Nigel Higson lectures 17-21 May 2010
Back to Focused Semester 6
Nigel Higson (
His first lecture and one by Terry Gannon (Alberta) will comprise a Spitalfields Day on the afternoon of 17 May.
Spitalfields Day (lectures will be held in room M/0.40):
Monday 13.00-14.00 Buffet lunch (room M/1.04)
14.00 Terry Gannon (
15.45 Nigel Higson
Rest of week (lectures will be held in room M/2.06):
Tuesday 14.00 Higson II
Wednesday 14.00 Gannon II
Thursday 14.00 Higson III
Friday 14.00 Higson IV
Nigel Higson: The Baum-Connes Conjecture and Group Representations
Operator algebra K-theory has well-known applications in topology and geometry stemming from the index theory of Dirac operators and the Baum-Connes conjecture. But the same techniques also resonate in various ways with Lie groups and representation theory. In this series of lectures I shall try to indicate how this comes about, focusing on some fairly new aspects of the relationship.
Lecture 1 (Spitalfields Talk): C*-algebras, unitary group representations and topology
C*-algebras and the theory of unitary group representations are both roughly sixty years old. The two subjects were practically one and the same during their first decade, but diverged soon after. I shall sketch some of the early history that C*-algebras and then describe developments coming from geometry and index theory that are reconnecting C*-algebras to group representations, particularly in ways that involve the topological structure of the space irreducible unitary group representations.
Lecture 2: Contractions of Lie groups and the Mackey analogy
Let K be the maximal compact subgroup of a connected Lie group G. The "contraction" of G along K is the semidirect product group associated to the adjoint action of K on the quotient of the Lie algebras of G and K. George Mackey proposed that when G is semisimple there ought to be an "analogy" between the unitary representation theories of G and its contraction. As I shall explain, Mackey's proposal is very closely related to the Baum-Connes conjecture for G. I shall examine the particular case of complex semisimple groups, and also briefly discuss the real case.
Lecture 3: Harish-Chandra homomorphisms
This is a continuation of the previous lecture. I shall look at the Mackey analogy for admissible rather than unitary representations, using convolution algebras of distributions on G rather than the group C*-algebra. From this point of view the analogy amounts to a certain generalization of the Harish-Chandra isomorphism theorem in Lie algebra theory.
Lecture 4: The Weyl character formula in KK-theory
Weyl's formula describes the characters of the irreducible representations of compact Lie groups. It has a beautiful relationship with K-theory and index theory, as was pointed out by Atiyah and Bott a long time ago. I shall revisit the subject from the perspective of Kasparov's KK-theory. There are interesting links to the Baum-Connes conjecture that in turn suggest interesting links between Baum-Connes and geometric representation theory.
For a list of participants, click here.
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Back to Focused Semester 6