Since the ancient Greeks, mathematicians have craved for understanding of the geometry lurking behind the most basic equations – polynomials. In the early 20th century, members of the famous Italian school of algebraic geometry finally managed to crack the case of surfaces, but the task of grasping the structure of higher-dimensional varieties seemed to be insurmountable.
The new era of higher-dimensional algebraic geometry started with the celebrated proof of the Hartshorne Conjecture by Shigefumi Mori. The technique which he introduced, called Mori’s bend and break, revolutionized algebraic geometry and allowed for the classification of algebraic threefolds.
In this talk, I will give a proof of Mori’s bend and break, and show some of its awe-inspiring applications.