Hodge theory is a branch of mathematics that studies the relationship between algebraic topology and differential geometry. It focuses on the decomposition of differential forms on a smooth manifold, allowing mathematicians to understand the structure of the manifold's cohomology groups. This theory provides tools to analyze the shapes and features of geometric objects. One of the key results in Hodge theory is the Hodge decomposition theorem, which states that any differential form can be uniquely expressed as the sum of an exact form, a co-exact form, and a harmonic form. This decomposition helps in solving various problems in both pure and applied mathematics, including string theory and mathematical physics.