4 November 2022, Cool Logic, Lingyuan Ye

Descartes completely changed our mathematical vision, by finding a correspondence between spaces (geometry) and equations (algebra). Starting from high school, we have learnt that x^2+y^2-1 can be viewed as the the unit circle in the Euclidean plane, and a lot of algebraic arguments we do have very natural geometric meaning behind them, and this geometric intuition serves as a guide for many algebraic solutions. Modern algebraic geometry extends this picture to also allow a geometric study of solutions on any rings other than R or C, like Z or Q. The powerful tools developed in topology suddenly applies to number theory, and derives powerful results.