Misha Shubin. 1944 -- 2020
The article describes the biography and manifold contributions to research in mathematics of Mikhail Aleksandrovich Shubin.
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arXiv subjects
L. Friedlander
Publications and source records attributed to L. Friedlander.
Relative torsion for representations in finite type Hilbert modules
For a closed manifold equipped with a Riemannian metric, a triangulation, a representation of its fundamental group on an Hilbert module of finite type (over of finite von Neumann algebra), and a Hermitian structure on the flat bundle associated to the representation, one defines a numerical invariant, the relative torsion. The relative torsion is a positive real number and unlike the analytic torsion or the Reidemeister torsion, which are defined only when the pair manifold- representation is of determinant class, is always defined. When the pair is of determinant class the relative torsionis equal to the quotient of the analytic and the Reidemeister torsion.We calculate the relative torsion.
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Torsions for manifolds with boundary and glueing formulas
We extend the definition of analytic and Reidemeister torsion from closed compact Riemannian manifolds to compact Riemannian manifolds with boundary $(M, \partial M)$, given a flat bundle $\Cal F$ of $\Cal A$-Hilbert modules of finite type and a decomposition of the boundary $\partial M =\partial_- M \cup \partial_+ M$ into disjoint components. In particular we extend the $L-2$ analytic and Reidemeister torsions to compact manifolds with boundary. If the system $(M,\partial_-M, \partial_+M, \Cal F)$ is of determinant class we compute the quotient of the analytic and the Reidemeister torsion and prove glueing formulas for both of them. In particular we answer positively Conjecture 7.6 in [LL]
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Analytic and Reidemeister torsion for representations in finite type Hilbert modules
For a closed Riemannian manifold we extend the definition of analytic and Reidemeister torsion associated to an orthogonal representation of fundamental group on a Hilbert module of finite type over a finite von Neumann algebra. If the representation is of determinant class we prove, generalizing the Cheeger-Müller theorem, that the analytic and Reidemeister torsion are equal. In particular, this proves the conjecture that for closed Riemannian manifolds with positive Novikov-Shubin invariants, the L2 analytic and Reidemeister torsions are equal.
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