Advances in Noncommutative Geometry: On the Occasion of Alain Connes' 70th Birthday

Foreword
Contents
A survey of spectral models of gravity coupled to matter
	1 Introduction
	2 Early days of the spectral Standard Model
		2.1 Noncommutative spaces and differential calculus
		2.2 Two-sheeted space-time
		2.3 Constructions beyond the Standard Model
		2.4 Coupling matter to gravity
	3 The spectral action principle
		3.1 Real structures on spectral triples
		3.2 The spectral action principle
	4 The spectral Standard Model
		4.1 Classification of irreducible geometries
		4.2 Noncommutative geometry of the Standard Model
		4.3 The gauge and scalar fields as inner fluctuations
		4.4 Spectral action
		4.5 Fermionic action in KO-dimension 6
		4.6 Phenomenological consequences
	5 Beyond the Standard Model with noncommutative geometry
		5.1 Resilience of the spectral Standard Model
		5.2 Pati–Salam unification and first-order condition
		5.3 Grand symmetry and twisted spectral triples
		5.4 Algebraic constraints on the finite geometry
	6 Volume quantization and uniqueness of SM
		6.1 Higher form of Heisenberg\'s commutation relations
		6.2 Volume quantization
	7 Outlook: towards quantization
	Appendix: Pati–Salam model: potential analysis
	References
The Riemann–Roch strategy
	1 Introduction
	2 The geometry behind the zeros of ζ
		2.1 Adelic approach
			2.1.1 Forward compactification
			2.1.2 Stability of noncommutative nature of quotients
			2.1.3 The need for the NCG point of view
			2.1.4 Classical orbit and cohomological meaning of the map E
		2.2 The Scaling Site
		2.3 Geometry of the adèle class space
	3 The Riemann–Roch strategy
		3.1 The role of the existence part of the Riemann–Roch formula in characteristic one
		3.2 Tropical descent
			3.2.1 Tropicalization in the p-adic case, Newton polygons
			3.2.2 Tropicalization in the Archimedean case, Jensen\'s formula
			3.2.3 Descent from characteristic zero to characteristic one
		3.3 The Hirzebruch–Riemann–Roch formula and the Index theorem
		3.4 Potential role of the complex lift of the Scaling Site
	4 Tropical descent and almost periodic functions
		4.1 Almost periodic functions
		4.2 From Jensen to Jessen and the tropical descent
		4.3 Discrete lift of a continuous divisor
	5 The complex lift of the Scaling Site
		5.1 Adelic almost periodic compactification of R
		5.2 The adelic complex lift
		5.3 The periodic orbits
		5.4 The classical orbit
	6 The moduli space interpretation
		6.1 Notations
		6.2 The relation with the GL(2)-system
		6.3 Commensurability classes of parabolic Q-lattices
		6.4 CQ and ΓQ as moduli spaces of elliptic curves
		6.5 Commensurability and isogenies
		6.6 The complex structure
		6.7 The right action of P()
		6.8 Boundary cases
	7 Lift of the Frobenius correspondences
		7.1 Witt construction in characteristic 1 and lift of the Ψλ
		7.2 Quantization
	References
The Baum–Connes conjecture: an extended survey
	1 Introduction
		1.1 Building bridges
		1.2 In a nutshell: without coefficients…
		1.3 …and with coefficients
		1.4 Structure of these notes
		1.5 What do we know in 2019?
		1.6 A great conjecture?
		1.7 Which mathematics are needed?
	2 Birth of a conjecture
		2.1 Elliptic (pseudo-) differential operators
		2.2 Square-integrable representations
		2.3 Enters K-theory for group C-algebras
		2.4 The Connes–Kasparov conjecture
		2.5 The Novikov conjecture
	3  Index maps in K-theory: the contribution of Kasparov
		3.1 Kasparov bifunctor
		3.2 Dirac induction in KK-theory
		3.3 The dual-Dirac method and the γ-element
		3.4 From K-theory to K-homology
		3.5 Generalization to the p-adic case
	4 Towards the official version of the conjecture
		4.1 Time-dependent left-hand side
