This course is an introduction to Algebraic Geometry, whose aim is to study the geometry underlying the set of common zeros of a collection of polynomial equations. It sets up the language of varieties and of morphisms between them, and studies their topological and manifold-theoretic properties. Commutative Algebra is the "calculus" that Algebraic Geometry uses. Therefore a prerequisite for this course would be a course in Algebra covering basic aspects of commutative rings and some field theory, as also a course on elementary Topology. However, the necessary results from Commutative Algebra and Field Theory would be recalled as and when required during the course for the benefit of the students.

Algebraic Geometry in its generality is connected to various areas of Mathematics such as Complex Analysis, PDE, Complex Manifolds, Homological Algebra, Field and Galois Theory, Sheaf Theory and Cohomology, Algebraic Topology, Number Theory, QuadraticForms, Representation Theory, Combinatorics, Commutative Ring Theory etc and also to areas of Physics like String Theory and Cosmology. Many of the Fields Medals awarded till date are for research in areas connected in a non-trivial way to Algebraic Geometry directly or indirectly. The Taylor-Wiles proof of Fermat's Last Theorem used the full machinery and power of the language of Schemes, the most sophisticated language of Algebraic Geometry developed over a couple of decades from the 1960s by Alexander Grothendieck in his voluminous expositions running to several thousand pages. The foundations laid in this course will help in a further study of the language of schemes.

Affine Varieties, Hilbert's Basis Theorem and the Hilbert Nullstellensatz, projective and quasi-projective varieties, morphisms, rational maps and function fields, nonsingularity, smooth varieties. The course will try to stress the nexus between Commutative Algebra and Algebraic Geometry. It begins the attempt to justify the philosophy that "Commutative Algebra is the Calculus for Algebraic Geometry" and illustrate the translation back and forth between concepts in Commutative Algebra and in Algebraic Geometry, in the spirit of Sophie Germaine's statement that "Algebra is none other than Geometry written down (in Mathematical language), and Geometry is none other than Algebra drawn out (as a beautiful picture)". For more details, please look at the lecture-wise titles, goals and keywords given below.

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Unit Number / Title

Lecture Number / Title

Unit 1: The Zariski Topology

Lecture 1:

What is Algebraic Geometry?

Lecture 2:

The Zariski Topology and Affine Space

Lecture 3:

Going back and forth between subsets and ideals

Unit 2: Irreducibility in the Zariski Topology

Lecture 4:

Irreducibility in the Zariski Topology

Lecture 5:

Irreducible Closed Subsets Correspond to Ideals Whose Radicals are Prime

Unit 3: Noetherianness in the Zariski Topology

Lecture 6:

Understanding the Zariski Topology on the Affine Line;
The Noetherian property in Topology and in Algebra

Lecture 7:

The Noetherian Decomposition of Affine Algebraic Subsets Into Affine Varieties

Unit 4: Dimension and Rings of Polynomial Functions

Lecture 8:

Topological Dimension, Krull Dimension and Heights of Prime Ideals

Lecture 9:

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 Localisation

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 Localisation into Geometry and Back

Lecture 27:

Adding a Variable is Undone by Homogenous 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 ?
Yes ! Possible Since Affines are Building Blocks and
Allow Algebraic Translation to Local Rings which
Arise from Geometrically Restricting Attention Close to a Point !

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 Localisations, Quotients, Homogenisation and Dehomogenisation !

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:

The Birth of Local Rings in Geometry and in Algebra

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

Lecture 33:

Global Regular Functions on Projective Varieties are Simply the Constants

Lecture 34:

The d-Uple Embedding and the Non-Intrinsic Nature
of the Homogeneous Coordinate Ring
of a Projective Variety

Unit 14: The Importance of Local Rings and Function Fields

Lecture 35:

The Importance of Local Rings - A Morphism is an Isomorphism
iff 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

Lecture 38:

Local Ring Isomorphism
Equals Function Field Isomorphism
Equals Birationality

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

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 Non-Smooth Boundary

Lecture 42:

Any Variety is a Smooth Hypersurface On an Open Dense Subset

A course in Algebra covering basic aspects of commutative rings and some field theory, and a course on elementary Topology

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1) Algebraic Geometry by Robin Hartshorne, Graduate Texts in Mathematics GTM 52, Springer

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2) The Red Book of Varieties and Schemes by David Mumford

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3) An Introduction to Commutative Algebra by M. F. Atiyah and I. G. Macdonald, Addison-Wesley

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