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A universal triangulation for flat tori

January 13, 2021 @ 9:00 am - 10:00 am

Speaker:Francis Lazarus, CNRS / Grenoble University

Title: A universal triangulation for flat tori

Abstract: A celebrated theorem of Nash completed by Kuiper implies that every smooth Riemannian surface has a C¹ isometric embedding in the Euclidean 3-space E³. An analogous result, due to Burago and Zalgaller, states that every polyhedral surface, obtained by gluing Euclidean triangles, has an isometric PL embedding in E³. In particular, this provides PL isometric embeddings for every flat torus (a quotient of E² by a rank 2 lattice). However, the proof of Burago and Zalgaller is partially constructive, relying on the Nash-Kuiper theorem. In practice, it produces PL embeddings with a huge number of vertices, moreover distinct for every flat torus. Based on a construction of Zalgaller and on recent works by Arnoux et al. we exhibit a universal triangulation with less than 10.000 vertices, admitting for any flat torus an isometric embedding that is linear on each triangle. Based on joint work with Florent Tallerie.


January 13, 2021
9:00 am - 10:00 am
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