		4.2 The classifying space for proper actions, and its K-homology
		4.3 The Baum–Connes–Higson formulation of the conjecture
		4.4 Generalizing the γ-element method
			4.4.1 The case of groups acting on bolic spaces
			4.4.2 Tu\'s abstract gamma element
			4.4.3 Nishikawa\'s new approach
		4.5 Consequences of the Baum–Connes conjecture
			4.5.1 Injectivity: the Novikov conjecture
			4.5.2 Injectivity: the Gromov–Lawson–Rosenberg conjecture
			4.5.3 Surjectivity: the Kadison–Kaplansky conjecture
			4.5.4 Surjectivity: vanishing of a topological Whitehead group
			4.5.5 Surjectivity: discrete series of semisimple Lie groups
	5 Full and reduced C-algebras
		5.1 Kazhdan vs. Haagerup: property (T) as an obstruction
		5.2 A trichotomy for semisimple Lie groups
		5.3 Flag manifolds and KK-theory
			5.3.1 The BGG complex
			5.3.2 The model: SO0(n,1)
			5.3.3 Generalization to other rank one groups
			5.3.4 Generalization to higher rank groups
	6 Banach algebraic methods
		6.1 Lafforgue\'s approach
			6.1.1 Banach KK-theory
			6.1.2 Bost conjecture and unconditional completions
			6.1.3 Application to the Baum–Connes conjecture
			6.1.4 The rapid decay property
		6.2 Back to Hilbert spaces
			6.2.1 Uniformly bounded and slow growth representations
			6.2.2 Cowling representations and γ
			6.2.3 Lafforgue\'s result for hyperbolic groups
		6.3 Strong property (T)
		6.4 Oka principle in Noncommutative Geometry
			6.4.1 Isomorphisms in K-theory
			6.4.2 Relation with the Baum–Connes conjecture
			6.4.3 Weighted group algebras
	7 The Baum–Connes conjecture for groupoids
		7.1 Groupoids and their C-algebras
		7.2 Counter-examples for groupoids
	8 The coarse Baum–Connes conjecture (CBC)
		8.1 Roe algebras
			8.1.1 Locality conditions on operators
			8.1.2 Paschke duality and the index map
		8.2 Coarse assembly map and Rips complex
			8.2.1 The Rips complex and its K-homology
			8.2.2 Statement of the CBC
			8.2.3 Relation to the Baum–Connes conjecture for groupoids
			8.2.4 The descent principle
		8.3 Expanders
		8.4 Overview of CBC
			8.4.1 Positive results
			8.4.2 Negative results
		8.5 Warped cones
	9 Outreach of the Baum–Connes conjecture
		9.1 The Haagerup property
		9.2 Coarse embeddings into Hilbert spaces
		9.3 Yu\'s property A: a polymorphous property
			9.3.1 Property A
			9.3.2 Boundary amenability
			9.3.3 Exactness
		9.4 Applications of strong property (T)
			9.4.1 Super-expanders
			9.4.2 Zimmer\'s conjecture
	References
	References
Lie groupoids, pseudodifferential calculus, and index theory
	1 Introduction
	2 Lie groupoids and their operator algebras
		2.1 Generalities on Lie groupoids
			2.1.1 Morita equivalence of Lie groupoids
		2.2 C*-algebra of a Lie groupoid
			2.2.1 Convolution*-algebra of smooth functions with compact support
		2.3 Norm and C*-algebra
		2.4 Deformation to the normal cone and blowup groupoids
			2.4.1 Deformation to the normal cone groupoid
			2.4.2 Blowup groupoid
	3 Pseudodifferential calculus on Lie groupoids
		3.1 Distributions on G conormal to G(0)
			3.1.1 Symbols and conormal distributions
			3.1.2 Convolution
			Push-forward, pull-back, product of distributions
			3.1.3 Pseudodifferential operators of order ≤0
			3.1.4 Analytic index
		3.2 Classical examples
		3.3 Analytic index via deformation groupoids
		3.4 Deformation to the normal cone, zooming action and PDO
			3.4.1 The zooming action of R+* on a deformation to the normal cone
			3.4.2 Integrals of smooth functions
			3.4.3 Almost equivariant distributions
		3.5 Some generalizations
			3.5.1 More general families of pseudodifferential operators
			3.5.2 Inhomogeneous calculus
			3.5.3 Fourier integral operators
	4 Constructions based on Lie groupoids and their deformations
