varieties |

Lecture 26 - Translating Homogeneous Localization into Geometry and Back |

Lecture 27 - Adding a Variable is Undone by Homogeneous Localization - What is the Geometric Significance of this Algebraic Fact? |

**Unit 11: The Local Ring of Germs of Functions at a Point** |

Lecture 28 - Doing Calculus without Limits in Geometry |

Lecture 29 - The Birth of Local Rings in Geometry and in Algebra |

Lecture 30 - The Formula for the Local Ring at a Point of a Projective Variety or Playing with Localizations, Quotients, Homogenization and Dehomogenization |

**Unit 12: The Function Field of Functions on Large Open Sets** |

Lecture 31 - The Field of Rational Functions or Function Field of a Variety - The Local Ring at the Generic Point |

Lecture 32 - Fields of Rational Functions or Function Field of Affine and Projective Varieties and Their Relationships with Dimensions |

**Unit 13: Two Facts about Rings of Functions on Projective Varieties** |

Lecture 33 - Global Regular Functions on Projective Varieties are Simply the Constants |

**Unit 14: The Importance of Local Rings and Function Fields** |

Lecture 34 - The d-Uple Embedding and the Non-intrinsic Nature of the Homogeneous Coordinate Ring of a Projective Variety |

Lecture 35 - The Importance of Local Rings - A Morphism is an Isomorphism if it is a Homeomorphism and Induces Isomorphisms at the Level of Local Rings |

Lecture 36 - The Importance of Local Rings - A Rational Function in Every Local Ring is Globally Regular |

Lecture 37 - Geometric Meaning of Isomorphism of Local Rings - Local Rings are Almost Global |

**Unit 15: Regular or Smooth Points and Manifold Varieties or Smooth Varieties** |

Lecture 38 - Local Ring Isomorphism - Equals Function Field Isomorphism - Equals Birationality |

Lecture 39 - Why Local Rings Provide Calculus without Limits for Algebraic Geometry Pun Intended |

Lecture 40 - How Local Rings Detect Smoothness or Nonsingularity in Algebraic Geometry |

Lecture 41 - Any Variety is a Smooth Manifold with or without Nonsmooth Boundary |

Lecture 42 - Any Variety is a Smooth Hypersurface on an Open Dense Subset |