• Non-commutative Geometry and its Applications
• Special topics series

• Description
• Participants
• Trimester Seminar
• Junior Seminar
• Summer School
• Workshop: Non-commutative geometry's interactions with mathematics
• Workshop: Quantum physics and non-commutative geometry
• Special topics series
• Plücker Lecture
• Workshop: Number theory and non-commutative geometry
• Workshop: Future directions for non-commutative geometry
• Poster
• Recorded Talks
• Publications & Preprints
• Final Report

This will be a series of five lectures - for the most part expository - on index theory. The overall point of view is that the Atiyah-Singer index theorem can be proved as a corollary of Bott periodicity.

Titles for the five lectures are:

#1. DIRAC OPERATOR
The Dirac operator of Rⁿ will be explicitly constructed and Spin-c manifolds will be introduced.

#2. ATIYAH-SINGER REVISITED
The Atiyah-Singer index theorem for Dirac operators will be proved as a corollary of Bott periodicity. In particular, this proves HRR (Hirzebruch-Riemann-Roch).

#3. WHAT IS K-HOMOLOGY ?
K-homology (i.e. the dual theory to K-theory) will be introduced and used to prove that Atiyah-Singer for Dirac operators implies the general case of Atiyah-Singer.

#4. BEYOND ELLIPTICITY
K-homology will be applied to solve the index problem for a class of Fredholm, but not elliptic, differential operators. This is joint work with E. van Erp.

#5. THE RIEMANN-ROCH THEOREM
K-homology will be used to state and prove the Riemann-Roch theorem for complex projective algebraic varieties which are allowed to have singularities. When restricted to non-singular projective varieties, GRR (Grothendieck-Riemann-Roch) is recovered. This is joint work with W. Fulton and R. MacPherson.

In these three lectures I will explain the basic notions of gauge theory for connections on vector bundles over non-commutative manifolds. In classical differential geometry, Simon Donaldson in particular showed how the gauge theory of a given four-dimensional manifold can be used to construct invariants of its smooth structure. I plan to explain as far as possible how one can detect the structure of a non-commutative four-manifold (in the sense of Connes) via its gauge theory.

I will review some recent work in which ideas like the Witten index are connected to NCG methods.

We will explain how to use some particular groupoids and their associated calculus to obtain explicit index formulas for some index problems for singular manifolds. In particular we will concentrate in the famous Atiyah-Patodi-Singer index (for a fully alliptic operator on a manifold with boundary) and we will use very explicit groupoids to do index theory only using classic algebraic topology methods. Behind this there are powerful tools and methods of Noncommutative geometry that we will try to discuss as well. These lectures are a natural sequel (or complement) of those by Eric van Erp at the summer school, I wont suppose however the audience attended those talks.

Pseudodifferential calculus can be used to compute recursively the heat coefficients of an elliptic positive differential operator on a closed manifold. After reviewing this method, the computation of the terms up to a₁₂ in the expansion of the spectral action for Robertson-Wlaker metrics and the proof of Chamseddine-Connes conjecture on rationality of the coefficients for general terms in the expansion will be explained (joint work with A. Ghorbanpour and M. Khalkhali). I will then turn my focus to noncommutative tori, with their flat geometry conformally perturbed by a Weyl factor. Connes' pseudodiffernetial calculus can be applied to compute their geometric invariants, such as scalar curvature, whose final formulas are accompanied by noncommutative features. I will explain my joint works with M. Khalkhali on the analog of Weyl's law, Connes' trace theorem, the extension of the Gauss-Bonnet theorem of Connes and Tretkoff to general translation-invariant conformal structures on noncommutative two-tori, computation of scalar curvature, and extrema of the Einstein-Hilbert action for noncommutative four-tori.

This series of three lectures will cover the topic of anti-Drinfeld doubles from the beginning till the state-of-the-art. Anti-Drinfeld doubles where discovered as algebras whose modules are coefficients of Hopf-cyclic (co)homology. They are Galois objects over the corresponding Drinfeld-double Hopf algebras. The anti-Drinfeld doubles are isomorphic as algebras (not as comodule algebras) with the corresponding Drinfeld-doubles if and only if there exists a modular pair in involution, i.e. a 1-dimensional stable anti-Yetter-Drinfeld module. Kauffman and Radford proved that a modular pair in involution exists for a finite-dimensional Hopf algebra if and only if its Drinfeld-double admits a ribbon element, an important tool to study the topology of knots. Quite recently more structure of anti-Drinfeld doubles was unraveled: they are always bi-Galois objects and, as such, can be used for braided noncommutative join constructions. The consequences of the existence of this bi-Galois structure in Hopf-cyclic (co)homology remains to be found. Time permitting, the course will also cover some aspects of anti-Drinfeld doubles of infinite-dimensional Hopf algebras.

