Morse Theory

Morse Theory is a mathematical framework that studies the shape and features of complex surfaces or spaces by analyzing smooth functions defined on them. Think of it like examining a landscape with hills, valleys, and passes: by looking at the highest and lowest points and the way the terrain changes, Morse Theory helps us understand the space's overall structure and how its parts connect. It connects the critical points of these functions (like peaks or dips) to the fundamental features of the space, allowing mathematicians to reveal insights into the topology— the underlying shape— of objects from a mathematical perspective.