THE TALKS

NIGEL HIGSON

Noncommutative geometry of the Satake compactification

Roughly speaking, noncommutative geometry is about the application of ideas from C*-algebra theory and Hilbert space theory to problems in other parts of mathematics. Very often, the path from C*-algebras to applications goes through groupoids, which is where the geometry is, and groupoid C*-algebras,  and in this talk I shall present such an example, worked out by Jacob Bradd, Bob Yuncken and myself.  The application is to representation theory in the sense of Harish-Chandra, and specifically to a basic principle of Harish-Chandra that describes the general form of the representations used in his famous Plancherel formula.  The groupoid comes from Satake’s compactification of the symmetric space that may be associated to any of Harish-Chandra’s real reductive groups.  The possibility that such an approach to Harish-Chandra’s principle might exist was suggested to us by Alain Connes; the groupoid itself was invented by Omar Mohsen. With the groupoid in hand, Harish-Chandra’s principle more or less proves itself.  I shall tell the first part of the story in my lecture, and Jacob will tell the tell the rest of it in his.

OMAR MOHSEN

TBA

ALAIN CONNES

Correspondences and Isospectrality

I will explain the role of the notion of correspondence in noncommutative geometry.

HERVE OYONO-OYONO

K-theoretical computability for groupoids

In this lecture, we discuss some K-theoretical computability problems for groupoid crossed-product algebras  in connection with the Baum-Connes conjecture. We first review some recent results of K. Mao extending the going-down technics developed for locally compact groups by J. Chabert, S. Echterhoff and myself  to the framework of étale groupoids. Inspired by some ideas of E. Guentner, R. Willet and  G. Yu, we then introduce the notion of geometric decomposition for groupoids and give applications to these computability questions.

EDWARD McDONALD

The trace theorem for Carnot manifolds

Connes' trace formula states that the Wodzicki residue of a pseudodifferential operator on a compact manifold is equal to its Dixmier trace. Carnot manifolds (also called regular filtered manifolds, among other names) have their own natural pseudodifferential calculus and an associated residue functional. I will explain why this associated residue is also given by a singular trace.

EVA-MARIA HEKKELMANN

The 1915 Edition of Connes' Integral Formula

The noncommutative integral in NCG is based on Connes' trace theorem from 1988. I will attempt to convince you that a theorem by Szegő from 1915 for Toeplitz matrices can be related to Connes' trace theorem on the circle.  Arguably more useful is that Szegő's result can be generalised way beyond the circle case, leading to a noncommutative version of Szegő's limit theorem. This is intimately connected to the Connes--Van Suijlekom study of spectrally truncated spectral triples. In particular, we obtain a formula for the noncommutative integral adapted to these spectral truncations. I will then highlight how the mathematics involved establishes a connection between NCG and the field of Quantum Ergodicity, and discuss local Weyl laws. Based on joint work with Edward McDonald.

BRAM MESLAND

A category of correspondences for spectral triples

In their work on the foliation index theorem, Connes and Skandalis showed that the KK-theory of manifolds can be described entirely by correspondences, and characterised the Kasparov product in KK-theory in terms of connections. In Connes’ paradigm of noncommutative geometry, manifolds are replaced by spectral triples, and Baaj-Julg put forward a definition of unbounded cycle for KK-theory. In this talk I will give a definition of correspondence of spectral triples as smooth KK-cycles equipped with a connection. Such correspondences can be composed on the nose (as opposed to up to equivalence) to form a category. This category is flexible enough to accommodate natural geometric and noncommutative examples, yet rigid enough that it comes equipped with a surjective functor onto KK-theory. It provides a natural context for the study of curvature, the bivariant Chern character and other fundamental notions of noncommutative geometry. This talk reports on unpublished work in progress. 

JACOB BRADD

The Satake groupoid and the Harish-Chandra principle

(Joint with Nigel Higson and Robert Yuncken.) This is effectively a continuation of Nigel's talk. There we saw (will see) the construction of the Satake groupoid associated to a real reductive group, as well as an outline of its structure. In this talk we will recall this and then explore more precisely the structure of the C*-algebra of this groupoid, and define an integration map (originally due to Omar Mohsen) that gives a morphism from the reduced group C*-algebra to this groupoid C*-algebra. As a consequence, and a demonstration of the utility of this groupoid, we will obtain an easy and geometric proof of the Harish-Chandra principle, which is the foundational fact that the representations appearing in the Plancherel formula of a real reductive group are either subrepresentations of L^2 ("discrete series"), or representations induced by such representations on smaller subgroups ("parabolically induced" representations, or principal series).

