Miniconference
NONCOMMUTATIVE GEOMETRY
AND QUANTUM GROUPS
2008

2 June – 6 June, 2008
Stefan Banach International Mathematical Center, Warsaw, Poland

Organizing Committee: P. M. Hajac (Warszawa), N. Higson (State College), R. Meyer (Göttingen)

2 June 2008 (10:15 - 11:45)
Banach Centre, Śniadeckich 8 (room 408)

CONTRACTIONS OF SEMISIMPLE GROUPS AND THE MACKEY ANALOGY

Suppose that G is a connected Lie group and that K is a maximal compact subgroup of G. There is a smooth family of Lie groups Gt, t a real number, such that Gt = G when t is not 0, and such that G0 is the semidirect product group associated to the adjoint action of K on the quotient of the Lie algebra of G by the Lie algebra of K. The group G0 is called a contraction of G, and in a 1975 paper Mackey proposed that, when G is semisimple, the unitary representation theories of G and G0 ought to be analogous to one another. Mackey's proposed analogy is very closely related to the Connes-Kasparov conjecture in C*-algebra K-theory. I shall briefly review this fact, and then examine the analogy from the related, but different, point of view of Harish-Chandra modules and Hecke algebras.

NIGEL HIGSON

3 June 2008 (12:00 - 13:30)
Institute of Mathematics, Warsaw University, Banacha 2 (room 5870)

NON-COMMUTATIVE TOPOLOGY

I will survey properties of bivariant K-theory and contrast them with the stable homotopy category. After that, I explain a general framework for doing homological algebra in triangulated categories, which is general enough to apply in the context of bivariant K-theory.

RALF MEYER

4 June 2008 (10:15 - 13:45)
Institute of Mathematics, Warsaw University, Banacha 2 (room 2180)

NONCOMMUTATIVE GEOMETRY AND QUANTUM GROUPS

NIGEL HIGSON

4 June 2008 (15:00 - 16:00)
Banach Centre, Śniadeckich 8 (room 106)

FOUNDATIONS OF SPECTRAL GEOMETRY

LUDWIK DĄBROWSKI

5 June 2008 (10:15 - 12:00)
Department of Mathematical Methods in Physics, Warsaw University, Hoża 74 (5th floor)

FROM POISSON TO QUANTUM GEOMETRY

NICOLA CICCOLI

5 June 2008 (13:15 - 15:00)
Department of Mathematical Methods in Physics, Warsaw University, Hoża 74 (5th floor)

QUANTUM LORENTZ GROUP AND DOUBLE GROUP CONSTRUCTION

STANISŁAW L. WORONOWICZ

6 June 2008 (12:00 - 12:30)
Banach Centre, Śniadeckich 8 (room 321)

THE EXISTENCE OF SINGULARITIES AND THE ORIGIN OF SPACE-TIME

MICHAŁ HELLER

6 June 2008 (12:45 - 13:15)
Banach Centre, Śniadeckich 8 (room 321)

WHAT IS THE ATIYAH-SINGER INDEX THEOREM?

I shall try to explain the connections between linear elliptic partial differential equations and topology that led to the formulation and proof of the famous Atiyah-Singer index theorem. I shall include all the necessary definitions during the talk (or at least informal versions of them) and as a result the lecture will be accessible to all graduate students.

NIGEL HIGSON

6 June 2008 (13:30 - 14:00)
Banach Centre, Śniadeckich 8 (room 321)

QUANTUM GROUPS

PIOTR M. SOŁTAN

6 June 2008 (14:15 - 14:45)
Banach Centre, Śniadeckich 8 (room 321)

FROM GROUPOIDS TO C*-ALGEBRAS: THE EXAMPLE OF ROTATION ALGEBRAS

The construction of C*-algebras for badly behaved quotient spaces proceeds via groupoids. Since this groupoid is, in general, only unique up to Morita equivalence, the resulting C*-algebras are also only unique up to Morita equivalence or, equivalently, stable isomorphism. I will illustrate this situation by considering rotation algebras. These describe the quotient space of the real numbers by a dense subgroup with two generators. Various equivalent descriptions of this non-commutative space use rotations of the circle with irrational angle and the Kronecker foliation of the 2-torus.

RALF MEYER