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 am a geometer interested in mathematical physics (especially spectral theory, symplectic invariants), dynamical systems (especially integrable systems), probability and related areas. Recently I have written papers about p-adic symplectic geometry, with an eye towards a possible future implementation with computer proof assistants in Voevodsky´s Univalent Foundations. I have published in 45 different mathematics journals, the ones where I have published at least three papers reflect the interdisciplinary nature of my research: Communications in Mathematical Physics (2006, 2012, 2014, 2025); Proceedings of the London Mathematical Society (2014, 2015, 2026); Bulletin of the American Mathematical Society (2011, 2014, 2017); Journal of Nonlinear Science (2016, 2019, 2025); RACSAM (2014, two in 2026). A more detailed description of my mathematical contributions can be found in the section Research.

Below is a selection of 10 articles, followed by a complete list.

10 SELECTED ARTICLES

For full citation details and downloads please refer to the complete list of articles below this selection.

  • The structure of monotone blow-ups in symplectic toric geometry and a question of McDuff (with Francisco Santos),
    Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 2026.
  • Moduli spaces of Delzant polytopes and symplectic toric manifolds (with Francisco Santos),
    Advances in Mathematics 2025.
  • The random Arnold Conjecture: a new probabilistic Conley-Zehnder Theory for symplectic maps (with Fraydoun Rezakhanlou),
    Communications in Mathematical Physics     2025.
    Described as ¨groundbreaking¨ in its AMS Mathscinet review.
  • Symplectic G-capacities and integrable systems (with Alessio Figalli and Joseph Palmer),
    Annali della Scuola Normale Superiore di Pisa, Classe di Scienze     2018.
  • The tropical momentum map: a classification of toric log symplectic manifolds (with Marco Gualtieri, Songhao Li, and Tudor Ratiu),
    Mathematische Annalen 2017.
    Takes position 25 out of 643 among most cited papers in AMS Mathscinet category “Momentum maps and symplectic reduction”, July 2026.
  • Semiclassical quantization and spectral limits of h-pseudodifferential and Berezin-Toeplitz operators (with Leonid Polterovich and San Vu Ngoc),
    Proceedings of the London Mathematical Society     2014.
  • Isospectrality for quantum toric integrable systems (with Laurent Charles and San Vu Ngoc),
    Annales Scientifiques de l’École Normale Supérieure 2013.
  • Constructing integrable systems of semitoric type (with San Vu Ngoc),
    Acta Mathematica     2011.
    Takes position 12 out of 643 among most cited papers in AMS Mathscinet category “Momentum maps and symplectic reduction”, July 2026.
  • Symplectic actions of 2-tori on 4-manifolds,
    Memoirs of the American Mathematical Society 2010.
  • Semitoric integrable systems on symplectic 4-manifolds (with San Vu Ngoc),
    Inventiones Mathematicae 2009.
    Takes position 7 out of 1301 among most cited papers in AMS Mathscinet category “Completely integrable systems, topological structure of phase space, integration methods”, July 2026.


COMPLETE LIST OF ARTICLES

Below is a complete list of my research articles, including sixteen articles (published, to appear or as arxiv preprints) in the 2023-2026 period. The following are highlights of these recent contributions: (i) proof of Random Arnold Conjecture (item 58 below, with Rezakhanlou); (ii) solution to well-known folklore problem originating in Oda´s classical 1978 work (item 60 below, with Santos); (iii) solution to problem of McDuff from 2011 about the polytopes of Gorenstein Fano toric varieties (item 61 below, with Santos); (iv) proof of a broad case of Ewald´s Conjecture from 1988 (item 62 below, with Crespo and Santos); (v) classification of normal forms of p-adic integrable systems (items 64 and 65 below, with Crespo); (vi) proof of the p-adic Darboux´s Theorem (item 66 below, with Crespo).

