Publications

My publications arranged according to area. The general area is Dynamics, Geometry, Topology. My work tends to be interdisciplinary and has an important intersection with at least another area.

I have published papers with authors from Australia, Belgium, Canada, China, France, Germany, Iran, Israel, Italy, New Zealand, Portugal, Romania, Russia, Spain, Switzerland, The Netherlands, 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 2017-2024), 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. Since about twenty years ago I have had  extensive collaborations with San Vu Ngoc (14 papers) and with Fraydoun Rezakhanlou (2 papers) with a focus on integrable and dynamical systems. More recently I have been working on combinatorial problems arising in symplectic and algebraic geometry with Francisco Santos (3 papers), and on p-adic symplectic geometry with my former PhD student from the University of Cantabria Luis Crespo (5 papers). The work with Crespo continues a research program which Vladimir Voevodsky, Michael A. Warren and myself started more than a decade ago, concerning a p-adic approach to symplectic geometry, with an eye towards implementing it in the language of homotopy type theory and Voevodsky’s Univalent Foundations program, using a computer proof assistant (we also developed computer code in the programming language/interative proof assistant Coq, which accompanies two of our papers, and is available below).

Throughout my career 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).

My contributions since 2023 (with date October 17, 2025) are spread over thirteen papers (published, to appear or as arxiv preprints) and distributed in approximately 600 pages. The following are highlights of these contributions, roughly in ascending chronological order:

  • Classification of non-simple semitoric integrable systems (item 57 below, with Palmer and Tang);
  • Proof of Random Arnold Conjecture in Stochastic Dynamics and Symplectic Topology (item 58 below, with Rezakhanlou);
  • Solution to well-known folklore problem originating in Oda´s classical work from 1978 (item 60 below, with Santos);
  • Solution to problem of McDuff from 2011 about the polytopes of Gorenstein Fano toric varieties (item 61 below, with Santos);
  • Proof of a broad case of Ewald´s Conjecture in algebraic/toric/symplectic geometry (item 62 below, with Crespo and Santos);
  • Classification of local normal forms of p-adic integrable systems (item 63 below, with Crespo) and in depth study of two fundamental examples: the Jaynes-Cummings model and the coupled angular momentum and (items 59 and 66 below, with Crespo);
  • Proof of the p-adic Darboux´s Theorem and ¨symplectic analog¨ of the p-adic Serre´s Classification (item 65 below, with Crespo).

A more in depth description of my research contributions, from the beginning of my career around 2005, till now, can be found in the Research section of the website.


