Geometric Aspects of Analysis and Mechanics: In Honor of the 65th Birthday of Hans Duistermaat (Progress in Mathematics Book 292)

Hans Duistermaat, an influential geometer-analyst, made substantial contributions to the theory of ordinary and partial differential equations, symplectic, differential, and algebraic geometry, minimal surfaces, semisimple Lie groups, mechanics, mathematical physics, and related fields. Written in his honor, the invited and refereed articles in this volume contain important new results as well as surveys in some of these areas, clearly demonstrating the impact of Duistermaat's research and, in addition, exhibiting interrelationships among many of the topics. The well-known contributors to this text cover a wide range of topics: semi-classical inverse problems; eigenvalue distributions; symplectic inverse spectral theory for pseudodifferential operators; solvability for systems of pseudodifferential operators; the Darboux process and a noncommutative bispectral problem; a proof of the Atiyah-Weinstein conjecture on the index of Fourier integral operators and the relative index of CR structures; relations between index theory and localization formulas of Duistermaat–Heckman; non-Abelian localization; symplectic implosion and nonreductive quotients; conjugation spaces; and Hamiltonian geometry. Also included are several articles in memory of Hans Duistermaat. Contributors include J.-M. Bismut, L. Boutet de Monvel, Y. Colin de Verdière, R.H. Cushman, N. Dencker, F.A. Grünbaum, V.W. Guillemin, J.-C. Hausmann, G. Heckman, T. Holm, L.C. Jeffrey, F. Kirwan, E. Leichtnam, B. McLellan, E. Meinrenken, P.-E. Paradan, J. Sjöstrand, X. Tang, S. Vũ Ngọc, A. Weinstein 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 Preface -- About J.j. Duistermaat -- Hans Duistermaat (1942-2010) -- Recollections Of Hans Duistermaat -- Classical Mechanics And Hans Duistermaat -- Duistermaat-heckman Formulas And Index Theory -- Asymptotic Equivariant Index Of Toeplitz Operators And Relative Index Of Cr Structures -- A Semi-classical Inverse Problem I: Taylor Expansions -- A Semi-classical Inverse Problem Ii: Reconstruction Of The Potential -- On The Solvability Of Systems Of Pseudodifferential Operators -- The Darboux Process And A Noncommutative Bispectral Problem: Some Explorations And Challenges -- Conjugation Spaces And Edges Of Compatible Torus Actions -- Non-abelian Localization For U(1) Chern-simons Theory -- Symplectic Implosion And Non-reductive Quotients -- Quantization Of Q-hamiltonian Su(2)-spaces -- Wall-crossing Formulas In Hamiltonian Geometry -- Eigenvalue Distributions And Weyl Laws For Semi-classical Non-self-adjoint Operators In 2 Dimensions -- Symplectic Inverse Spectral Theory For Pseudodifferential Operators. Erik P. Van Den Ban, Johan A.c. Kolk, Editors. Includes Bibliographical References.