PAOLO PIAZZA

Witt pseudomanifolds, Gysin homomorphisms in K-homology and a formula for the Goresky-MacPherson G-signature

Gysin homomorphisms in K-theory and K-homology and their functoriality properties have played a major role in index theory (on smooth manifolds, Lipschitz manifolds and foliations). Using stable homotopy theory,  Banagl has defined Gysin homomorphisms in K-homology for Witt pseudomanifolds and proved that they preserve the  Sullivan-Siegel orientation class. Examples of Witt pseudomanifols include complex projective varieties. In this talk I will explain an analytic approach to these results, using KK-theory and the signature operator.  This analytic approach will in fact allow for much more general results. I will also discuss an application of these techniques to the formulation and proof of an Atiyah-Segal-Singer formula for the Goresky-MacPherson G-signature of a Witt pseudomanifold. The first part of the talk is joint work with Pierre Albin and Markus Banagl. The second part, on the G-signature formula, is joint work with Markus Banagl and Eric Leichtnam.

SARA AZZALI

Traces in KK-theory and index pairings

Traces on C*-algebras play an important role in index theory, for instance they allow to extract numerical  invariants from classes in K-theory. When introducing real coefficients in Kasparov bivariant K-theory, traces  naturally give rise classes in KK- theory with real coefficients. In this talk, we explain these constructions  and some of their applications. In particular, we present a natural class that represents the Godbillon-Vey  invariant of a foliation of codimension one. This work is in collaboration with Paolo Antonini (Università del  Salento) and Georges Skandalis (Université Paris Cité).

ATABEY KAYGUN

The Quantum van Est

The van Est isomorphism reduces Lie group (co)homology to Lie algebra (co)homology. In this talk, we investigate the quantum group/enveloping algebra analog of the reduction, and the advancements (the proverbial raising sea) of the last 25 years that led to this result. (Joint work with Serkan Sütlü).

MARCELO LACA

Toeplitz C*-algebras of product systems of correspondences

Given a product system of correspondences over a sub-monoid of a group, we show that, under some technical  assumptions, the fixed-point subalgebra of the gauge action on the Toeplitz C*-algebra is nuclear if and only if the coefficient algebra is nuclear. When the group is amenable, we conclude that this happens if and only if the Toeplitz algebra is nuclear. I will discuss several classes of product systems for which nuclearity of the Toeplitz C*-algebra follows from nuclearity of the coefficient algebra.  

This is joint work with E. G. Katsoulis and C.F. Sehnem.

EVGENIOS KAKARIADIS

Symmetrisation products and systems of operators

Operator systems have a central role in the theory of operator algebras. Lately, the community has been actively considering selfadjoint operator subspaces, but which need not be unital. Those appear naturally, for example when comparing operator systems up to their stabilisations as in the work of Connes and van Suijlekom. The absence of a unit disrupts the link between the norm and the matrix order structure, creating significant difficulties in obtaining even fundamental theorems like the celebrated Arveson's Extension Theorem.

In the first part of this talk I will present how such problems are resolved by working with morphisms that their unitisation (in the sense of Werner) is completely isometric. Furthermore I will provide (old and new) characterisations of such maps in terms of extensions and gauge isometries in the sense of Russell), with a focus on approximately positively generated or singly generated spaces.

Stabilisations can be seen as an incarnation of Morita equivalence for operator systems, similar to Rieffel's notion for C*-algebras introduced in the 60s. In the second part of this talk I will give equivalent characterizations via Morita contexts, bihomomoprhisms and a symmetrisation product, while I will highlight properties that are invariant. Time permitted I will provide applications to rigid systems, function systems and non-commutative graphs.

This talk is based on joint works with Alexandros Chatzinikolaou, Joseph Dessi, George Eleftherakis, Se-Jin Kim, Apollonas Paraskevas, and Ivan Todorov.

JEAN RENAULT

Boolean inverse semigroups and groupoids

The intimate connection between groupoids and inverse semigroups has been observed long time ago. In the case of Boolean groupoids and Boolean inverse semigroups, this connection can be expressed as an equivalence of category which extends Stone duality. In view of applications to operator algebras, two complements will be given: a twisted version of this equivalence and the realization of any separable measure inverse semigroup as the full inverse semigroup of a second countable Boolean groupoid.