characteristic classes

Characteristic classes are mathematical tools used to study and distinguish complex geometric shapes called vector bundles, which are collections of vectors attached smoothly to each point of a space. They capture intrinsic features of these shapes—like twisting or curvature—that remain unchanged under continuous transformations. Think of them as "labels" or "signatures" that identify how a bundle is assembled over a space. Characteristic classes help mathematicians understand the underlying structure of geometric and topological objects, aiding in classification and analysis of complex spaces and their properties.