Chern classes are characteristic classes. They are topological invariants associated to vector bundles on a smooth manifold. If you describe the same vector bundle on a manifold in two different ways, the Chern classes will be the same. When are two ostensibly different vector bundles the same? When are they different? These questions can be quite hard to answer. But the Chern classes provide a simple test: if the Chern classes of a pair of vector bundles do not agree, then the vector bundles are different.In topology, differential geometry, and algebraic geometry, it is often important to count how many linearly independent sections a vector bundle has. The Chern classes offer some information about this through, for instance, the Riemann-Roch theorem and the Atiyah-Singer index theorem. Chern classes are therefore useful in modern mathematics.

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