  1. p-adic integrable systems: from biquadratic equations to local models (with Luis Crespo),
    Preprint 2026, 31 pages.
    arxiv:2606.24363
  2. Group actions on p-adic symplectic manifolds (with Luis Crespo),
    Preprint 2025, 27 pages.
    arxiv:2512.15575
  3. p-adic angular momentum coupling in symplectic geometry (with Luis Crespo),
    Preprint 2025, 43 pages.
    arxiv:2510.13415
  4. Darboux’s Theorem in p-adic symplectic geometry (with Luis Crespo),
    Preprint 2025, 32 pages.
    arxiv:2508.15443
  5. p-adic symplectic geometry of integrable systems and Weierstrass-Williamson theory II (with Luis Crespo),
    Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas RACSAM, to appear, 66 pages.
    Publisher Website       arxiv:2602.09898
  6. p-adic symplectic geometry of integrable systems and Weierstrass-Williamson theory I (with Luis Crespo),
    Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas RACSAM, to appear, 61 pages.
    Publisher Website   arxiv:2501.14444
  7. Rigidity and flexibility in p-adic symplectic geometry (with Luis Crespo),
    Revista Matemática Complutense (2026), 64 pages, https://doi.org/10.1007/s13163-026-00566-8
    Publisher Website (OPEN ACCESS)   arxiv:2505.07663
    (This paper appears mentioned in general news media such as Fundación para el Conocimiento madri+d)
  8. Ewald’s Conjecture and integer points in algebraic and symplectic toric geometry (with Luis Crespo and Francisco Santos),
    Proceedings of the London Mathematical Society     132 (2026) e70144, 42 pages.
    Publisher Website (OPEN ACCESS)   arxiv:2310.10366
  9. The structure of monotone blow-ups in symplectic toric geometry and a question of McDuff (with Francisco Santos),
    Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, to appear, 14 pages.
    Publisher Website   arxiv:2308.03085
  10. Moduli spaces of Delzant polytopes and symplectic toric manifolds (with Francisco Santos),
    Advances in Mathematics 482, Part B (2025), article number 110624, 56 pages.
    Publisher Website   arxiv:2303.02369
  11. The p-adic Jaynes-Cummings model in symplectic geometry (with Luis Crespo),
    Journal of Nonlinear Science 35 (2025), article number 66, 76 pages. 
    Publisher Website (OPEN ACCESS)       arxiv:2406.18415
  12. The random Arnold Conjecture: a new probabilistic Conley-Zehnder Theory for symplectic maps (with Fraydoun 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)
  13. Semitoric systems of non-simple type (with Joseph Palmer and Xiudi Tang),
    Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas RACSAM 118, 161 (2024), 32 pages.
    Publisher Website   arXiv:1909.03501
  14. Symplectic geometry: applications and challenges (in Spanish),
    La Gaceta de la Real Sociedad Matemática Española     27 (2024) 285-311.
    Publisher Website (OPEN ARCHIVE)
  15. Vu Ngoc’s Conjecture on focus-focus singular fibers with multiple pinched points (with Xiudi Tang),
    Journal of Fixed Point Theory and Applications 26, 6 (2024), 34 pages.
    Publisher Website   arXiv:1803.00998
  16. 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
  17. 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
  18. Moser-Greene-Shiohama stability for families (with Xiudi Tang),
    Journal of Symplectic Geometry 17 (2019) 1427-1446.
    Publisher Website (OPEN ACCESS)
  19. Symplectic stability on manifolds with cylindrical ends (with Sean Curry and Xiudi Tang),
    The Journal of Geometric Analysis     29 (2019) 1660-1675.
    Publisher Website   arXiv:1704.08659
  20. Symplectic geometry and spectral properties of classical and quantum coupled angular momenta (with Yohann Le Floch),
    Journal of Nonlinear Science 29 (2019) 655-708.
