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Hans Duistermaat

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Johannes Jisse (Hans) Duistermaat (The Hague, December 20, 1942 – Utrecht, March 19, 2010) was a Dutch mathematician.[1][2]

Hans Duistermaat
Duistermaat in Berkeley (1981)
Born(1942-12-20)December 20, 1942
The Hague
DiedMarch 19, 2010(2010-03-19) (aged 67)
Utrecht, Netherlands
Nationality Netherlands
Alma materUniversity of Utrecht
Known forDuistermaat–Heckman formula
Scientific career
FieldsMathematics
InstitutionsUniversity of Utrecht
Thesis Energy and Entropy as Real Morphisms for Addition and Order  (1968)
Doctoral advisorHans Freudenthal
Doctoral studentsHenk Broer
Gert Heckman [de]
Michael Ruzhansky [de; fr]

Biography

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Duistermaat attended primary school in Jakarta, at the time capital of the Dutch East Indies, where his family moved after the end of World War II. In 1957, a few years after the Indonesian independence, they came back to the Netherlands and Duistermaat completed his high school studies in Vlaardingen.[3][4]

From 1959 to 1965 he studied mathematics at Utrecht University, and he obtained his PhD degree at the same institution in 1968, with a thesis on the mathematical structures of thermodynamics entitled "Energy and Entropy as Real Morphisms for Addition and Order".[5] His original supervisor was the applied mathematician Günther K. Braun, who passed away one year before the thesis defense, so the official supervision was taken over by geometer Hans Freudenthal.[6][7][8][9]

After a postdoctoral stay in Lund (1969–70), Duistermaat returned to the Netherlands in 1971 and became in 1972 full professor in Nijmegen. In 1974 he returned to Utrecht, where he was offered the chair of Freudenthal.

In 2004 he was awarded a special academy professorship by the KNAW, allowing him to focus exclusively on research, even after his official retirement in 2007.[6][8][10][4] He died in March 2010, due to non-Hodgkin lymphoma and pneumonia.[11]

He was elected a member of the Royal Netherlands Academy of Arts and Sciences in 1982,[12] and of Academia Europaea in 1993.[13] In 2007 he became a Knight of the Order of the Netherlands Lion.[14][8][15]

Apart from being an eminent mathematician, Duistermaat was also a good chess player. In a simultaneous match of 10 against Anatoly Karpov in 1977, Duistermaat was the only one who did not lose.[4][7][16]

Research

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Duistermaat defined himself a geometric analyst,[8] but his research covered many different areas in analysis, geometry and mathematical physics, including classical mechanics, symplectic geometry, Fourier integral operators, partial differential equations, algebraic geometry, harmonic analysis, and dynamical systems.

Duistermaat dropped thermodynamics after his PhD, due to dissents between mathematicians and physicists in Utrecht. Nevertheless, during his PhD research he read the work by Sophus Lie, and in particular encountered contact transformations. These topics exerted an important influence for his future pioneering works in microlocal analysis,[6][8][10] most prominently during his collaboration with Lars Hörmander, when they developed the theory of Fourier integral operators and proved the Propagation of singularities theorem [de].[17] This led him also to the work with Victor Guillemin on the link between spectra of elliptic operators and periodic bicharacteristics.[18]

Duistermaat introduced the notion of monodromy in Hamiltonian systems[8][9][7] as obstruction to the existence of global action-angle coordinates,[19] later, together with Richard Cushman, he extended it to the quantum systems.[20]

In symplectic geometry he is well known[9][15][7] for his article with his PhD student Gert Heckman [de] on the Duistermaat–Heckman formula,[21] which will later be placed in the more general framework of equivariant cohomology,[22] independently by Berline and Vergne[23] and by Atiyah and Bott.[24] Other contributions in this field include a generalisation of the Morse index theorem[25] and a contribution to the problem of quantisation commutes with reduction.[26]

In his book on Lie groups, together with his PhD student Johan Kolk,[27] he provided an alternative proof of the third Lie theorem, which will turn out to be crucial for proving the analogous theorem for Lie groupoids and for its applications to Poisson geometry.[6][8][7]

His work with Alberto Grünbaum on the bispectral problem[28] was influential for integrable systems and noncommutative algebraic geometry.[7] In the last stages of his life he became interested in algebraic geometry[8][15] and wrote a book on QRT maps and elliptic surfaces.[29]

Duistermaat contributed also to applied mathematics.[6][8][10] He was a consultant to Royal Dutch Shell, which led to the thesis of Christiaan Stolk on the inversion of seismic data.[30] He also worked on barrier functions in convex programming,[31] collaborated with biomedical technologists on computer vision[32] and with geophysicists on modeling the polarity reversals of the Earth's magnetic field.[33]

He supervised 25 PhD students.[5]

