Given a nested pair X and Y of complex projective varieties, there is a single positive integer e which measures the singularity type of X inside Y. This is called the Hilbert-Samuel multiplicity of Y along X, and it appears in the formulations of several standard intersection-theoretic constructions including Segre classes, Euler obstructions, and various other multiplicities. The standard method for computing e requires knowledge of the equations which define X and Y, followed by a (super-exponential) Grobner basis computation. In this talk we will connect the HS multiplicity to complex links, which are fundamental invariants of (complex analytic) Whitney stratified spaces. Thanks to this connection, the enormous computational burden of extracting e from polynomial equations reduces to a simple exercise in clustering point clouds. In fact, one doesn't even need the polynomials which define X and Y: it suffices to work with dense point samples. This is joint work with Martin Helmer.

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