| Abstract: |
Any real or complex singularity \( (X, 0) \) is equipped with two
natural metrics. The outer metric, which is the restriction of the
ambient euclidean metric, and the inner metric, which is the metric
associated with a riemannian metric on the germ. Up to bilipschitz
equivalence these metrics does not depends on the choices of analytic
embedding. The inner and outer metrics are in general not bilipschitz
equivalent, and one says that \( (X, 0) \) is Lipschitz normally
embedded if they are. In this talk we will give an overview of the
subject and discuss the current state of the question about which
singularities are Lipschitz normally embedded. From the beginning of
the modern study done by Birbrair, Fernandes and Neumann, over our
joint work with Neumann and Pichon on proving that minimal surfaces
singularities are Lipschitz normally embedded, and our work with
Kerner and Ruas on which matrix singularities are Lipschitz normally
embedded, to work still in progress joint with Fantini, Pichon and
Schober on surface singularities and with Langois on toric
singularities.
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