Selected works

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  • Duistermaat, J. J. (2011) [1996], Fourier integral operators, Modern Birkhäuser Classics, Birkhäuser/Springer, New York, doi:10.1007/978-0-8176-8108-1, ISBN 978-0-8176-8107-4, MR 1362544
  • Duistermaat, J. J. (2011), The heat kernel Lefschetz fixed point formula for the Spinc dirac operator, Boston: Birkhäuser, ISBN 978-0-8176-8247-7; Duistermaat, J. J. (1996). 1st edition. ISBN 0-8176-3865-2.[34]
  • Duistermaat, J. J.; Guillemin, V. W. (1975), "The spectrum of positive elliptic operators and periodic bicharacteristics" (PDF), Inventiones Mathematicae, 29 (1): 39–79, Bibcode:1975InMat..29...39D, doi:10.1007/BF01405172, hdl:10338.dmlcz/126178, MR 0405514, S2CID 189832135
  • Duistermaat, J. J.; Heckman, G. J. (1982), "On the variation in the cohomology of the symplectic form of the reduced phase space", Inventiones Mathematicae, 69 (2): 259–268, Bibcode:1982InMat..69..259D, doi:10.1007/BF01399506, MR 0674406, S2CID 119943006
  • Duistermaat, J. J.; Grünbaum, F. A. (1986), "Differential equations in the spectral parameter", Communications in Mathematical Physics, 103 (2): 177–240, Bibcode:1986CMaPh.103..177D, doi:10.1007/bf01206937, MR 0826863, S2CID 121915958

