CONFERENCE TALKS xiii

Hopf-cyclic cohomology and Connes-Moscovici characteristic map

Atabey Kaygun

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-

cyclic cohomology.

Holomorphic structures on the quantum projective line

Masoud Khalkhali

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

1

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

Giovanni Landi

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

John Lott

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.