I have published papers with authors from Australia, Belgium, Canada, China, France, Germany, Iran, Israel, Italy, Netherlands, New Zealand, Portugal, Romania, Russia, Spain, Switzerland, United Kingdom, and USA. My recent work includes collaborations with my former PhD students at the University of California San Diego Joseph Palmer and Xiudi Tang (several papers in the period 2017-2020), with Alessio Figalli (Fields Medal 2018, 2 papers), with Vladimir Voevodsky (Fields Medal 2002, 1 paper) and with Tudor Ratiu (6 papers). Between the years 2003 and 2006 I worked almost exclusively on symplectic group actions, in collaboration with Johannes J. Duistermaat (4 papers), a topic to which I have been coming back throughout my career and written papers about regularly. Since 2006 I have worked primarily on finite dimensional classical and quantum integrable systems, having had an extensive collaboration with San Vu Ngoc (14 papers) which continues today, as well as on probabilistic symplectic geometry with Fraydoun Rezakhanlou (2 papers). More recently I have been working on combinatorial problems arising in symplectic geometry with Luis Crespo and Francisco Santos.

I have been particularly interested in the construction of symplectic invariants and inverse spectral problems, mainly in dimension four. Among the papers which resulted from the collaboration with Vu Ngoc are two papers in which we gave a symplectic classification of semitoric systems (Inventiones Mathematicae 2009 and Acta Mathematica 2011), a paper where we solved, with Laurent Charles, the inverse spectral problem for quantum toric integrable systems on compact manifolds (Annales Scientifiques de l’École Normale Supérieure 2013) and another paper with Yohann Le Floch and Vu Ngoc which extends this to certain quantum semitoric systems (Mathematische Annalen 2016),  all of which are displayed below. I gave a short general audience introduction to these ideas at the 2011 Board of Trustees Meeting at the Institute for Advanced Study in Princeton, see Talk at Board of Trustees Meeting. An introducion to the basic ideas of symplectic geometry is given in this article (in Spanish, written for the BBVA Foundation).


  1. Ewald’s Conjecture and integer points in algebraic and symplectic toric geometry (with L. Crespo and F. Santos), Preprint.
  2. The structure of monotone blow-ups in symplectic toric geometry and a question of McDuff (with F. Santos), Preprint.
  3. The random Arnold Conjecture: a new probabilistic Conley-Zehnder Theory for symplectic maps (with F. Rezakhanlou), Preprint.
  4. Moduli spaces of Delzant polytopes and symplectic toric manifolds (with F. Santos), Preprint.
  5. Semitoric systems of non-simple type (with J. Palmer and X. Tang), Preprint.
  6. Symplectic geometry: applications and challenges (in Spanish), La Gaceta de la Real Sociedad Matemática Española 27 (2024) 285-311.
    Publisher Website
  7. Vu Ngoc’s Conjecture on focus-focus singular fibers with multiple pinched points (with X. Tang), Journal of Fixed Point Theory and Applications 26, 6 (2024).
    Publisher Website   arXiv:1803.00998
  8. Symplectic and inverse spectral geometry of integrable systems: a glimpse and open problems. Topology and its Applications 339 Part A (2023) 108577.
    Publisher Website
  9. Symplectic invariants of semitoric systems and the inverse problem for quantum systems, Indagationes Mathematicae 32 (2021) 246-274.
    Publisher Website   arXiv:2002.05306
  10. Moser-Greene-Shiohama stability for families (with X. Tang), Journal of Symplectic Geometry 17 (2019) 1427-1446.
    Publisher Website
  11. Symplectic stability on manifolds with cylindrical ends (with S. Curry and X. Tang), Journal of Geometric Analysis 29 (2019) 1660-1675.
    Publisher Website   arXiv:1704.08659
  12. Symplectic geometry and spectral properties of classical and quantum coupled angular momenta (with Y. Le Floch), Journal of Nonlinear Science 29 (2019) 655-708.
    Publisher Website   arXiv:1607.05419
  13. Spectral asymptotics of semiclassical unitary operators (with Y. Le Floch), Journal of Mathematical Analysis and Applications 473 (2019) 1174-1202
    Publisher Website   arXiv:1506.02873
  14. Moser stability for volume forms on noncompact fiber bundles (with X. Tang), Differential Geometry and its Applications 63 (2019) 120-136.