References

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  1. ^ "Hans Duistermaat". Geometry and Quantum Theory (GQT). Retrieved 2022-08-18.
  2. ^ "In memoriam: Prof. dr. Johannes Jisse Duistermaat". Academia Europaea. Retrieved 2022-08-16.
  3. ^ de Waard, Peter (2010-04-12). "De man van de formule" [The man of the formula]. De Volkskrant (in Dutch).
  4. ^ a b c Kreulen, Edwin (2010-04-19). "De wiskunde bleef altijd roepen. Hans Duistermaat 1942-2010" [Mathematics always keeps calling. Hans Duistermaat 1942-2010] (PDF). Trouw (in Dutch). p. 25.
  5. ^ a b Johannes Jisse Duistermaat at the Mathematics Genealogy Project
  6. ^ a b c d e van den Ban, Erik; Kolk, Johan (2010). "In Memoriam Hans Duistermaat (1942-2010): Grasping the essence" (PDF). Nieuw Archief voor Wiskunde. 11 (5): 235–237.
  7. ^ a b c d e f Sjamaar, Reyer (2011-10-25). "Hans Duistermaat's contributions to Poisson geometry". arXiv:1110.5627 [math.HO].
  8. ^ a b c d e f g h i Guillemin, Victor; Pelayo, Álvaro; Vũ Ngọc, San; Weinstein, Alan, eds. (2011). "Remembering Johannes J. Duistermaat (1942–2010)" (PDF). Notices of the American Mathematical Society. 58 (6): 794–802.
  9. ^ a b c Cushman, Richard (2011). "Hans Duistermaat (1942-2010)" (PDF). Newsletter of the European Mathematical Society. 79. European Mathematical Society: 17–19.
  10. ^ a b c van der Vorst, H.A. (2011). "Levensbericht Johannes Jisse Duistermaat" [Life story of Johannes Jisse Duistermaat] (PDF). Jaarboek - Koninklijke Nederlandse Akademie van Wetenschappen. Royal Netherlands Academy of Arts and Sciences.
  11. ^ "Overlijden Hans Duistermaat" [Death of Hans Duistermaat]. Utrecht University (in Dutch). Retrieved 2022-08-16.
  12. ^ "J.J. Duistermaat (1942 - 2010)" (in Dutch). Royal Netherlands Academy of Arts and Sciences. Retrieved 14 July 2015.
  13. ^ "Johannes Jisse Duistermaat". Academia Europaea. Retrieved 2022-08-16.
  14. ^ "Lintjesregen: De opvallendste feiten op een rij" [Annual award celebration: the most notable facts in a row]. RTV Utrecht (in Dutch). 2007-04-27.
  15. ^ a b c Vũ Ngọc, San (2011). "Johannes Jisse (dit Hans) Duistermaat (1942-2010)" (PDF). La Gazette des mathématiciens (in French). 127. Société mathématique de France: 84–91.
  16. ^ Beekman, Robert (2016). "Simultaan Karpov (1977)" [Simultaneous Karpov (1977)]. Schaakclub Oud Zuylen Utrecht (in Dutch). Retrieved 2022-08-16.
  17. ^ Duistermaat, J. J.; Hörmander, L. (1972). "Fourier integral operators. II". Acta Mathematica. 128 (none): 183–269. doi:10.1007/BF02392165. ISSN 0001-5962. S2CID 189785151.
  18. ^ Duistermaat, J. J.; Guillemin, V. W. (1975-02-01). "The spectrum of positive elliptic operators and periodic bicharacteristics". Inventiones Mathematicae. 29 (1): 39–79. Bibcode:1975InMat..29...39D. doi:10.1007/BF01405172. ISSN 1432-1297. S2CID 189832135.
  19. ^ Duistermaat, J. J. (1980). "On global action-angle coordinates". Communications on Pure and Applied Mathematics. 33 (6): 687–706. doi:10.1002/cpa.3160330602.
  20. ^ Cushman, R.; Duistermaat, J. J. (1988). "The quantum mechanical spherical pendulum". Bulletin of the American Mathematical Society. 19 (2): 475–479. doi:10.1090/S0273-0979-1988-15705-9. ISSN 0273-0979.
  21. ^ Duistermaat, J. J.; Heckman, G. J. (1982-06-01). "On the variation in the cohomology of the symplectic form of the reduced phase space". Inventiones Mathematicae. 69 (2): 259–268. Bibcode:1982InMat..69..259D. doi:10.1007/BF01399506. ISSN 1432-1297. S2CID 119943006.
  22. ^ Heckman, Gert (2010). "In Memoriam Hans Duistermaat (1942-2010): Recollections of a godsend talent" (PDF). Nieuw Archief voor Wiskunde. 11 (5): 240–241.
  23. ^ Berline, Nicole; Vergne, Michèle (1982). "Classes caracteristiques equivariantes. Formule de localisation en cohomologie equivariante" [Equivariant characteristic classes. Localisation formula in equivariant cohomology]. Comptes rendus de l'Académie des sciences (in French). 295: 539–541.
  24. ^ Atiyah, M.F.; Bott, R. (1984). "The moment map and equivariant cohomology". Topology. 23 (1): 1–28. doi:10.1016/0040-9383(84)90021-1.
  25. ^ Duistermaat, J. J (1976-08-01). "On the Morse index in variational calculus". Advances in Mathematics. 21 (2): 173–195. doi:10.1016/0001-8708(76)90074-8. ISSN 0001-8708.
  26. ^ Duistermaat, Hans; Guillemin, Victor; Meinrenken, Eckhard; Wu, Siye (1995). "Symplectic reduction and Riemann-Roch for circle actions". Mathematical Research Letters. 2 (3): 259–266. doi:10.4310/MRL.1995.v2.n3.a3.
  27. ^ J. J., Duistermaat; Kolk, J. A. C. (2000). Lie Groups. Springer Verlag. doi:10.1007/978-3-642-56936-4. ISBN 978-3-540-15293-4.
  28. ^ Duistermaat, J. J.; Grünbaum, F. A. (1986-06-01). "Differential equations in the spectral parameter". Communications in Mathematical Physics. 103 (2): 177–240. Bibcode:1986CMaPh.103..177D. doi:10.1007/BF01206937. ISSN 1432-0916. S2CID 121915958.
  29. ^ Duistermaat, J. J. (2010). Discrete integrable systems : QRT Maps and Elliptic Surfaces. New York: Springer. ISBN 978-0-387-72923-7. OCLC 676697713.
  30. ^ Stolk, Christiaan (2000). On the Modeling and Inversion of Seismic Data (PDF) (PhD thesis). Utrecht: Universiteit Utrecht. ISBN 90-393-2551-0.
  31. ^ Duistermaat, Hans (2001). "The universal barrier function of a convex polytope". Circumspice. Various Papers in Around Mathematics in Honour of Arnoud van Rooij. Katholieke Universiteit Nijmegen: 207–220.
  32. ^ Loog, Marco; Duistermaat, Johannes; Florack, Luc M. J. (2001). Kerckhove, Michael (ed.). "On the Behavior of Spatial Critical Points under Gaussian Blurring A Folklore Theorem and Scale-Space Constraints". Scale-Space and Morphology in Computer Vision. Berlin, Heidelberg: Springer: 183–192. doi:10.1007/3-540-47778-0_15. ISBN 978-3-540-47778-5.
  33. ^ Hoyng, P.; Duistermaat, J. J. (2004-10-01). "Geomagnetic reversals and the stochastic exit problem". Europhysics Letters. 68 (2): 177. Bibcode:2004EL.....68..177H. doi:10.1209/epl/i2004-10243-1. ISSN 0295-5075. S2CID 250865891.
  34. ^ Freed, Daniel S. (1997). "Review: The heat kernel Lefschetz fixed point formula for the Spinc dirac operator, by J. J. Duistermaat" (PDF). Bull. Amer. Math. Soc. (N.S.). 34 (1): 73–78. doi:10.1090/s0273-0979-97-00698-8.
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