		4.1 The associated index map
			4.1.1 The \"Dirac element\" of a Lie group
			4.1.2 Foliation and shriek map for immersions
			4.1.3 On the computation of the index map in some cases
			4.1.4 Full index
		4.2 Groupoids using deformation constructions
			4.2.1 Pseudodifferential calculi on singular manifolds
			4.2.2 Inhomogeneous pseudodifferential calculus
			4.2.3 Poincaré dual of a stratified manifold
	5 Related topics and further questions
		5.1 Relation to Roe algebras
		5.2 Singular foliations and \"singular Lie groupoids\"
		5.3 Computations using cyclic cohomology
		5.4 Behavior of the resolvent of an elliptic operator
		5.5 Relations with the Boutet de Monvel calculus
		5.6 Algebroids and integrability
		5.7 Fourier integral operators
	References
Cyclic homology in a special world
	1 Cyclic homology
		1.1 Prehistory
		1.2 Cyclic homology
		1.3 The ``topological\'\' version
		1.4 The cyclotomic trace
		1.5 The original construction of TC
		1.6 The Nikolaus–Scholze approach
	2 The special version
		2.1 -spaces as a generalization of symmetric monoids
			2.1.1 Smash as a generalization of tensor
			2.1.2 Special -spaces
		2.2 Symmetries on -spaces
			2.2.1 Fixed points
			2.2.3 Model structures
			2.2.7 Special fibrant replacements and geometric fixed points
		2.3 Fixed points of smash powers
			2.3.2 Smash powers
			2.3.4 The Geometric diagonal
	3 TC in a special world
		3.1 Cyclic objects
		3.2 Edgewise subdivision
		3.3 (Topological) Hochschild homology
		3.4 Topological cyclic homology
	4 On modules and monoids
		4.1 Linearization
			4.1.4 The special path monoid
	References
Curvature in noncommutative geometry
	1 Introduction
	2 Curvature in noncommutative geometry
		2.1 A brief history of curvature
		2.2 Laplace type operators and Gilkey\'s theorem
		2.3 Noncommutative Chern-Weil theory
		2.4 From spectral geometry to spectral triples
	3 Pseudodifferential calculus and heat expansion
		3.1 Classical pseudodifferential calculus
		3.2 Small-time heat kernel expansion
		3.3 Pseudodifferential calculus and heat kernel expansion for noncommutative tori
	4 Gauss-Bonnet theorem and curvature for noncommutative 2-tori
		4.1 Scalar curvature and Gauss-Bonnet theorem for Tθ2
		4.2 The Laplacian on (1, 0)-forms on Tθ2 with curved metric
	5 Noncommutative residues for noncommutative tori and curvature of noncommutative 4-tori
		5.1 Noncommutative residues
		5.2 Scalar curvature of the noncommutative 4-torus
	6 The Riemann curvature tensor and the term a4 for noncommutative tori
		6.1 Functional relations
		6.2 A partial differential system associated with the functional relations
		6.3 Action of cyclic groups in the differential system, invariant expressions and simple flow of the system
		6.4 Gradient calculations leading to functional relations
		6.5 The term a4 for non-conformally flat metrics on noncommutative four tori
	7 Twisted spectral triples and Chern-Gauss-Bonnet theorem for ergodic C*-dynamical systems
		7.1 Twisted spectral triples
		7.2 Conformal perturbation of a spectral triple
		7.3 Conformal perturbation of the flat metric on Tθ2
		7.4 Conformally twisted spectral triples for C*-dynamical systems
		7.5 The Chern-Gauss-Bonnet theorem for C*-dynamical systems
	8 The Ricci curvature
		8.1 A Weitzenböck formula
		8.2 Ricci curvature as a spectral functional
		8.3 Spectral zeta function and the Ricci functional
		8.4 The de Rham spectral triple for the noncommutative two torus
		8.5 The twisted de Rham spectral triple
		8.6 Ricci functional and Ricci curvature for the curved noncommutative torus
	9 Beyond conformally flat metrics and beyond dimension four
		9.1 Rearrangement lemma revisited
		9.2 A new idea