    Publisher Website   arXiv:1607.03800
  15. Classifying toric and semitoric fans by lifting equations from SL(2,Z) (with D. Kane and J. Palmer), Symmetry, Integrability, and Geometry: Methods and Applications SIGMA 14 (2018), 016, 43 pages.
    Publisher Website   arXiv:1502.07698
  16. Minimal models of compact symplectic semitoric manifolds (with D. Kane and J. Palmer) Journal of Geometry and Physics 125 (2018) 49-74.
    Publisher Website   arXiv:1610.05423 
  17. Symplectic G-capacities and integrable systems (with A. Figalli and J. Palmer), Annali della Scuola Normale Superiore di Pisa (5) Vol. XVIII (2018) 65-103.
    Publisher Website   arXiv:1511.04499
  18. Poincare-Birkhoff theorems in random dynamics (with F. Rezakhanlou), Transactions of the American Mathematical Society 370 (2018) 601-639.
    Publisher Website   arXiv:1306.0821
  19. Hamiltonian and symplectic symmetries: an introduction, Bulletin of the American Mathematical Society 54 (2017) 383-436.
    Publisher Website   arXiv:1610.09718
  20. The affine invariant of proper semitoric integrable systems (with T. Ratiu and S. Vu Ngoc), Nonlinearity 30 (2017) 3993-4028.
    Publisher Website   arXiv:1307.7516 
  21. The tropical momentum map: a classification of toric log symplectic manifolds (with M. Gualtieri, S. Li, and T. Ratiu), Mathematische Annalen 367 (2017) 1217-1258.
    Publisher Website   arXiv:1407.3300
  22. Generating hyperbolic singularities in semitoric systems via Hopf bifurcations (with H. Dullin), Journal of Nonlinear Science 26 (2016) 787-811.
    Publisher Website   arXiv:1503.01534 
  23. On the continuity of ball packing density on moduli spaces of toric manifolds (with A. Figalli), Advances in Geometry 16 (2016) 291-300.
    Publisher Website   arXiv:1408.1462 
  24. Spectral limits of semiclassical commuting self-adjoint operators (with S. Vu Ngoc), A Mathematical Tribute to J.M. Montesinos, Editorial Universidad Complutense de Madrid (2016) 527-546.
    Publisher Website   arXiv:1506.04591
  25. Inverse spectral theory for semiclassical Jaynes-Cummings systems (with Y. Le Floch and S. Vu Ngoc), Mathematische Annalen 364 (2016) 1393-1413. Erratum in Mathematische Annalen 375 (2019) 917-920.
    Publisher Website   arXiv:1407.5159
  26. Sharp symplectic embeddings of cylinders (with S. Vu Ngoc), Indagationes Mathematicae N.S. 27 (2016) 307-317.
    Publisher Website   arXiv:1304.5250
  27. Euler-MacLaurin formulas via differential operators (with Y. Le Floch) Advances in Applied Mathematics 76 (2016) 99-124.
    Publisher Website   arXiv:1312.5711
  28. Fermat and the number of fixed points of periodic flows (with L. Godinho and S. Sabatini), Communications in Number Theory and Physics 9 (2015) 643-687.
    Publisher Website  arXiv:1404.4541
  29. A univalent formalization of the p-adic numbers (with V. Voevodsky and M. Warren), Mathematical Structures in Computer Science 25 (2015) 1147-1171.
    Publisher Website   arXiv:1302.1207
  30. Fiber connectivity and bifurcation diagrams for almost toric systems (with T. Ratiu and S. Vu Ngoc) Journal of Symplectic Geometry 13 (2015) 343-386.
    Publisher Website   arXiv:1108.0328
  31. Hofer’s question on intermediate symplectic capacities (with S. Vu Ngoc), Proceedings of the London Mathematical Society 110 (2015) 787-804.
    Publisher Website   arXiv:1210.1537
  32. L^2-Cohomology and complete Hamiltonian manifolds (with R. Mazzeo and T. Ratiu), Journal of Geometry and Physics 87 (2015) 305-313.
