Intersection Numbers: Algebraic Counting Solutions

This lecture covers the concept of intersection numbers, which provide a way of counting solutions to polynomial equations algebraically. The instructor explains how these numbers are crucial for correctly counting solutions and introduces the notion of perturbation-invariant. The lecture delves into the significance of intersection numbers in determining the number of intersection points between polynomial curves. It also explores the application of intersection numbers in algebraic geometry, specifically in the context of projective curves. The discussion extends to the general statement about the set of solutions of homogeneous polynomials. Geometrically, intersection numbers mark the beginning of intersection theory and enumerative geometry, exemplified by Apollonius Circles from ancient times.