Geometric Aspects of Analysis and Mechanics

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. Vu Ng c, A. Weinstein.

Preface. - About J. J. Duistermaat. - Hans Duistermaat (1942-2010). - Recollections of Hans Duistermaat. - Recollections of Hans Duistermaat. - 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.