Jonathan Block

Non-commutative tori and Fourier–Mukai duality

The classical Fourier–Mukai duality establishes an equivalence of categories between the derived categories of sheaves on dual complex tori. In this article we show that this equivalence extends to an equivalence between two dual objects. Both of these are generalized deformations of the complex tori. In one case, a complex torus is deformed formally in a non-commutative direction specified by a holomorphic Poisson structure. In the other, the dual complex torus is deformed in a B-field direction to a formal gerbe. We show that these two deformations are Fourier–Mukai equivalent. Contents

Milnor descent for cohesive dg-categories

We show that the functor from curved differential graded algebras to differential graded categories, defined by the second author in [B], sends Cartesian diagrams to homotopy Cartesian diagrams, under certain reasonable hypotheses. This is an extension to the arena of dg categories of a construction of projective modules due to Milnor. As an example, we show that the functor satisfies descent for certain partitions of a complex manifold.

Homotopy invariance of Novikov-Shubin invariants and ² Betti numbers

We give short proofs of the Gromov-Shubin theorem on the homotopy invariance of the Novikov-Shubin invariants and of the Dodziuk theorem on the homotopy invariance of the L2 Betti numbers of the universal covering of a closed manifold in this paper. We show that the homotopy invariance of these invariants is no more difficult to prove than the homotopy invariance of ordinary homology theory.

Some Remarks Concerning the Baum-Connes Conjecture

-algebra of Γ. So far, there is quite little evidencefor this conjecture. For example, there is not a single property T group forwhich it is known to be true. In this note we show that, in some sense, thehomological algebra of their conjecture is correct. In many cases, the periodiccyclic homology of certain dense subalgebras suggests what the K-theoryshould be. In the case of a discrete group Γ, the periodic cyclic homology ofthe algebraic group algebra CΓ is quite easy to calculate. Let !Γ

Explicit homotopy limits of dg-categories and twisted complexes

In this paper we study the homotopy limits of cosimplicial diagrams of dg-categories. We first give an explicit construction of the totalization of such a diagram and then show that the totalization agrees with the homotopy limit in the following two cases: (1) the complexes of sheaves of

Weyl Character Formula in KK-Theory

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String-Math 2011

-modules on the Cech nerve of an open cover of a ringed space

André–Quillen cohomology and rational homotopy of function spaces

(X, mathcal O)

The higher Riemann–Hilbert correspondence

; (2) the complexes of sheaves on the simplicial nerve of a discrete group

Cohesive DG Categories I: Milnor Descent

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