What is Algebraic Geometry? 
What is Algebraic Geometry?
by IIT Madras
Video Lecture 1 of 30
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Date Added: August 16, 2016

Lecture Description

In this beginning lecture, we introduce Algebraic Geometry as the study of the geometry of the set of common zeros of a collection of polynomials. We indicate that this would involve setting up a dictionary of sorts between the Geometric side and the Commutative Algebra side.

Course Index

  1. What is Algebraic Geometry?
  2. The Zariski Topology and Affine Space
  3. Going back and forth between subsets and ideals
  4. Irreducibility in the Zariski Topology
  5. Irreducible Closed Subsets Correspond to Ideals Whose Radicals are Prime
  6. Understanding the Zariski Topology on the Affine Line
  7. The Noetherian Decomposition of Affine Algebraic Subsets Into Affine Varieties
  8. Topological Dimension, Krull Dimension and Heights of Prime Ideals
  9. The Ring of Polynomial Functions on an Affine Variety
  10. Geometric Hypersurfaces are Precisely Algebraic Hypersurfaces
  11. Why Should We Study Affine Coordinate Rings of Functions on Affine Varieties ?
  12. Capturing an Affine Variety Topologically
  13. Analyzing Open Sets and Basic Open Sets for the Zariski Topology
  14. The Ring of Functions on a Basic Open Set in the Zariski Topology
  15. Quasi-Compactness in the Zariski Topology
  16. What is a Global Regular Function on a Quasi-Affine Variety?
  17. Characterizing Affine Varieties
  18. Translating Morphisms into Affines as k-Algebra maps
  19. Morphisms into an Affine Correspond to k-Algebra Homomorphisms
  20. The Coordinate Ring of an Affine Variety
  21. Automorphisms of Affine Spaces and of Polynomial Rings - The Jacobian Conjecture
  22. The Various Avatars of Projective n-space
  23. Gluing (n+1) copies of Affine n-Space to Produce Projective n-space in Topology
  24. Translating Projective Geometry into Graded Rings and Homogeneous Ideals
  25. Expanding the Category of Varieties
  26. Translating Homogeneous Localisation into Geometry and Back
  27. Adding a Variable is Undone by Homogenous Localization
  28. Doing Calculus Without Limits in Geometry
  29. The Birth of Local Rings in Geometry and in Algebra
  30. The Formula for the Local Ring at a Point of a Projective Variety

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.

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