Differential topology is a branch of mathematics that studies the properties of differentiable functions on differentiable manifolds. It focuses on the ways in which these functions can be manipulated and transformed while preserving certain structures, such as smoothness. This field combines concepts from both differential geometry and topology to explore how shapes and spaces behave under continuous transformations. One of the key concepts in differential topology is the idea of a manifold, which is a space that locally resembles Euclidean space. Researchers in this field investigate various properties of manifolds, such as homotopy and homology, to understand their structure and classification. Differential topology has applications in areas like physics, particularly in the study of general relativity and string theory.