    Publisher Website   arXiv:1402.0098
  33. Log-concavity and symplectic flows (with Y. Lin), Mathematical Research Letters 22 (2015) 501-527.
    Publisher Website   arXiv:1207.1335
  34. Semiclassical quantization and spectral limits of h-pseudodifferential and Berezin-Toeplitz operators (with L. Polterovich and S. Vu Ngoc), Proceedings of the London Mathematical Society 109 (2014) 676-696.
    Publisher Website   arXiv:1302.0424
  35. Homotopy type theory in context (with M. Warren), Gazette des mathématiciens Société Mathématique de France 42 (2014) 95-110.
    Publisher Website
  36. Homotopy type theory and Voevodsky’s univalent foundations (with M. Warren), Bulletin of the American Mathematical Society 51 (2014) 597-648
    Publisher Website  arXiv:1210.5658
  37. Semiclassical inverse spectral theory for singularities of focus-focus type (with S. Vu Ngoc), Communications in Mathematical Physics 329 (2014) 809-820.
    Publisher Website   arXiv:1302.2260
  38. Moduli spaces of toric manifolds (with A. Pires, T. Ratiu, and S. Sabatini), Geometriae Dedicata 169 (2014) 323-341.
    Publisher Website   arXiv:1207.0092
  39. Isospectrality for quantum toric integrable systems (with L. Charles and S. Vu Ngoc), Annales Scientifiques de l’École Normale Supérieure 43 (2013) 815-849.
    Publisher Website   arXiv:1111.5985
  40. Voevodsky’s Univalence Axiom in homotopy type theory (with S. Awodey and M. Warren), Notices of the American Mathematical Society 60 (2013) 1164-1167.
    Publisher Website   arXiv:1302.4731
  41. Symplectic spectral geometry of semiclassical operators, Bulletin of the Belgian Mathematical Society 20 (2013) 405-415.
    Publisher Website   arXiv:1303.2570
  42. Circle valued momentum maps for symplectic periodic flows (with T. S. Ratiu), L’Enseignement Mathématique 58 (2012) 205-219.
    Publisher Website 
  43. First steps in symplectic and spectral theory of integrable systems (with S. Vu Ngoc), Discrete and Continuous Dynamical Systems, Series A 32 (2012) 3325-3377.
    Publisher Website   arXiv:1306.0124
  44. Hamiltonian dynamics and spectral theory for spin-oscillators (with S. Vu Ngoc), Communications in Mathematical Physics 309 (2012) 123-154.
    Publisher Website   arXiv:1005.0439
  45. Symplectic geometry on moduli spaces of J-holomorphic curves (with J. Coffey and L. Kessler), Annals of Global Analysis and Geometry 41 (2012) 265-280.
    Publisher Website   arXiv:0905.3009
  46. Symplectic theory of completely integrable Hamiltonian systems (with S. Vu Ngoc) Bulletin of the American Mathematical Society 48 (2011) 409-455.
    Publisher Website   arXiv:1306.0115
  47. Constructing integrable systems of semitoric type (with S. Vu Ngoc), Acta Mathematica 206 (2011) 93-125.
    Publisher Website (OPEN ACCESS)   arXiv:0903.3376
    (This paper takes the position 15 out of 505 among the most cited papers in Mathscinet since 1940 in the category 53D20 of “Momentum maps and symplectic reduction”, with date April 28, 2023.)
  48. Fourier integral operators, spectral theory, and Hamiltonian systems, Gaceta de la Real Sociedad Matemática Española 14 (2011) 247-262.
    Publisher Website
  49. Fixed points of symplectic periodic flows (with S. Tolman), Ergodic Theory and Dynamical Systems 31 (2011) 1237-1247.
    Publisher Website   arXiv:1003.4787
  50. Complex structures on 4-manifolds with symplectic 2-torus actions (with J.J. Duistermaat), International Journal of Mathematics 2 (2011) 449-463.
    Publisher Website
  51. Non-Kahler symplectic manifolds with toric symmetries (with Y. Lin), Quarterly Journal of Mathematics 62 (2011) 103-114.
