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SUMMARY:Geometric Whitney problem: reconstruction of a manifold from a poi
 nt cloud - Matti Lassas\, Helsinki
DTSTART:20160217T160000Z
DTEND:20160217T170000Z
UID:TALK61646@talks.cam.ac.uk
CONTACT:Ivan Smith
DESCRIPTION:We study the geometric Whitney problem on how a Riemannian man
 ifold (M\,g) can be constructed to approximate a metric space (X\,d_X).  T
 his  problem is closely related to  manifold interpolation (or manifold le
 arning) where a smooth n-dimensional surface S in Euclidean m-space\, m>n\
 , needs to be constructed to approximate a point cloud in m-space. These q
 uestions are  encountered in differential geometry\, machine learning\, an
 d in many inverse problems encountered in applications. The determination 
 of a Riemannian manifold includes the construction of its topology\, diffe
 rentiable structure\, and metric. \n\nWe give constructive solutions to th
 e above problems. Moreover\, we characterize the metric spaces that can be
  approximated\, by Riemannian manifolds with bounded geometry: we give suf
 ficient conditions to ensure that a metric space  can be approximated\, in
  the Gromov-Hausdorff or quasi-isometric sense\, by a Riemannian manifold 
 of a fixed dimension and with bounded diameter\, sectional curvature\, and
  injectivity radius. Also\, we show that similar conditions\, with modifie
 d values of parameters\, are necessary.\n\nMoreover\, we characterise the 
 subsets of Euclidean spaces that can be approximated in the Hausdorff metr
 ic by submanifolds of a fixed dimension\nand with bounded principal curvat
 ures and normal injectivity radius.\n\nThe above interpolation problems   
 are also studied for unbounded metric sets and manifolds. The results for 
 Riemannian manifolds are based on a generalisation of the Whitney embeddin
 g construction where approximative coordinate charts are embedded in Eucli
 dean m-space and interpolated to a smooth surface.\nWe also give algorithm
 s that solve the problems for finite data.\n
LOCATION:MR13
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