		9.3 Newton divided differences
		9.4 Laplace type h-differential operators and asymptotic expansions
		9.5 Functional metrics and scalar curvature
		9.6 Twisted product, warped product, and scalar curvature
		9.7 Dimension two and Gauss-Bonnet theorem
	10 Matrix Gauss-Bonnet
		10.1 Matrix curvature
		10.2 The Gauss-Bonnet theorem
		10.3 Higher genus matrix Gauss-Bonnet
	11 Curvature of the determinant line bundle
		11.1 The determinant line bundle
		11.2 The canonical trace and noncommutative residue
		11.3 Log-polyhomogeneous symbols
		11.4 Cauchy-Riemann operators on noncommutative tori
		11.5 The curvature of the determinant line bundle for Aθ
		11.6 Variations of LogDet and curvature form
	12 Open problems
	References
Index theory and noncommutative geometry: a survey
	1 Introduction
	2 Atiyah and Singer
	3 The Gysin map
	4 The tangent groupoid
	5 K-homology
		5.1 Analytic K-homology
		5.2 Geometric K-homology
	6 Elliptic families
	7 The adiabatic groupoid
	8 Foliations
	9 The Baum–Connes conjecture
	10 Cohomological index formulas and local index theory
	11 Cyclic complexes
	12 The longitudinal index formula in cyclic theory
	13 Finitely summable Fredholm modules
	14 Unbounded picture
	15 Locality and spectral invariants
	16 Pseudodifferential calculus for spectral triples
	17 Dimension Spectrum
	18 The local index formula in the odd case
	19 Renormalization
	20 The even case
	References
Modular Gaussian curvature
	1 Introduction
	2 Curvature of modular spectral triples
		2.1 Flat spectral triples
		2.2 Modular spectral triples
		2.3 Modular curvature
		2.4 Modular Gauss–Bonnet formula
		2.5 Variation of determinant and modular Gaussian curvature
	3 Morita invariance of the modular curvature
		3.1 Foliation algebras and Heisenberg bimodules
		3.2 Modular Heisenberg spectral triples
		3.3 Ray–Singer determinant vs. Yang–Mills functional
		3.4 Invariance of the Gaussian curvature
	4 Pseudodifferential multipliers and symbol calculus
		4.1 Ordinary DO in Rn from the point of view of C*-dynamical systems
			4.1.1 Standard representation on the L2-space (GNS space)
			4.1.2 Pseudodifferential multipliers
		4.2 Pseudodifferential multipliers on twisted crossed products
		4.3 Differential multipliers
		4.4 Differential multipliers of order 1 and 2
			4.4.1 Differential multipliers in dimension n=2
	5 The resolvent expansion and trace formula
		5.1 Second heat coefficient
		5.2 Effective pseudodifferential operators and trace formulas
	References
Advances in Dixmier traces and applications
	1 Introduction
	2 Preliminaries
		2.1 Ideals and singular values
		2.2 Trace class and the trace
		2.3 Weak trace class and Dixmier\'s trace
			2.3.1 Weak trace class
			2.3.2 Extended limits
			2.3.3 Dixmier trace
	3 Existence and construction of traces
		3.1 Existence of traces and the commutator subspace
			3.1.1 Calkin correspondence
			3.1.2 Commutator subspace
			3.1.3 Existence of traces
		3.2 Construction of traces and dyadic averages
			3.2.1 Dixmier trace
			3.2.2 Dyadic averages
			3.2.3 Characterisation of traces
		3.3 Spectrality and expectation values
			3.3.1 Spectral formulation and limits of expectation values at infinity
			3.3.2 Log submajorisation
	4 Calculation and independence from the singular trace
		4.1 Zeta functions and heat kernels
			4.1.1 Zeta function and residues
			4.1.2 Heat kernel
			4.1.3 Calculation of leading terms
			4.1.4 The second term
		4.2 Measurability of operators
			4.2.1 Measurability
			4.2.2 Universal measurability
			4.2.3 Products
			4.2.4 Dyadic averaging and remainders
			4.2.5 Examples from fractals
		4.3 Fubini theorem
	5 Recent applications
		5.1 Integral operators and symbols
			5.1.1 Noncommutative residue
