The formality theorem for Hochschild chains of the algebra of functions on a smooth manifold gives us a version of the trace density map from the zeroth Hochschild homology of a deformation quantization algebra to the zeroth Poisson homology. We propose a version of the algebraic index theorem for a Poisson manifold which is based on this trace density map. 1991 MSC: 19K56, 16E40.

We compare the domain of the assembly map in algebraic K – theory with respect to the family of finite subgroups with the domain of the assembly map with respect to the family of virtually cyclic subgroups and prove that the former is a direct summand of the later. AMS Classification 19D50; 19A31, 19B28

Using pure entangled Schmidt states, we show that m-positivity of a map is bounded by the ranks of its negative Kraus matrices. We also give an algebraic condition for a map to be m-positive. We interpret these results in the context of positive maps as entanglement witnesses, and find that only 1-positive maps are needed for testing entanglement.

Let ⊗ be the map which classifies the tensor product of two line bundles, an extension of this map to the space of all codimension 1 algebraic cycles is constructed. It is proved that this extension cannot exist in codimension greater than 1.

In this paper, firstly, a generalized subconvexlike set-valued map involving the relative algebraic interior is introduced in ordered linear spaces. Secondly, some properties of a generalized subconvexlike set-valued map are investigated. Finally, the optimality conditions of set-valued optimization problem are established.

A covalence sequence of a vertex-transitive map is a cyclic sequence of covalences around a vertex of the map. We explain the algebraic background for a study of the relationship between covalence sequences of maps on compact orientable surfaces and covalence sequences of (infinite) plane tessellations that both exhibit the same ‘level of transitivity’ of automorphism groups on vertices or edges.

We study the algebraic Gauss-Manin system and the algebraic Brieskorn module associated to a polynomial mapping with isolated singularities. Since the algebraic GaussManin system does not contain any information on the cohomology of singular fibers, we first construct a non quasi-coherent sheaf which gives the cohomology of every fiber. Then we study the algebraic Brieskorn module, and show tha...

The original Novikov conjecture concerns the (oriented) homotopy invariance of higher signatures of manifolds and is equivalent to the rational injectivity of the assembly map in surgery theory. The integral injectivity of the assembly map is important for other purposes and is called the integral Novikov conjecture. There are also assembly maps in other theories and hence related Novikov and i...