Maths - Algebraic Geometry

Algebraic geometry is about applying geometrical constructions to algebraic problems.

Or, to turn this around, its about geometric structures defined by polynomials.

Set theory has functions. Topology has continuous functions. Differential geometry has smooth functions. Analytic geometry has analytic functions. Algebraic geometry has polynomial functions.

Main objects in algebraic geometry are algebraic varieties.

Videos

Fields Medal Lecture: Classification of algebraic varieties — Caucher Birkar — ICM2018 Youtube.

Algebraic geometry - Richard E. BORCHERDS

1. Introduction - https://www.youtube.com/watch?v=JZKDmTIFR7A
2. Two cubic curves - https://www.youtube.com/watch?v=SxGqKIALcgI
3. Bezout, Pappus, Pascal - https://www.youtube.com/watch?v=-9qugwEZDJs
4. Kakeya sets - https://www.youtube.com/watch?v=0I4ThQGwr3Y
5. Affine space and the Zariski topology - https://www.youtube.com/watch?v=GQTwEMhbQ4k
6. Noetherian spaces - https://www.youtube.com/watch?v=D_eJ8BWLb24
7. weak nullstellensatz - https://www.youtube.com/watch?v=hBQXqIkUUyY

Computational Algebraic Geometry

wiki