Morse theory is a branch of mathematics that studies the topology of manifolds using smooth functions. It focuses on the critical points of these functions, which are points where the function's derivative is zero. By analyzing these critical points, mathematicians can gain insights into the shape and structure of the manifold. The theory connects the geometry of a manifold to its topology, allowing for the classification of spaces based on their critical points. This approach has applications in various fields, including differential topology, algebraic topology, and even theoretical physics, where it helps in understanding complex systems.