ON PERIODS OF NONCOMMUTATIVE ALGEBRAS

Connes' theory of integration says that a finitely summable Fredholm module defines a linear functional on periodic cyclic homology of an algebra defined over a subfield of complex numbers. Let us assume that the algebra is defined over rational numbers. Then one can show that, in this way, one can obtain all periods of algebraic varieties over number fields. In my talk, I will speculate about the topological K-theory of general finitely generated algebras, and provide evidence that, in the noncommutative world, one should encounter "motives" of infinite ranks and new transcendental periods.