    Publisher Website   arXiv:1607.05419
  21. Spectral asymptotics of semiclassical unitary operators (with Yohann Le Floch),
    Journal of Mathematical Analysis and Applications 473 (2019) 1174-1202
    Publisher Website (OPEN ARCHIVE)   arXiv:1506.02873
  22. Moser stability for volume forms on noncompact fiber bundles (with Xiudi Tang),
    Differential Geometry and its Applications     63 (2019) 120-136.
    Publisher Website (OPEN ARCHIVE)   arXiv:1607.03800
  23. Classifying toric and semitoric fans by lifting equations from SL(2,Z) (with Daniel Kane and Joseph Palmer),
    Symmetry, Integrability, and Geometry: Methods and Applications SIGMA     14 (2018), 016, 43 pp.
    Publisher Website (OPEN ACCESS)   arXiv:1502.07698
  24. Minimal models of compact symplectic semitoric manifolds (with Daniel Kane and Joseph Palmer), 
    Journal of Geometry and Physics     125 (2018) 49-74.
    Publisher Website (OPEN ARCHIVE)   arXiv:1610.05423
  25. Symplectic G-capacities and integrable systems (with Alessio Figalli and Joseph Palmer),
    Annali della Scuola Normale Superiore di Pisa, Classe di Scienze     (5) Vol. XVIII (2018) 65-103.
    Publisher Website (OPEN ARCHIVE)   arXiv:1511.04499
  26. Poincare-Birkhoff theorems in random dynamics (with Fraydoun Rezakhanlou),
    Transactions of the American Mathematical Society     370 (2018) 601-639.
    Publisher Website (FREELY AVAILABLE)   arXiv:1306.0821
  27. Hamiltonian and symplectic symmetries: an introduction,
    Bulletin of the American Mathematical Society 54 (2017) 383-436.
    Publisher Website (FREELY AVAILABLE)   arXiv:1610.09718
  28. The affine invariant of proper semitoric integrable systems (with Tudor Ratiu and San Vu Ngoc),
    Nonlinearity     30 (2017) 3993-4028.
    Publisher Website   arXiv:1307.7516 
  29. The tropical momentum map: a classification of toric log symplectic manifolds (with Marco Gualtieri, Songhao Li, and Tudor Ratiu),
    Mathematische Annalen 367 (2017) 1217-1258.
    Publisher Website   arXiv:1407.3300
    (This paper takes the position 25 out of 643 among the most cited papers in Mathscinet in the category 53D20 of “Momentum maps and symplectic reduction”, with date July 28, 2026; it is the most cited paper among the 209 papers published since 2016.)
  30. Generating hyperbolic singularities in semitoric systems via Hopf bifurcations (with Holger Dullin),
    Journal of Nonlinear Science     26 (2016) 787-811.
    Publisher Website   arXiv:1503.01534
  31. On the density function on moduli spaces of toric 4-manifolds (with Alessio Figalli),
    Advances in Geometry 16 (2016) 291-300.
    Publisher Website   arXiv:1408.1462
  32. Spectral limits of semiclassical commuting self-adjoint operators (with San Vu Ngoc),
    A Mathematical Tribute to J.M. Montesinos, Editorial Universidad Complutense de Madrid (2016) 527-546.
    Publisher Website (FREELY AVAILABLE)   arXiv:1506.04591
  33. Inverse spectral theory for semiclassical Jaynes-Cummings systems (with Yohann Le Floch and San Vu Ngoc),
    Mathematische Annalen 364 (2016) 1393-1413. Erratum in vol. 375 (2019) 917-920.
    Publisher Website   arXiv:1407.5159
  34. Sharp symplectic embeddings of cylinders (with San Vu Ngoc),
    Indagationes Mathematicae N.S. 27 (2016) 307-317.
    Publisher Website (OPEN ARCHIVE)   arXiv:1304.5250
  35. Euler-MacLaurin formulas via differential operators (with Yohann Le Floch),
    Advances in Applied Mathematics 76 (2016) 99-124.
    Publisher Website (OPEN ARCHIVE)   arXiv:1312.5711
  36. Fermat and the number of fixed points of periodic flows (with Leonor Godinho and Silvia Sabatini),
    Communications in Number Theory and Physics     9 (2015) 643-687.
