Ajay C. Ramadoss

Some notes on the Feigin-Losev-Shoikhet integral conjecture

Given a holomorphic vector bundle E on a smooth connected compact complex manifoldX, Feigin, Losev and Shoikhet (FLS) use a notion of completed Hochschild homology c HH of Diff.E/ such that c HH0.Diff.E// is isomorphic to H 2n .X; C/. On the other hand, they construct a linear functional on c HH0.Diff.E//. This therefore gives rise to a linear functional IE on H 2n .X; C/. They show that this functional is R X if E has non-zero Euler characteristic. They conjecture that this functional is R X for all E. In this article it is proved thatIE DIF for any pair.E; F/ of holomorphic vector bundles onX. In particular, ifX has one vector bundle with non-zero Euler characteristic, thenIE D R X

The Picard group of a noncommutative algebraic torus

We compute the Picard group

The relative Riemann-Roch theorem from Hochschild homology

Pic(A_q)

THE MUKAI PAIRING AND INTEGRAL TRANSFORMS IN HOCHSCHILD HOMOLOGY

of the noncommutative algebraic 2-torus

THE BIG CHERN CLASSES AND THE CHERN CHARACTER

A_q

Derived Representation Schemes and Noncommutative Geometry

, describe its action on the space

Noncommutative Poisson structures, derived representation schemes and Calabi-Yau algebras

R(A_q)

Stable representation homology and Koszul duality

of isomorphism classes of rk 1 projective modules and classify the algebras Morita equivalent to

Representation homology, Lie algebra cohomology and the derived Harish-Chandra homomorphism

A_q

A generalized Hirzebruch Riemann―Roch theorem

. Our computations are based on a quantum version of the Calogero-Moser correspondence relating projective