ARTICLES

  1. p-adic angular momentum coupling in symplectic geometry (with L. Crespo), Preprint 2025, 43 pages.
    arxiv:2510.13415
  2. Darboux’s Theorem in p-adic symplectic geometry (with L. Crespo), Preprint 2025, 32 pages.
    arxiv:2508.15443
  3. Rigidity and flexibility in p-adic symplectic geometry (with L. Crespo), Preprint 2025, 52 pages.
    arxiv:2505.07663
  4. p-adic symplectic geometry of integrable systems and Weierstrass-Williamson theory (with L. Crespo), Preprint 2025, 107 pages.
    arxiv:2501.14444
  5. Ewald’s Conjecture and integer points in algebraic and symplectic toric geometry (with L. Crespo and F. Santos), Preprint 2023, 37 pages.
    arxiv:2310.10366
  6. The structure of monotone blow-ups in symplectic toric geometry and a question of McDuff (with F. Santos), to appear in Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 14 pages.
    Publisher Website   arxiv:2308.03085
  7. Moduli spaces of Delzant polytopes and symplectic toric manifolds (with F. Santos), Advances in Mathematics 482, Part B (2025), article number 110624, 57 pages.
    Publisher Website   arxiv:2303.02369
  8. The p-adic Jaynes-Cummings model in symplectic geometry (with L. Crespo), Journal of Nonlinear Science 35 (2025), article number 66, 76 pages. 
    Publisher Website (OPEN ACCESS)       arxiv:2406.18415
  9. The random Arnold Conjecture: a new probabilistic Conley-Zehnder Theory for symplectic maps (with F. Rezakhanlou),  Communications in Mathematical Physics 406 (2025), article number 35, 61 pages.
    Publisher Website (OPEN ACCESS)   arxiv:2306.15586
    (This paper is described as ¨groundbreaking¨ in the American Mathematical Society website Mathscinet review. The paper also appears mentioned in general news media such as Europa Press, Fundación para el Conocimiento madri+d, and Diario Siglo XXI)
  10. Semitoric systems of non-simple type (with J. Palmer and X. Tang), Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 118, 161 (2024), 32 pages.
    Publisher Website   arXiv:1909.03501
  11. Symplectic geometry: applications and challenges (in Spanish), La Gaceta de la Real Sociedad Matemática Española 27 (2024) 285-311.
    Publisher Website
  12. 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), 34 pages.
    Publisher Website   arXiv:1803.00998
  13. Symplectic and inverse spectral geometry of integrable systems: a glimpse and open problems. Topology and its Applications 339 Part A (2023) 108577, 35 pages.
    Publisher Website
  14. Symplectic invariants of semitoric systems and the inverse problem for quantum systems, Indagationes Mathematicae 32 (2021) 246-274.
    Publisher Website (OPEN ARCHIVE)  arXiv:2002.05306
  15. Moser-Greene-Shiohama stability for families (with X. Tang), Journal of Symplectic Geometry 17 (2019) 1427-1446.
    Publisher Website (OPEN ACCESS)
  16. 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
  17. 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
  18. Spectral asymptotics of semiclassical unitary operators (with Y. Le Floch), Journal of Mathematical Analysis and Applications 473 (2019) 1174-1202
    Publisher Website (OPEN ARCHIVE)   arXiv:1506.02873
  19. Moser stability for volume forms on noncompact fiber bundles (with X. Tang), Differential Geometry and its Applications 63 (2019) 120-136.
    Publisher Website (OPEN ARCHIVE)   arXiv:1607.03800
  20. 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 pp.
    Publisher Website (OPEN ACCESS)   arXiv:1502.07698
  21. Minimal models of compact symplectic semitoric manifolds (with D. Kane and J. Palmer) Journal of Geometry and Physics 125 (2018) 49-74.
    Publisher Website (OPEN ARCHIVE)   arXiv:1610.05423
  22. 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 (OPEN ARCHIVE)   arXiv:1511.04499
  23. Poincare-Birkhoff theorems in random dynamics (with F. Rezakhanlou), Transactions of the American Mathematical Society 370 (2018) 601-639.
    Publisher Website (FREELY AVAILABLE)   arXiv:1306.0821
  24. Hamiltonian and symplectic symmetries: an introduction, Bulletin of the American Mathematical Society 54 (2017) 383-436.
    Publisher Website (FREELY AVAILABLE)   arXiv:1610.09718
  25. 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 
  26. 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
  27. 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
  28. On the density function on moduli spaces of toric 4-manifolds (with A. Figalli), Advances in Geometry 16 (2016) 291-300.
    Publisher Website   arXiv:1408.1462
  29. 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 (FREELY AVAILABLE)   arXiv:1506.04591
  30. 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
  31. Sharp symplectic embeddings of cylinders (with S. Vu Ngoc), Indagationes Mathematicae N.S. 27 (2016) 307-317.
    Publisher Website (OPEN ARCHIVE)   arXiv:1304.5250
  32. Euler-MacLaurin formulas via differential operators (with Y. Le Floch) Advances in Applied Mathematics 76 (2016) 99-124.
    Publisher Website (OPEN ARCHIVE)   arXiv:1312.5711
  33. 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 (OPEN ACCESS)  arXiv:1404.4541
  34. 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 (ACCESS)    arXiv:1302.1207
    This paper comes with an associated Coq code (Coq is an interative proof assistant/programming language) by the same authors, which is distributed in several chapters:
    Formal Power Series (24 pp);
    The Heyting Field of Fractions for an Apartness Domain (8 pp);
    Fixing Notation, Terminology and Basic Lemmas (15 pp);
    p-Adic Numbers (29 pp);
    Integers mod p (28pp).
  35. 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 (OPEN ACCESS)   arXiv:1108.0328
  36. 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
  37. L^2-Cohomology and complete Hamiltonian manifolds (with R. Mazzeo and T. Ratiu), Journal of Geometry and Physics 87 (2015) 305-313.
    Publisher Website (OPEN ARCHIVE)   arXiv:1402.0098
  38. Log-concavity and symplectic flows (with Y. Lin), Mathematical Research Letters 22 (2015) 501-527.
    Publisher Website (OPEN ACCESS)   arXiv:1207.1335
  39. 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
  40. Homotopy type theory in context (with M. Warren), Gazette des mathématiciens Société Mathématique de France 42 (2014) 95-110.
    Publisher Website
  41. Homotopy type theory and Voevodsky’s univalent foundations (with M. Warren), Bulletin of the American Mathematical Society 51 (2014) 597-648.
    This paper comes with an associated Coq code tutorial, 22pp, by the same authors (Coq is an interative proof assistant/programming language).
    Publisher Website (FREELY AVAILABLE)  arXiv:1210.5658
  42. 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
  43. Moduli spaces of toric manifolds (with A. Pires, T. Ratiu, and S. Sabatini), Geometriae Dedicata 169 (2014) 323-341.
    Publisher Website   arXiv:1207.0092
  44. 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 (OPEN ACCESS)   arXiv:1111.5985
  45. 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 (FREELY AVAILABLE)   arXiv:1302.4731
  46. Symplectic spectral geometry of semiclassical operators, Bulletin of the Belgian Mathematical Society 20 (2013) 405-415.
    Publisher Website (OPEN ACCESS)   arXiv:1303.2570
  47. Circle valued momentum maps for symplectic periodic flows (with T. S. Ratiu), L’Enseignement Mathématique 58 (2012) 205-219.
    Publisher Website (OPEN ARCHIVE) 
  48. 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
  49. 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
  50. 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
  51. Symplectic theory of completely integrable Hamiltonian systems (with S. Vu Ngoc) Bulletin of the American Mathematical Society 48 (2011) 409-455.
    Publisher Website (FREELY AVAILABLE)   arXiv:1306.0115
  52. 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 13 out of 539 among the most cited papers in Mathscinet since 1940 in the category 53D20 of “Momentum maps and symplectic reduction”, with date February 14, 2025.)
  53. Fourier integral operators, spectral theory, and Hamiltonian systems, Gaceta de la Real Sociedad Matemática Española 14 (2011) 247-262.
    Publisher Website (OPEN ARCHIVE)
  54. Fixed points of symplectic periodic flows (with S. Tolman), Ergodic Theory and Dynamical Systems 31 (2011) 1237-1247.
    Publisher Website   arXiv:1003.4787
  55. Complex structures on 4-manifolds with symplectic 2-torus actions (with J.J. Duistermaat), International Journal of Mathematics 2 (2011) 449-463.
    Publisher Website
  56. Non-Kahler symplectic manifolds with toric symmetries (with Y. Lin), Quarterly Journal of Mathematics 62 (2011) 103-114.
    Publisher Website
  57. Topology of symplectic torus actions with symplectic orbits (with J.J. Duistermaat), Revista Matemática Complutense 24 (2011) 59-81.
    Publisher Website (OPEN ACCESS)
  58. 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
  59. 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
  60. 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 9 out of 1138 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 February 14, 2025.)
  61. Maximal ball packings of symplectic toric manifolds (with B. Schmidt) International Mathematics Research Notices IMRN (2008), 24p, ID rnm139.
    Publisher Website   arXiv:0704.1036
  62. Symplectic torus actions with coisotropic principal orbits (with J.J. Duistermaat), Annales de l’Institut Fourier 57 (2007) 2239-2327.
    Publisher Website (OPEN ACCESS)   arXiv:math/0511676
  63. Topology of spaces of equivariant symplectic embeddings, Proceedings of the American Mathematical Society 35 (2007) 277-288.
    Publisher Website (FREELY AVAILABLE)  arXiv:0704.1033
  64. 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 
  65. Toric symplectic ball packing, Topology and Its Applications 157 (2006) 3633-3644.
    Publisher Website (OPEN ARCHIVE)   arXiv:0704.1034
  66. Unounded trajectories of dynamical systems (with F. G. Gascón and D. Peralta-Salas), Applied Mathematics Letters 17 (2004) 253-259.
    Publisher Website (OPEN ARCHIVE) 

