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SUMMARY:Flagfolds: multi-dimensional varifolds to handle discrete surfaces
DTSTART:20250519T120000Z
DTEND:20250519T130000Z
DTSTAMP:20260317T021900Z
UID:indico-event-13819@indico.math.cnrs.fr
DESCRIPTION:Speakers: Blanche Buet (Paris-Saclay)\n\nWe propose a natural 
 framework for the study of surfaces and their different discretizations ba
 sed on varifolds. Varifolds have been introduced by Almgren to carry out t
 he study of minimal surfaces. Though mainly used in the context of rectifi
 able sets\, they turn out to be well suited to the study of discrete type 
 objects as well. While the structure of varifold is flexible enough to ada
 pt to both regular and discrete objects\, it allows to define variational 
 notions of mean curvature and second fundamental form based on the diverge
 nce theorem.\nThanks to a regularization of these weak formulations\, we p
 ropose a notion of discrete curvature (actually a family of discrete curva
 tures associated with a regularization scale) relying only on the varifold
  structure. We performed numerical computations of mean curvature and Gaus
 sian curvature on 3D point clouds to illustrate this approach. Though flex
 ible\, varifolds require the knowledge of the dimension of the shape to be
  considered. By interpreting the product of the Principal Component Analys
 is\, that is the covariance matrix\, as a sequence of nested subspaces nat
 urally coming with weights according to the level of approximation they pr
 ovide\, we are able to embed all d-dimensional Grassmannians into a strati
 fied space of covariance matrices.\nBuilding upon the proposed embedding o
 f Grassmannians into the space of covariance matrices\, we generalize the 
 concept of varifolds to what we call flagfolds in order to model multi-dim
 ensional shapes.\nThe first part of the talk will be dedicated to introduc
 ing varifolds (providing definition and examples\, we will not need any re
 gularity theory) and explaining how they can be used to model discrete sur
 faces and approximate their curvature.\nIn the second part\, we will focus
  on multi-dimensional varifolds (flagfolds) that rely on the embedding of 
 Grassmannians of different dimensions into a common stratified space.\nJoi
 nt works with G.P. Leonardi\, S. Masnou and X. Pennec.\n\nhttps://indico.m
 ath.cnrs.fr/event/13819/
LOCATION:Amphi Schwartz
URL:https://indico.math.cnrs.fr/event/13819/
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