Differential topology is a branch of mathematics that studies the properties of differentiable functions on differentiable manifolds. It combines concepts from both topology and calculus, focusing on how shapes and spaces can be analyzed through smooth transformations. This field is essential for understanding the geometric structure of various mathematical objects. One key concept in differential topology is the idea of a manifold, which is a space that locally resembles Euclidean space. Researchers often explore smooth maps and differentiable structures to understand how these manifolds can be manipulated and classified, leading to insights in areas like theoretical physics and robotics.