The Zariski Topology and Affine Space
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### Lecture Description

In this lecture, we describe how the Zariski topology is defined using sets of common zeros of collections of polynomials as closed sets, also called algebraic sets. We explain that it is more natural to study ideals generated by subsets of polynomials and indicate how the Hilbert Basis Theorem (or Emmy Noether's Theorem) makes sure that we deal only with finitely many polynomials at a time

### Course Description

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.