Vidit Nanda

Landfall Geometry

I will describe some new results concerning negative gradient flows of analytic functions defined on compact analytic Riemannian manifolds without imposing any nondegeneracy or transversality assumptions. The key idea is to place small subanalytic blocks around (connected components of) the critical set and to follow orbits for only a bounded time: from the moment they exit one block until they make landfall in the interior of another, recording which other blocks are encountered en route. These contacts assemble into a topological category, called the landfall category, whose classifying space is simple homotopy equivalent to the underlying manifold, compatibly with the natural filtration coming from the flow partial order. This filtered homotopy type does not depend on the choice of blocks. Triangulating the blocks produces an explicit finite landfall chain complex that recovers the homology of the manifold, with one generator for each simplex away from the exit sets. The same finite model computes the Morse–Conley spectral sequence and can be further compressed into a connection matrix for the gradient flow. All of this is joint work with Frances Kirwan.