Louis Boutet de Monvel, Selected Works

Cover
Progress in Mathematics 292
Geometric Aspects of Analysis and Mechanics
ISBN 9780817682439
Preface
	About J.J. Duistermaat
		Ph.D. students of J.J. Duistermaat
		List of publications by J.J. Duistermaat
		Articles
		Not (yet) published preprints
		Books
		Published lectures
		Book reviews
	Hans Duistermaat (1942–2010)
	Recollections of Hans Duistermaat
		References
	Recollections of Hans Duistermaat
	Recollections of Hans Duistermaat
		References
	Classical mechanics and Hans Duistermaat
		References
Contents
Duistermaat–Heckman formulas and index theory
	Introduction
	1 The Duistermaat–Heckman formula
		1.1 The localization formula
		1.2 Functoriality of the Duistermaat–Heckman formula
		1.3 Transgression currents and localization
		1.4 Localization formulas and integration along the fibre
		1.5 Transgression formulas and integration along the fibre
		1.6 Equivariant projections and adiabatic limits
		1.7 Localization formulas and complex manifolds
		1.8 Complex manifolds and integration along the fibre
	2 Heat equation and measures on the loop space
		2.1 Finite-dimensional traces
		2.2 The heat kernel on a compact Riemannian manifold
		2.3 The Feynman–Kac formula
	3 Index theory and differential forms on the loop space
		3.1 Clifford algebras
		3.2 Spin manifolds and the Dirac operator
		3.3 The index of DX+ and the McKean–Singer formula
		3.4 The fantastic cancellations
		3.5 The heat kernel for DX,2 as a path integral
		3.6 The index as a well-defined path integral
		3.7 The loop space and the action of S1
		3.8 The remark of Atiyah and Witten for the Dirac operator on spinors
		3.9 An extension to general Dirac operators
		3.10 The fantastic cancellations and the localization formulas
		3.11 Formal versus rigorous arguments
		3.12 Hamiltonian–Lagrangian correspondence and index theory
	4 From localization formulas to Hermitian K-theory
		4.1 Families index theorem and equivariant integration along the fibre
		4.2 The local Getzler operator and superconnections
		4.3 η forms and the currents ε
		4.4 Holomorphic torsion forms and Bott–Chern currents
		4.5 Lefschetz formulas
		4.6 Towards the hypoelliptic Laplacian
	References
Asymptotic equivariant index of Toeplitz operators and relative index of CR structures
