Title: Lipschitz normal embeddings in the space of matrices

2019.01.10 |

Date | Wed 30 Jan |

Time | 14:15 — 15:15 |

Location | TBA |

**Abstract:**

The germ of an algebraic variety (X,0) is naturally equipped with two different metrics: the outer metrics which is the restriction of the Euclidean metrics and the inner metrics which is defined using lengths of paths in X. The bilipschitz equivalence class of these metrics is an intrinsic property of the variety and do not depend of the choice of embeddings. If the inner metric is bilipschitz equivalent to the outer metric, we say that (X,0) is Lipschitz normally embedded. In this talk we prove Lipschitz normally embeddedness of some algebraic subsets of the space of matrices. Let X be a subset of the space of matrices, then let X(r) be the set of matrices in X of rank r, and X(<r) be the set of matrices in X of rank less than r. We prove that if X is either the space of all matirces, the space of symmetric matrices or the space of anti-symmetric matrices, then X(r) and X(<r) are Lipschitz normally embedded for any r. We also prove that the set of upper triangular matrices of determinant 0, is Lipschitz normally embedded and that transversal intersections of linear spaces and X(<r) where X is the space of all matrices are Lipschitz normally embedded for any r. This last result have generalizations in the language of determinantal singularities which we will briefly discuss.