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John Milnor works across topology, geometry and dynamical systems. His expositions connect local calculations with global structure: Morse Theory by John Milnor develops the topology of a manifold through critical points, while his complex-dynamics lectures study iteration of holomorphic maps.^[ref][ref]

Near a nondegenerate critical point, the Morse lemma reduces a smooth function exactly to a signed quadratic form in suitable coordinates. Its index controls the topological change in sublevel sets. With James D. Stasheff, Milnor also wrote Characteristic Classes by John Milnor and James D. Stasheff, developing invariants of vector bundles.

Let $f:M\to \mathbb{R}$ be smooth, and let $p$ be a critical point whose Hessian is nondegenerate. There are smooth local coordinates $y_1,\dots ,y_n$, centered at $p$, in whichThe integer $\lambda$ is the Morse index. This is an exact local identity, rather than a quadratic approximation with a remainder.^[ref]

To see the mechanism, apply the integral form of Taylor's theorem twice at the critical point, writing $f-f(p)={\sum }_{i,j}{x}_i{x}_j{h}_{ij}(x)$ with smooth symmetric coefficients. Nondegeneracy allows successive smooth completion of squares; the resulting change of coordinates has derivative given by an invertible linear map, so the inverse function theorem makes it a valid coordinate system. Sylvester's law of inertia makes the number of negative squares independent of the coordinates.

For $f(x,y)=x^2-y^2$, the origin has index one and is a saddle point. For $f(x)=x^3$, the origin is critical but its Hessian is zero; no smooth coordinate change makes it a nondegenerate quadratic critical point. The normal form implies that every nondegenerate critical point is isolated, and supplies the local model for the Morse handle-attachment theorem.