	1 Introduction
	2 Equivariant trace and index
		2.1 Equivariant Toeplitz Operators
		2.2 G-trace
		2.3 G index
	3 K-theory and embedding
		3.1 A short digression on Toeplitz algebras
		3.2 Asymptotic trace and index
		3.3 E-modules
		3.4 Embedding
	4 Relative index
		4.1 Holomorphic setting
		4.2 Collar isomorphisms
		4.3 Embedding
		4.4 Index
		4.5 Appendix
			4.5.1 Contact isomorphisms and base symplectomorphisms
			4.5.2 Example
		4.6 Final remarks
	References
A semi-classical inverse problem I: Taylor expansions
	1 Introduction
	2 A counterexample for a general Hamiltonian
	3 Review of the Moyal product
	4 The Weyl algebra
	5 Moyal versus functional QBNF
	6 Useful lemmas
	7 TheQBNF
	8 The first terms
	9 The induction
	10 Getting the QBNF from the density of states in case of a local extremum of the potential
		10.1 ћ-dependent distributions
		10.2 Density of states
		10.3 Singularity of the density of states near a local maximum of the potential
			10.3.1 The singularity of the density of states and the QBNF
			10.3.2 Computing some singularities
			10.3.3 End of the proof of Theorem 10.5
		10.4 The case of a local minimum
	11 Open problems
	12 Homogeneity properties of the QBNF
	References
A semi-classical inverse problem II: reconstruction of the potential
	1 Introduction
	2 Motivation I: surfaces of revolution
	3 Motivation II: effective surface waves Hamiltonian
	4 Schrödinger operators and spectra
	5 A theorem for one-well potentials
	6 One-well potentials: Bohr–Sommerfeld rules and a pseudodifferential trace formula
	7 Two potentials with the same semi-classical spectra
	8 One-well potentials: the proof of Theorem 5.1
		8.1 Some useful lemmas
		8.2 Rewriting V using F and G
		8.3 How to get V from S0 and S2
	9 Taylor expansions
	10 A theorem for a potential with several wells
		10.1 The genericity assumptions
			10.1.1 Assumption on critical points
			10.1.2 A generic symmetry defect
			10.1.3 Separation of the wells
		10.2 Quartic potentials
		10.3 Statement of the result
	11 The semi-classical trace formula
	12 The case of several wells: the proof of Theorem 10.1
		12.1 What can be read from Weyl’s asymptotics?
		12.2 The scheme of the reconstruction
		12.3 Separation of spectra
		12.4 Limit values of some integrals
	13 Extensions to other operators
		13.1 The statement
		13.2 The Weyl symbol and the actions
		13.3 Recovering G
		13.4 Recovering ±F
	Appendix: Abel’s result
	References
On the solvability of systems of pseudodifferential operators
	1 Introduction
	2 Statement of results
	3 The multiplier estimates
	4 Proof of Theorem 2.7
	5 The symbol classes and weights
	6 The Wick quantization
	7 The lower bounds
	References
The Darboux process and a noncommutative bispectral problem: some explorations and challenges
	1 The bispectral problem
	2 What is the main purpose of this paper?
	3 The Bochner–Krall problem
	4 A matrix-valued version of the Darboux process for a difference operator
	5 Fancier versions of the Darboux process
	6 Matrix-valued orthogonal polynomials
	7 A few examples
	8 A few Jacobi-type examples
	9 An explicit differential operator
	10 Toda flows with matrix-valued time
	11 Electrostatics: Heine, Stieltjes, Darboux
	12 Markov chains
	13 Things that appear before their time
	14 The multivariable case
	15 Conclusion
	References
Conjugation spaces and edges of compatibletorus actions
	1 Introduction
	2 A review of conjugation spaces
	3 Conjugation spaces and 1-skeleta
	4 Preliminaries
		4.1 Compatibility
		4.2 Equivariantly formal spaces
		4.3 The image of the 1-skeleton
	5 Proof of the main results
	6 Remarks
		6.1 A case when the skeleta differ
		6.2 The relationship to previous work
	References
Nonabelian localization for U(1) Chern–Simons theory
	1 Introduction
	2 k-dependence
	3 Reidemeister torsion and symplectic volume
	References
Symplectic implosion and nonreductive quotients
	1 Introduction
		1.1 Index of notation
	2 Symplectic reduction and geometric invariant theory
		2.1 Symplectic reduction
		2.2 Mumford’s geometric invariant theory
	3 Symplectic implosion and quotients by nonreductive groups
		3.1 Symplectic implosion for a maximal unipotent subgroup
		3.2 Symplectic implosion for the unipotent radical of a parabolicsub group
		3.3 Wonderful compactifications, symplectic cuts, and partial desingularisations
	4 Nonreductive geometric invariant theory
		4.1 Background
		4.2 Some examples of reductive envelopes
		4.3 Symplectic implosion for U = (C+)r ≤ SL(r + 1;C) actions
	References
Quantization of q-Hamiltonian SU(2)-spaces
	1 Introduction
	2 The fusion ring Rk(SU(2))
		2.1 First description
		2.2 Second description
		2.3 Third description
	3 The twisted equivariant K-homology of SU(2)
		3.1 G-Dixmier–Douady bundles
		3.2 The Dixmier–Douady bundle over SU(2)
		3.3 The equivariant Cartan 3-form on SU(2)
		3.4 Twisted K-homology
		3.5 The K-homology fundamental class
	4 q-Hamiltonian SU(2)-spaces
		4.1 Basic definitions
		4.2 Example: The 4-sphere
	5 Cross-sections
	6 The canonical “twisted Spinc-structure”
	7 Prequantization of q-Hamiltonian SU(2)-spaces
	8 Quantization of q-Hamiltonian SU(2)-spaces
	9 Localization
	10 Quantization commutes with reduction
	11 Examples
		11.1 The double
		11.2 Conjugacy classes
		11.3 The 4-sphere
		11.4 Moduli spaces of flat SO(3)-bundles
	References
Wall-crossing formulas in Hamiltonian geometry
	1 Introduction
	Acknowledgments
	Notation
	2 Duistermaat–Heckman measures
		2.1 Equivariant cohomology and localization
		2.2 Localization of DH(M)
		2.3 Polynomial behavior
		2.4 Wall-crossing formulas
	3 Quantum version of Duistermaat–Heckman measures
		3.1 Elliptic and transversally elliptic symbols
		3.2 Localization of the Riemann–Roch character
		3.3 Periodic polynomial behavior of the multiplicities
		3.4 Riemann–Roch–Kawasaki theorem
		3.5 Wall-crossing formulas for the mc
	4 Multiplicities of group representations
		4.1 Borel–Weil Theorem
		4.2 Critical points of Ф : K · λ → t∗
		4.3 Main theorems
		4.4 The case of SU(n)
	5 Vector partition functions
		5.1 Quantization of Cd
		5.2 Transversally elliptic symbols on Cd
		5.3 Localization in a noncompact setting
		5.4 Proof of Theorem 5.1
		5.5 Proof of Theorem 5.2
	References
Eigenvalue distributions and Weyl laws for semiclassical non-self-adjoint operators in 2 dimensions
	1 Introduction
	2 The integrable case
	3 The general case
	References
Symplectic inverse spectral theory for pseudodifferential operators
	1 Introduction
	Acknowledgements
	2 The setting
	3 Singularities
	4 Topology
		4.1 Connected components
		4.2 Singular fibres
		4.3 Global topology
	5 Symplectic geometry
		The lengths
		The Taylor series at the bifurcation points
		Symplectic equivalence
	References