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 which continues today, with San Vu Ngoc (14 papers).
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.
- Semitoric systems of non-simple type (with J. Palmer and X. Tang), Preprint.
- Vu Ngoc’s Conjecture on focus-focus singular fibers with multiple pinched points (with X. Tang), Preprint.
- Symplectic invariants of semitoric systems and the inverse problem for quantum systems, Indagationes Mathematicae 32 (2021) 246-274.
Publisher Website arXiv:2002.05306
- Moser-Greene-Shiohama stability for families (with X. Tang), Journal of Symplectic Geometry 17 (2019) 1427-1446.
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- Poincare-Birkhoff theorems in random dynamics (with F. Rezakhanlou), Transactions of the American Mathematical Society 370 (2018) 601-639.
Publisher Website arXiv:1306.0821
- Hamiltonian and symplectic symmetries: an introduction, Bulletin of the American Mathematical Society 54 (2017) 383-436.
Publisher Website arXiv:1610.09718
- 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
- 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
- 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
- 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
- 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
- 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
- Sharp symplectic embeddings of cylinders (with S. Vu Ngoc), Indagationes Mathematicae N.S. 27 (2016) 307-317.
Publisher Website arXiv:1304.5250
- Euler-MacLaurin formulas via differential operators (with Y. Le Floch) Advances in Applied Mathematics 76 (2016) 99-124.
Publisher Website arXiv:1312.5711
- 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
- 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
- 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
- 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
- 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
- Log-concavity and symplectic flows (with Y. Lin), Mathematical Research Letters 22 (2015) 501-527.
Publisher Website arXiv:1207.1335
- 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
- Homotopy type theory in context (with M. Warren), Gazette des mathématiciens Société Mathématique de France 42 (2014) 95-110.
- 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
- 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
- Moduli spaces of toric manifolds (with A. Pires, T. Ratiu, and S. Sabatini), Geometriae Dedicata 169 (2014) 323-341.
Publisher Website arXiv:1207.0092
- 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
- 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
- Symplectic spectral geometry of semiclassical operators, Bulletin of the Belgian Mathematical Society 20 (2013) 405-415.
Publisher Website arXiv:1303.2570
- Circle valued momentum maps for symplectic periodic flows (with T. S. Ratiu), L’Enseignement Mathématique 58 (2012) 205-219.
- 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
- 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
- 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
- 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
- Constructing integrable systems of semitoric type (with S. Vu Ngoc), Acta Mathematica 206 (2011) 93-125.
Publisher Website (OPEN ACCESS)
(This paper takes the position 15 out of 468 among the most cited papers in Mathscinet since 1940 in the category 53D20 of “Momentum maps and symplectic reduction”, with date January 18, 2021.)
- Fourier integral operators, spectral theory, and Hamiltonian systems, Gaceta de la Real Sociedad Matemática Espanola 14 (2011) 247-262.
- Fixed points of symplectic periodic flows (with S. Tolman), Ergodic Theory and Dynamical Systems 31 (2011) 1237-1247.
Publisher Website arXiv:1003.4787
- Complex structures on 4-manifolds with symplectic 2-torus actions (with J.J. Duistermaat), International Journal of Mathematics 2 (2011) 449-463.
- Non-Kahler symplectic manifolds with toric symmetries (with Y. Lin), Quarterly Journal of Mathematics 62 (2011) 103-114.
- Topology of symplectic torus actions with symplectic orbits (with J.J. Duistermaat), Revista Matemática Complutense 24 (2011) 59-81.
- 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
- 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
- Semitoric integrable systems on symplectic 4-manifolds (with S. Vu Ngoc), Inventiones Mathematicae 177 (2009) 571-597.
Publisher Website (OPEN ACCESS)
(This paper takes the position 14 out of 1054 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 January 18, 2021.)
- Maximal ball packings of symplectic toric manifolds (with B. Schmidt) International Mathematics Research Notices IMRN (2008), 24p, ID rnm139.
Publisher Website arXiv:0704.1036
- 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
- Topology of spaces of equivariant symplectic embeddings, Proceedings of the American Mathematical Society 35 (2007) 277-288.
Publisher Website arXiv:0704.1033
- 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.
- Toric symplectic ball packing, Topology and Its Applications 157 (2006) 3633-3644.
Publisher Website arXiv:0704.1034
- Unounded trajectories of dynamical systems (with F. G. Gascón and D. Peralta-Salas), Applied Mathematics Letters 17 (2004) 253-259.
Reviews of these publications may be found in the American Mathematical Society website Mathscinet.
- 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.
- PhD Thesis: Symplectic Torus Actions. 2007. Advisor: Yael Karshon.
- 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.
Everyone is the son of his own works.
M. de Cervantes