3,141 research outputs found
Why Use Sobolev Metrics on the Space of Curves
We study reparametrization invariant Sobolev metrics on spaces of regular curves. We discuss their completeness properties and the resulting usability for applications in shape analysis. In particular, we will argue, that the development of efficient numerical methods for higher order Sobolev type metrics is an extremely desirable goal
Equivariant embedding theorems and topological index maps
The construction of topological index maps for equivariant families of Dirac
operators requires factoring a general smooth map through maps of a very simple
type: zero sections of vector bundles, open embeddings, and vector bundle
projections. Roughly speaking, a normally non-singular map is a map together
with such a factorisation. These factorisations are models for the topological
index map. Under some assumptions concerning the existence of equivariant
vector bundles, any smooth map admits a normal factorisation, and two such
factorisations are unique up to a certain notion of equivalence. To prove this,
we generalise the Mostow Embedding Theorem to spaces equipped with proper
groupoid actions. We also discuss orientations of normally non-singular maps
with respect to a cohomology theory and show that oriented normally
non-singular maps induce wrong-way maps on the chosen cohomology theory. For
K-oriented normally non-singular maps, we also get a functor to Kasparov's
equivariant KK-theory. We interpret this functor as a topological index map
Clemens-Schmid exact sequence in characteristic p
For a semistable family of varieties over a curve in characteristic , we
prove the existence of a "Clemens-Schmid type" long exact sequence for the
-adic cohomology. The cohomology groups appearing in such a long exact
sequence are defined locall
Homological stability for moduli spaces of high dimensional manifolds. II
We prove a homological stability theorem for moduli spaces of manifolds of
dimension , for attaching handles of index at least , after these
manifolds have been stabilised by countably many copies of .
Combined with previous work of the authors, we obtain an analogue of the
Madsen--Weiss theorem for any simply-connected manifold of dimension .Comment: 60 pages, 4 figures. Final accepted versio
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