    Publisher Website (OPEN ACCESS)  arXiv:1404.4541
  37. A univalent formalization of the p-adic numbers (with Vladimir Voevodsky and Michael 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).
  38. Fiber connectivity and bifurcation diagrams for almost toric systems (with Tudor Ratiu and San Vu Ngoc),
    Journal of Symplectic Geometry 13 (2015) 343-386.
    Publisher Website (OPEN ACCESS)   arXiv:1108.0328
  39. Hofer’s question on intermediate symplectic capacities (with San Vu Ngoc),
    Proceedings of the London Mathematical Society     110 (2015) 787-804.
    Publisher Website   arXiv:1210.1537
  40. L^2-Cohomology and complete Hamiltonian manifolds (with Rafe Mazzeo and Tudor Ratiu),
    Journal of Geometry and Physics     87 (2015) 305-313.
    Publisher Website (OPEN ARCHIVE)   arXiv:1402.0098
  41. Log-concavity and symplectic flows (with Yi Lin),
    Mathematical Research Letters 22 (2015) 501-527.
    Publisher Website (OPEN ACCESS)   arXiv:1207.1335
  42. Semiclassical quantization and spectral limits of h-pseudodifferential and Berezin-Toeplitz operators (with Leonid Polterovich and San Vu Ngoc),
    Proceedings of the London Mathematical Society     109 (2014) 676-696.
    Publisher Website   arXiv:1302.0424
  43. Homotopy type theory in context (with Michael Warren),
    Gazette des mathématiciens Société Mathématique de France 42 (2014) 95-110.
    Publisher Website
  44. Homotopy type theory and Voevodsky’s univalent foundations (with Michael 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
  45. Semiclassical inverse spectral theory for singularities of focus-focus type (with San Vu Ngoc),
    Communications in Mathematical Physics 329 (2014) 809-820.
    Publisher Website   arXiv:1302.2260
  46. Moduli spaces of toric manifolds (with Ana Pires, Tudor Ratiu, and Silvia Sabatini),
    Geometriae Dedicata     169 (2014) 323-341.
    Publisher Website   arXiv:1207.0092
  47. Isospectrality for quantum toric integrable systems (with Laurent Charles and San Vu Ngoc),
    Annales Scientifiques de l’École Normale Supérieure 43 (2013) 815-849.
    Publisher Website (OPEN ACCESS)   arXiv:1111.5985
  48. Voevodsky’s Univalence Axiom in homotopy type theory (with Steve Awodey and Michael Warren),
    Notices of the American Mathematical Society 60 (2013) 1164-1167.
    Publisher Website (FREELY AVAILABLE)   arXiv:1302.4731
  49. Symplectic spectral geometry of semiclassical operators,
    Bulletin of the Belgian Mathematical Society     20 (2013) 405-415.
    Publisher Website (OPEN ACCESS)   arXiv:1303.2570
  50. Circle valued momentum maps for symplectic periodic flows (with Tudor Ratiu),
    L’Enseignement Mathématique     58 (2012) 205-219.
    Publisher Website (OPEN ARCHIVE) 
  51. First steps in symplectic and spectral theory of integrable systems (with San Vu Ngoc),
    Discrete and Continuous Dynamical Systems, Series A 32 (2012) 3325-3377.
    Publisher Website   arXiv:1306.0124
  52. Hamiltonian dynamics and spectral theory for spin-oscillators (with San Vu Ngoc),
    Communications in Mathematical Physics     309 (2012) 123-154.
    Publisher Website   arXiv:1005.0439
  53. Symplectic geometry on moduli spaces of J-holomorphic curves (with Joseph Coffey and Liat Kessler),
    Annals of Global Analysis and Geometry     41 (2012) 265-280.
    Publisher Website   arXiv:0905.3009
  54. Symplectic theory of completely integrable Hamiltonian systems (with San Vu Ngoc)
    Bulletin of the American Mathematical Society 48 (2011) 409-455.
    Publisher Website (FREELY AVAILABLE)   arXiv:1306.0115
  55. Constructing integrable systems of semitoric type (with San Vu Ngoc),
    Acta Mathematica     206 (2011) 93-125.