In representation theory (as developed by Harish-Chandra and friends) it is typical to concentrate on irreducible group representations. In contrast, noncommutative geometers often deal with representations that are far from irreducible, but instead are projective (as modules over an appropriate group convolution algebra); so to speak they are continuous families of irreducible representations. In these lectures I want to examine some foundational points in Harish-Chandra's tempered representation theory from a perspective suggested by noncommutative geometry. The emphasis will be on parabolic induction, which creates continuous families of representations, and its adjoint functor of parabolic restriction. It appears that operator algebraic methods can play an illuminating role here.

In this series of lectures I will give an introduction to the current developments in unbounded KK-theory. The starting point for these investigations is to find explicit unbounded representatives for interior Kasparov products in bounded KK-theory. An example would here be to represent a K-homology class by an explicit spectral triple. This turns out to be deeply linked to the understanding of differentiable structures in Hilbert C*-modules.

After having reviewed the general framework I will focus on a situation of particular interest for the theory: One could consider an ideal in a C*-algebra which already carries a spectral triple (for example an open subset in n-dimensional Euclidean space). The problem of computing the unbounded Kasparov product then amounts to (the highly non-trivial task of) restricting the spectral triple to the ideal in question.

The construction of new spectral triples via the unbounded Kasparov product has in recent years produced a myriad of examples of operator spaces. This is of particular interest for purely infinite C*-algebras, such as Cuntz-Krieger algebras, boundary crossed products and quantum groups. The phenomenon is however already present in the case of manifolds, eg the 3-dimensional Hopf fibration. The examples will serve as a guide in the lectures. More abstractly, the lifiting problem from bounded to unbounded KK-theory will be addressed in the context of differential approximate identities.

Apart from the usual unbounded operators defining KK-theory cycles and spectral triples, a host of objects with unbounded differential norms naturally appear. Instead of disposing of these objects as untractable, it is possible to cast their analysis in the framework operator space theory. In this way, solid meaning can be given to clean algebraic expressions arising from the Kasparov product. In the context of approximate identities, the notion of quasicentrality can be developed on a differential level, resulting in a geometric approach to the lifting problem.

1- Geometric: A review is given on the characteristic invariants of foliations from Hopf cyclic cohomology point of view. Indication  is given on the works of Bott, Connes-Moscovici, and Moscovici-R.

2- Algebraic: Development of algebraic set up for Hopf cyclic cohomology is presented. We start with the general set up of Hopf cyclic cohomology with coefficients i.e. the works of Hajac-Khalkhali-R-Sommerhauser. We then present  Lie-Hopf algebras as the main tool for our study of geometric Hopf algebras at the works of R-Sutlu.

3- Topologic: It is shown that Hopf cyclic cohomology is much more applicable and understandable when we consider topology. By considering topological Hopf algebras we are able to answer questions left open in the algebraic and geometric cases. The ultimate coefficients space for Hopf cyclic cohomology is presented and manifested. The Lie Hopf algebras are now defined for infinite Lie algebras. The topological Weil complex of Hopf algebras is defined. This part is based on the recent work in progress of R-Sutlu.

The subject of my lectures will be the interplay between topological dynamics and C*-algebras. In particular, we shall see how the construction of the C*-algebra of a groupoid can be made functorial. This requires to define the morphisms as correspondences. The notion of C*-correspondence is now classical; there is a suitable notion of groupoid correspondence which will be presented. As examples and applications, we shall discuss induced representations, hypergroupoids, and Hecke C*-algebras.

I will address the following question. If is a unital -algebra Poincaré dual to its opposite algebra , can we lift this Poincaré duality to Cuntz-Pimsner algebras of -correspondences over .

To obtain a concrete Poincaré dual for , we need to belong to a class of -bimodules which we introduce here, along with their basic properties and many examples. The potential Poincaré dual algebra comes from the opposite bimodule structure.

To actually prove Poincaré duality for these algebras requires further compatibility between the dynamics defined by and the geometry encoded in explicit representatives of the Poincaré duality classes for .

This work was inspired by trying to find a Kasparov module picture of the Poincaré duality results of Kaminker-Putnam for Cuntz-Krieger algebras, and uses the unbounded approach to the Kasparov product to make the constructions as concrete as possible. This is joint work with Dave Robertson and Aidan Sims.