			5.1.2 Noncommutative integral and symbols
			5.1.3 Noncommutative symbols
			5.1.4 Noncommutative torus and noncommutative plane
		5.2 Integration of functions and Cwikel estimates
			5.2.1 Integration of functions
			5.2.2 Cwikel estimates
		5.3 Integration of forms
			5.3.1 Quantum differentiability
			5.3.2 Hochschild character formula
	References
Commutants mod normed ideals
	1 Introduction
	2 Background on normed ideals
	3 The theorems of Weyl–von Neumann–Kuroda and of Kato–Rosenblum
	4 The theorem of Voiculescu
	5 The invariant kI(τ)
	6 Some uses of kI(τ)
	7 Perturbations of commuting n-tuples of Hermitian operators
	8 k-p(τ) at the endpoint p = ∞ and dynamical entropy
	9 Finitely generated groups and supramenability
	10 The commutant mod a normed ideal E(τ;I) and its compact ideal K(τ;I)
	11 Banach space dualities
	12 Multipliers
	13 Countable degree -1-saturation
	14 K-theory aspects
	15 The hybrid generalization
	16 Unbounded Fredholm modules
	17 Sample open problems
	References
Quantum field theory on noncommutative spaces
	1 Introduction
		1.1 Quantum field theory and gravity
		1.2 Noncommutativity
		1.3 Structure of the survey
		1.4 Disclaimer
	2 Quantum field theory
		2.1 Axiomatic and algebraic quantum field theory
		2.2 Euclidean QFT
		2.3 The free Euclidean scalar field
		2.4 The interacting scalar field
		2.5 Feynman graphs and Feynman integrals
	3 Euclidean quantum fields on noncommutative geometries
		3.1 Nuclear AF Fréchet algebras
		3.2 The free Euclidean scalar field on a noncommutative geometry
		3.3 Towards an interacting scalar field on noncommutative geometry
	4 Some noncommutative geometries for QFT
		4.1 Simplest example: Moyal algebra
		4.2 Quantum fields on the Moyal algebra
		4.3 4-Dimensional Moyal space
		4.4 Gauge models
		4.5 Fuzzy spaces
		4.6 A non-example: the noncommutative torus
		4.7 Other (non-) examples
	5 QFT on NCG: the first years
		5.1 Very short overview about QFT on deformed Minkowski space
		5.2 Perturbative QFT on deformed Euclidean space
		5.3 Numerical simulations
	6 Renormalisation of noncommutative phi^4-theory to all orders
		6.1 QFT with harmonic oscillator covariance on Moyal space
		6.2 The beta-function
		6.3 Constructive renormalisation
		6.4 Other developments
		6.5 Tensor models
	7 Structures and techniques in matrix models
		7.1 Riemann surfaces and ribbon graphs
		7.2 The Kontsevich model
		7.3 The Ward–Takahashi identity in matrix models
	8 Exact solution of the Phi^3-model
		8.1 Preliminary remarks
		8.2 Solution of the planar sector
		8.3 The non-planar sector
		8.4 Summary
	9 Exact solution of the Phi^4-model
		9.1 The planar sector
		9.2 Exact solution of the planar 2-point function
		9.3 Outlook
	10 Osterwalder–Schrader axioms
		10.1 Previous approaches to reflection positivity
		10.2 A proposal
	References
Higher invariants in noncommutative geometry
	1 Introduction
	2 Geometric C-algebras
	3 Higher index theory and localization
		3.1 K-homology
		3.2 K-theory and boundary maps
		3.3 Higher index map and local index map
	4 The Baum–Connes assembly and a local-global principle
	5 The Novikov conjecture
		5.1 Non-positively curved groups and hyperbolic groups
		5.2 Amenable groups, groups with finite asymptotic dimension and coarsely embeddable groups
		5.3 Gelfand-Fuchs classes, the group of volume preserving diffeomorphisms, Hilbert–Hadamard spaces
	6 Secondary invariants for Dirac operators and applications
	7 Higher index, higher rho and positive scalar curvature at infinity
	8 Secondary invariants of the signature operators and topological non-rigidity
	9 Non-rigidity of topological manifolds and reduced structure groups
	10 Cyclic cohomology and higher rho invariants
	References