    Publisher Website (OPEN ACCESS)   arXiv:0903.3376
    (This paper takes the position 12 out of 643 among the most cited papers in Mathscinet in the category 53D20 of “Momentum maps and symplectic reduction”, with date July 28, 2026; it is the most cited paper among the 387 papers published since 2008.)
  56. Fourier integral operators, spectral theory, and Hamiltonian systems,
    Gaceta de la Real Sociedad Matemática Española 14 (2011) 247-262.
    Publisher Website (OPEN ARCHIVE)
  57. Fixed points of symplectic periodic flows (with Susan Tolman),
    Ergodic Theory and Dynamical Systems     31 (2011) 1237-1247.
    Publisher Website   arXiv:1003.4787
  58. Complex structures on 4-manifolds with symplectic 2-torus actions (with Johannes J. Duistermaat),
    International Journal of Mathematics     2 (2011) 449-463.
    Publisher Website
  59. Non-Kahler symplectic manifolds with toric symmetries (with Yi Lin),
    The Quarterly Journal of Mathematics 62 (2011) 103-114.
    Publisher Website
  60. Topology of symplectic torus actions with symplectic orbits (with Johannes J. Duistermaat),
    Revista Matemática Complutense     24 (2011) 59-81.
    Publisher Website (OPEN ACCESS)
  61. 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
  62. Reduced phase space and toric variety coordinatizations of Delzant spaces (with Johannes J. Duistermaat),
    Mathematical Proceedings of the Cambridge Philosophical Society     146 (2009) 695-718.
    Publisher Website   arXiv:0704.0430
  63. Semitoric integrable systems on symplectic 4-manifolds (with San Vu Ngoc),
    Inventiones Mathematicae 177 (2009) 571-597.
    Publisher Website (OPEN ACCESS)   arXiv:0806.1946
    (This paper takes the position 7 out of 1301 among the most cited papers in Mathscinet in the category 37J35 of “Completely integrable systems, topological structure of phase space, integration methods”, with date July 28, 2026; it is the most cited paper among the 884 papers published since 2004.)
  64. Maximal ball packings of symplectic toric manifolds (with Benjamin Schmidt),
    International Mathematics Research Notices IMRN     (2008), 24p, ID rnm139.
    Publisher Website   arXiv:0704.1036
  65. Symplectic torus actions with coisotropic principal orbits (with Johannes J. Duistermaat),
    Annales de l’Institut Fourier 57 (2007) 2239-2327.
    Publisher Website (OPEN ACCESS)   arXiv:math/0511676
  66. Topology of spaces of equivariant symplectic embeddings,
    Proceedings of the American Mathematical Society 35 (2007) 277-288.
    Publisher Website (FREELY AVAILABLE)  arXiv:0704.1033
  67. A geometric approach to the classification of the equilibrium shapes of self-gravitating fluids (with Daniel Peralta-Salas),
    Communications in Mathematical Physics     267 (2006) 93-115.
    Publisher Website 
  68. Toric symplectic ball packing,
    Topology and Its Applications 157 (2006) 3633-3644.
    Publisher Website (OPEN ARCHIVE)   arXiv:0704.1034
  69. Unounded trajectories of dynamical systems (with Francisco G. Gascón and Daniel 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.


RELATED WORKS

  • Extended abstract (2024): On Ewald’s and Nill’s Conjectures about smooth polytopes (with Luis Crespo and Francisco Santos), Conference Discrete Mathematics Days, Alcalá de Henares, July 3-5, 2024.
    Publisher Website
  • 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
  • Memorial article (2011): Remembering Johannes J. Duistermaat 1942-2010 (Edited by Victor Guillemin, Alvaro Pelayo, San Vu Ngoc, and Alan Weinstein), Notices of the American Mathematical Society 58 (2011) 794-802.
    Publisher Website
  • 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