CONFERENCE TALKS xiii
Hopf-cyclic cohomology and Connes-Moscovici characteristic map
In 1998 Alain Connes and Henri Moscovici invented a cohomology theory for
Hopf algebras and a characteristic map associated with the cohomology theory
in order to solve a speciﬁc technical problem in transverse index theory. In the
following decade, the cohomology theory they invented developed on its own under
the name Hopf-cyclic cohomology. But the history of Hopf-cyclic cohomology and
the characteristic map they invented remained intricately linked. In this survey
talk, I will give an account of the development of the characteristic map and Hopf-
Holomorphic structures on the quantum projective line
In this talk I report on our joint ongoing work with Giovanni Landi and Walter
van Suijlekom. We deﬁne a notion of holomorphic structure in terms of a bigrading
of a suitable diﬀerential calculus over the quantum sphere. Realizing the quantum
sphere as a principal homogeneous space of the quantum group SUq(2) plays an
important role in our approach. We deﬁne a notion of holomorphic vector bundle
and endow the canonical line bundles over the quantum sphere with a holomor-
phic structure. We also deﬁne the quantum homogeneous coordinate ring of the
projective line CPq
and identify it with the coordinate ring of the quantum plane.
Finally I shall formulate an analogue of Connes’ theorem, characterizing holomor-
phic structures on compact oriented surfaces in terms of positive currents, to our
noncommutative context. The notion of twisted positive Hochschild cocycles plays
an important role here.
Monopoles connections on the quantum projective plane
We present several results on the geometry of the quantum projective plane.
They include: explicit generators for the K-theory and the K-homology; a real
calculus with a Hodge star operator; anti-selfdual connections on line bundles with
explicit computation of the corresponding invariants; quantum invariants via equi-
variant K-theory and q-indices; and more.
An index theorem in diﬀerential K-theory
Diﬀerential K-theory is a reﬁnement of the usual K-theory of a manifold. Its
objects consist of a vector bundle with a Hermitian inner product, a compatible
connection and an auxiliary diﬀerential form. Given a ﬁber bundle with a Riemann-
ian structure on its ﬁbers, and a diﬀerential K-theory class on the total space, I
will deﬁne two diﬀerential K-theory classes on the base. These can be considered
to be topological and analytic indices. The main result is that they are the same.
This is joint work with Dan Freed.