    Publisher Website
  52. Topology of symplectic torus actions with symplectic orbits (with J.J. Duistermaat), Revista Matemática Complutense 24 (2011) 59-81.
    Publisher Website
  53. Symplectic actions of 2-tori on 4-manifolds, Memoirs of the American Mathematical Society 204 (2010) no. 959, viii+ 91p.
    Publisher Website   arXiv:math/0609848
  54. Reduced phase space and toric variety coordinatizations of Delzant spaces (with J.J. Duistermaat), Mathematical Proceedings of the Cambridge Philosophical Society 146 (2009) 695-718.
    Publisher Website   arXiv:0704.0430
  55. Semitoric integrable systems on symplectic 4-manifolds (with S. Vu Ngoc), Inventiones Mathematicae 177 (2009) 571-597.
    Publisher Website (OPEN ACCESS)   arXiv:0806.1946
    (This paper takes the position 15 out of 1101 among the most cited papers in Mathscinet since 1940 in the category 37J35 of “Completely integrable systems, topological structure of phase space, integration methods”, with date April 28, 2023.)
  56. Maximal ball packings of symplectic toric manifolds (with B. Schmidt) International Mathematics Research Notices IMRN (2008), 24p, ID rnm139.
    Publisher Website   arXiv:0704.1036
  57. Symplectic torus actions with coisotropic principal orbits (with J.J. Duistermaat), Annales de l’Institut Fourier 57 (2007) 2239-2327.
    Publisher Website   arXiv:math/0511676
  58. Topology of spaces of equivariant symplectic embeddings, Proceedings of the American Mathematical Society 35 (2007) 277-288.
    Publisher Website  arXiv:0704.1033
  59. A geometric approach to the classification of the equilibrium shapes of self-gravitating fluids (with D. Peralta-Salas), Communications in Mathematical Physics 267 (2006) 93-115.
    Publisher Website 
  60. Toric symplectic ball packing, Topology and Its Applications 157 (2006) 3633-3644.
    Publisher Website   arXiv:0704.1034
  61. Unounded trajectories of dynamical systems (with F. G. Gascón and D. Peralta-Salas), Applied Mathematics Letters 17 (2004) 253-259.
    Publisher Website 

Reviews of these publications may be found in the American Mathematical Society website Mathscinet.



  1. Article at the invitation of the BBVA Foundation (February 2024): Geometría simpléctica: de los movimientos planetarios a las misiones espaciales (en Español/in Spanish).
    BBVA Foundation Website
  2. Tribuna in newspaper El País (October 2023): La inminente y necesaria transformación de las universidades públicas (con Francisco Marcellán, en Español/in Spanish).
    Publisher Website (Text)
  3. Tribuna in newspaper El País (April 2023): La investigación básica en matemáticas: un motor esencial para España (en Español/in Spanish).
    Publisher Website (Text)
  4. Opinion article in Bulletin of the Royal Spanish Mathematical Society (May 2022): The internationalization of mathematics in Spain (en Español/in Spanish), Boletín de la Real Sociedad Matemática Española 760 (2022) 3-4.
    Publisher Website (Text)    Publisher Website (PDF)



  1. Extended abstract (2024): On Ewald’s and Nill’s Conjectures about smooth polytopes (with L. Crespo, F. Santos), Conference Discrete Mathematics Days, Alcalá de Henares, July 3-5, 2024.
  2. Book (2013): Homotopy Type Theory: Univalent Foundations of Mathematics, The Univalent Foundations Program, Institute for Advanced Study, 2013, 473 pages. I was a program participant of the Univalent Foundations Program.
    Publisher Website
  3. Memorial article (2011): Remembering Johannes J. Duistermaat 1942-2010 (Edited by V. Guillemin, A. Pelayo, S. Vu Ngoc, and A.Weinstein), Notices of the American Mathematical Society 58 (2011) 794-802.
    Publisher Website
  4. PhD Thesis at the University of Michigan (2007): Symplectic Torus Actions. 2007. Advisor: Yael Karshon.

Everyone is the son of his own works.
M. de Cervantes