geometric analysis
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Research topics investigated in Warsaw:
Noncommutative geometry of Galois type extensions.
In particular, applications
of the Noncommutative Index Theorem to the classification of stable
isomorphism
classes of projective modules associated with Hopf or coalgebra Galois extensions
of noncommutative algebras (quantum associated vector bundles).
P.M. Hajac
Extension of the Baum-Connes conjecture from locally compact topological
groups to locally compact quantum groups.
P.M. Hajac
Phenomena and effects in topology following from Weil conjectures:
A parallel theory which can lead to the similar results is the
theory of mixed Hodge modules developed by Saito. I have studied
ways of constructing intersection homology morphism which is
associated to a map of algebraic varieties. I have found a simple
proof of its existence and I have also related it to the hard
Lefschetz theorem. I have formulated a topological condition
''local Hard Lefschetz'', which is crucial for existence of such
morphism. By Saito's theory it is satisfied for analytic
varieties.
A. Weber
In a joint work with J-P. Brasselet we have constructed in canonical
way Chern classes in intersection homology. They are computable in
very few cases.
One should find their meaning and effective ways of computation.
A. Weber
Koszul duality: The topological version of this duality states
that, in derived category, the ordinary cohomology of G-space
(where G is compact Lie group) is dual to the equivariant
cohomology. The duality in question is the one established by
Beilinson--Gelfand--Gelfand; it relates $H_*(G)$--modules and
$H^*(BG)$--modules. I have managed to prove this duality in pure
algebraic context, when a reductive Lie algebra g acts on
differential graded module. Further generalizations are possible
for super Lie algebras with differential. The appropriate language
for this problem is Hohschield cohomology, deformation theory,
operads. This is joint work with Marcin Chalupnik and Tomasz
Maszczyk.
A. Weber
Residue theory:
I have studied behavior of the Leray residue form in
neighbourhoods of singular points. Such a form may give element
in local intersection homology. It depends on vanishing of certain
obstructions which are described in terms of the spectrum of a
singular point (in the case of isolated singularities). The
problem of lift of Leray residue to intersection homology may be
rephrased in the language of D-modules, and it seems that in
this setup one should find a general answer.
A. Weber
The isomorphism of intersection cohomology and $L^p$-cohomology:
I have investigated Riemannian
pseudomanifolds, i.e. the manifolds with singularities and with a
metric on the nonsingular part, which is conical in neighbourhoods
of singular points. I was interested in problems of duality
between $L^p$ and $L^q$-cohomology (with p and q complementary).
I've found a local topological condition (''negligible boundary'')
which is equivalent to the Verdier duality of the sheaves on forms.
A. Weber
Related seminars: