dott. Michor's seminar at SISSA (with abstract)

[Emanuele Tuillier Illingworth]

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 MATHEMATICAL PHYSICS SECTOR ACTIVITIES
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Peter Michor
(University of Vienna)

will hold a seminar on:
"Riemannian Geometries on shape spaces"

Abstract:
A version of shape space is the space of all smooth regular closed plane
curves, i.e., the space of orbits of the action of the diffeomorphism group
$Diff(S^1)$ on the space Imm(S^1,R^2)$ of smooth mappings $c:S^1\to R^2$
$with $c'$ never 0.
The aim is to find geodesic distance functions on shape space which make
sense for applications in imaging sciences. The simples solution does not
work because the $L^2$ or $H^0$ metric on the space of smooth plane regular
closed curves induces vanishing geodesic distance on the quotient
$Imm(S^1,R^2)/Diff(S^1)$.
This is a general phenomenon and holds on all full diffeomorphism groups and
spaces $Imm(M,N)/Diff(M)$ for a compact manifold $M$ and a Riemanninan
manifold $N$. Thus we have to consider more complicated Riemannian metrics
using lenght or curvature, and we do this in a systematic Hamiltonian way,
we derive geodesic equations and split them into horizontal and vertical
parts, and compute all conserved quantities via the momentum mappings of
several invariance groups (Reparameterizations, motions, and even scalings).
The resulting equations are relatives of well known completely integrable
systems (Burgers, Camassa Holm, Hunter Saxton). A certain scale invariant
geometry on $Imm(S^1,\mathbb R^2)$ is diffeomorphic to to the Grassmannian
of 2-planes in a pre-Hilbert space. Since for the latter one can write down
geodescs explicitly, following a Neretin from 2000, we have explicit
solutions for the distance, geodesics, and curvature.  This gives a
computatable distance on shape space $Imm(S^1,\mathbb R^2)/Diff(S^1)$, with
curvature. Some of these metrices have already found applications in
computational anatomy: One can compute from the tomographic data of hearts
whether the person is prone to develop cardiac rhythmic disturbances.



Wednesday, June 18, 14:30, room D