26. CJM 2001 (vol 53 pp. 278)
 Helminck, G. F.; van de Leur, J. W.

Darboux Transformations for the KP Hierarchy in the SegalWilson Setting
In this paper it is shown that inclusions inside the SegalWilson
Grassmannian give rise to Darboux transformations between the
solutions of the $\KP$ hierarchy corresponding to these planes. We
present a closed form of the operators that procure the transformation
and express them in the related geometric data. Further the
associated transformation on the level of $\tau$functions is given.
Keywords:KP hierarchy, Darboux transformation, Grassmann manifold Categories:22E65, 22E70, 35Q53, 35Q58, 58B25 

27. CJM 2001 (vol 53 pp. 212)
 Puppe, V.

Group Actions and Codes
A $\mathbb{Z}_2$action with ``maximal number of isolated fixed
points'' ({\it i.e.}, with only isolated fixed points such that
$\dim_k (\oplus_i H^i(M;k)) =M^{\mathbb{Z}_2}, k = \mathbb{F}_2)$
on a $3$dimensional, closed manifold determines a binary selfdual
code of length $=M^{\mathbb{Z}_2}$. In turn this code determines
the cohomology algebra $H^*(M;k)$ and the equivariant cohomology
$H^*_{\mathbb{Z}_2}(M;k)$. Hence, from results on binary selfdual
codes one gets information about the cohomology type of $3$manifolds
which admit involutions with maximal number of isolated fixed points.
In particular, ``most'' cohomology types of closed $3$manifolds do
not admit such involutions. Generalizations of the above result are
possible in several directions, {\it e.g.}, one gets that ``most''
cohomology types (over $\mathbb{F}_2)$ of closed $3$manifolds do
not admit a nontrivial involution.
Keywords:Involutions, $3$manifolds, codes Categories:55M35, 57M60, 94B05, 05E20 

28. CJM 2000 (vol 52 pp. 695)
 Carey, A.; Farber, M.; Mathai, V.

Correspondences, von Neumann Algebras and Holomorphic $L^2$ Torsion
Given a holomorphic Hilbertian bundle on a compact complex manifold, we
introduce the notion of holomorphic $L^2$ torsion, which lies in the
determinant line of the twisted $L^2$ Dolbeault cohomology and
represents a volume element there. Here we utilise the theory of
determinant lines of Hilbertian modules over finite von~Neumann
algebras as developed in \cite{CFM}. This specialises to the
RaySingerQuillen holomorphic torsion in the finite dimensional case.
We compute a metric variation formula for the holomorphic $L^2$
torsion, which shows that it is {\it not\/} in general independent of
the choice of Hermitian metrics on the complex manifold and on the
holomorphic Hilbertian bundle, which are needed to define it. We
therefore initiate the theory of correspondences of determinant lines,
that enables us to define a relative holomorphic $L^2$ torsion for a
pair of flat Hilbertian bundles, which we prove is independent of the
choice of Hermitian metrics on the complex manifold and on the flat
Hilbertian bundles.
Keywords:holomorphic $L^2$ torsion, correspondences, local index theorem, almost KÃ¤hler manifolds, von~Neumann algebras, determinant lines Categories:58J52, 58J35, 58J20 

29. CJM 1999 (vol 51 pp. 1123)
 Arnold, V. I.

First Steps of Local Contact Algebra
We consider germs of mappings of a line to contact space and
classify the first simple singularities up to the action of
contactomorphisms in the target space and diffeomorphisms of the
line. Even in these first cases there arises a new interesting
interaction of local commutative algebra with contact structure.
Keywords:contact manifolds, local contact algebra, Diracian, contactian Categories:53D10, 14B05 

30. CJM 1999 (vol 51 pp. 585)
 Mansfield, R.; MovahediLankarani, H.; Wells, R.

Smooth Finite Dimensional Embeddings
We give necessary and sufficient conditions for a normcompact subset
of a Hilbert space to admit a $C^1$ embedding into a finite dimensional
Euclidean space. Using quasibundles, we prove a structure theorem
saying that the stratum of $n$dimensional points is contained in an
$n$dimensional $C^1$ submanifold of the ambient Hilbert space. This
work sharpens and extends earlier results of G.~Glaeser on paratingents.
As byproducts we obtain smoothing theorems for compact subsets of
Hilbert space and disjunction theorems for locally compact subsets
of Euclidean space.
Keywords:tangent space, diffeomorphism, manifold, spherically compact, paratingent, quasibundle, embedding Categories:57R99, 58A20 