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

 

ARTICLES FOR A GENERAL PUBLIC

  1. Tribuna in newspaper La Razón (September 2025): Nuevo curso escolar: nuevos retos, pero merece la pena (en Español/in Spanish).
    La Razón Website (Text)
  2. Opinion Article in newspaper El Diario Montañés (September 2025): Cuidemos nuestra universidad más singular: la UIMP (en Español/in Spanish).
    El Diario Montañés Website (Text)
  3. Tribuna in newspaper La Razón (August 2025): Simbiosis Universidad-empresa: lo que falta  (en Español/in Spanish).
    La Razón Website (Text)
  4. Tribuna in newspaper La Razón (July 2025): Educar para pensar desde la infancia: el camino hacia la creatividad y el razonamiento crítico (en Español/in Spanish).
    La Razón Website (Text)
  5. Tribuna in newspaper La Razón (July 2025): La atracción de talento a las universidades (en Español/in Spanish).
    La Razón Website (Text)
  6. Opinion Article in newspaper El País (June 2025): Apoyar más la ciencia y la tecnología: una tarea inaplazable (en Español/in Spanish).
    El País Website (Text)
  7. Article at the invitation of the BBVA Foundation in Journal Sigma-FBBVA (January 2025): El poder de la interacción entre la investigación básica y aplicada (en Español/in Spanish).
    Red Leonardo (of BBVA Foundation) Website (Text)
  8. Article in newspaper ABC (January 2025): Muere Pedro Luis García Pérez, un histórico de las matemáticas y pionero de la geometría diferencial en España (con Francisco Marcellán, en Español/in Spanish).
    ABC Website (Text)
  9. 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 (Text)
  10. Tribuna in newspaper El País (October 2023): La inminente y necesaria transformación de las universidades públicas (with Francisco Marcellán, en Español/in Spanish).
    El País Website (Text)
  11. Tribuna in newspaper El País (April 2023): La investigación básica en matemáticas: un motor esencial para España (en Español/in Spanish).
    El País Website (Text)
  12. Opinion article in Bulletin of the Royal Spanish Mathematical Society (May 2022): La internacionalización de las matemáticas en España (en Español/in Spanish), Boletín de la Real Sociedad Matemática Española 760 (2022) 3-4.
    Publisher Website (Text)    Publisher Website (PDF)

 

RELATED WORKS

  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.
    Publisher Website
  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