Index theorems are mathematical results that connect the analysis of differential operators to topological properties of manifolds. They provide a way to compute the index of an operator, which is a measure of the difference between the dimensions of its kernel and cokernel. This relationship reveals deep insights into the geometry and topology of the underlying space. One of the most famous index theorems is the Atiyah-Singer Index Theorem, which links the index of elliptic differential operators to the topology of the manifold, specifically through characteristic classes. These theorems have significant implications in various fields, including geometry, physics, and mathematics.