Part 1: Quotients by actions of finite groups
Part 2: Noncommutative circle bundles

There is a plenitude of interesting manifolds, which are obtained as quotient manifolds by an action of a finite group. A good example are quotients of tori, which are so-called Bieberbach manifolds or quotients of spheres (lens spaces). In the classical case there are only two 2-dimensional Bieberbach manifolds, however, it appears that in the noncommutative case a pillow (orbifold with four corners) is more regular. I will discuss the problem how to distingush noncommutative "manifolds" from "orbifolds" using various methods. The quotients, which come from the noncommutative circle bundles offer a unique chance to relate the algebraic construction of (strong) connections on quantum principal bundles with the geometric construction of Dirac operators and spectral triples and give a new insight which classes of Dirac operators one can consider.

Part 1: I will introduce some basic local and non-local spectral invariants of Dirac type and Laplace type operators on compact Riemannian manifolds and discuss the relation between them and the underlying geometry. This will include a short discussion of the Duistermaat Guillemin wavetrace formula and derived Weyl theorems.

Part 2: I will review properties of the microlocalized counting function and will explain how it relates to the spectral zeta function and the heat expansions. I will show how some error estimates are obtained from Fourier Tauberian theorems. After a brief discussion of non-local spectral invariants I will review the notions of Quantum Ergodicity and Quantum unique ergodicity and discuss some recent results.

Part 3: This lecture will be about spectral methods on non-compact manifold. I will discuss stationary scattering theory on manifolds with fixed structure at infinity. The properties of the resolvent of the Laplace operator play an important role in scattering theory. I will give some examples of expansions that can be obtained in this context and will explain the notion of zero resonance states and how they relate to the spectral decomposition. I will also briefly review some recent results obtained for low energy expansions of the resolvent.

We describe nonassociative deformations of geometry probed by closed strings in non-geometric flux compactifications of string theory. We show that these non-geometric backgrounds can be geometrised through the dynamics of open membranes whose boundaries propagate in the phase space of the target space compactification, equiped with a twisted Poisson structure. The effective membrane target space is determined by the standard Courant algebroid over the target space twisted by an abelian gerbe in momentum space. Quantization of the membrane sigma-model leads to a proper quantization of the non-geometric background, which we relate to Kontsevich’s formalism of global deformation quantization that constructs a noncommutative nonassociative star product on phase space. We construct Seiberg–Witten type maps between associative and nonassociative backgrounds, and show how they may realise a nonassociative deformation of gravity. We also explain how this approach is related to the quantization of certain Lie 2-algebras canonically associated to the twisted Courant algebroid, and cochain twist quantization using suitable quasi-Hopf algebras of symmetries in the phase space description of R-space which constructs a Drinfel’d twist with non-trivial 3-cocycle. We illustrate and apply our formalism to present a consistent phase space formulation of nonassociative quantum mechanics.

The signature is a fundamental homotopy invariant for topological manifolds. However, for spaces with singularities, this usual notion of signature ceases to exist, since, in general, spaces with singularities fail the usual Poincaré duality. A generalized Poincaré duality theorem for spaces with singularities was proven by Goresky and MacPherson using intersection homology. The classical signature was then extended to Witt spaces by Siegel using this generalized Poincaré duality. Witt spaces are a natural class of spaces with singularities. For example, all complex algebraic varieties are Witt spaces. In these talks, I will describe a combinatorial approach to the higher signature of Witt spaces, using methods of noncommutative geometry. This is based on joint work with Nigel Higson.

We plan to give a gentle (and informal) introduction to some areas of number theory surrounding the absolute Galois group of the rational numbers, i.e. the symmetry group of algebraic integers, and some of its invariants, such as L-functions. Many outstanding problems in number theory are related to these structures but only very little is known so far, despite the efforts of many generations of mathematicians. We’re going to argue that non-linear structures are underlying the absolute Galois group of the rational numbers and explain why this necessitates the appearance of Noncommutative Geometry. (Thereto it will be absolutely crucial to understand how all three main flavors of NCG, the french, the japanese and the russian, are related to these non-linear structures).

Our aim for these lectures is not to give a thorough introduction into number theory or noncommutative geometry, this wouldn’t be possible in a handful of lectures, instead our goal is to give an idea of the "big picture", which is certainly missing in the literature. We’ll give plenty of examples, references and open problems and hope to inspire in particular the young participants.

We expect a certain mathematical maturity from the audience but otherwise we’re not going to expect any prerequisites. The course will be based on explicit examples and all notions, in particular from number theory, are going to be introduced.