Research

Research Interests

  • Noncommutative geometry.
  • Global analysis (index theory, geometric spectral theory).
  • Microlocal analysis (pseudodifferential operators).
  • Differential geometry (conformal geometry, Cauchy-Riemann geometry, contact geometry, sub-Riemannian geometry).
  • Homological algebra (cyclic homology).

Research Articles

  1. Dixmier trace formulas and negative eigenvalues of Schrödinger operators on curved noncommutative tori (with Edward McDonald) (arXiv). Submitted, 32 pages.
  2. Cwikel estimates and negative eigenvalues of Schrödinger operators on noncommutative tori (with Edward McDonald) (arXiv). Submitted, 27 pages.
  3. Periodicity and cyclic homology. Para-S-modules and perturbation lemmas (arXiv) (doi). J. Noncomm. Geom. 14 (2020), 1365–1444.
  4. Noncommutative residue and canonical trace on noncommutative tori. Uniqueness results (arXiv) (doi). SIGMA (Symmetry Integrability Geom. Methods Appl.) 16 (2020), 061, 31 pages.
  5. Functional calculus for elliptic operators on noncommutative tori, I (with Gihyun Lee) (arXiv) (doi). J. Pseudo-Differ. Oper. Appl. 11 (2020), 935–1004.
  6. Connes’s trace theorem for curved noncommutative tori. Application to scalar curvature (arXiv) (doi). J. Math. Phys. 61 (2020), 042301 (27 pages).
  7. Laplace-Beltrami operators on noncommutative tori (with Hyunsu Ha) (arXiv)(doi). J. Geom. Phys. 150 (2020), 103594 (25 pages).
  8. Pseudodifferential calculus on noncommutative tori, II. Main properties (with Hyunsu Ha and Gihyun Lee) (arXiv) (doi). Int. J. Math. 30(8) (2019), 1950034 (73 pages).
  9. Pseudodifferential calculus on noncommutative tori, I. Oscillating integrals (with Hyunsu Ha and Gihyun Lee) (arXiv) (doi). Int. J. Math. 30(8) (2019), 1950033 (74 pages).
  10. Tangent maps and tangent groupoid for Carnot manifolds (with Woocheol Choi) (arXiv) (doi). Differential Geom. Appl. 62 (2019), 136–183.
  11. Privileged coordinates and nilpotent approximation for Carnot manifolds, II. Carnot coordinates (with Woocheol Choi) (arXiv) (doi). J. Dyn. Control Syst. 25 (2019), 631–670.
  12. Privileged coordinates and nilpotent approximation for Carnot manifolds, I. General results (with Woocheol Choi) (arXiv) (doi). J. Dyn. Control Syst. 25 (2019), 109–157.
  13. Cyclic homology and group actions (arXiv) (doi). J. Geom. Phys. 123 (2018), 30–52.
  14. Noncommutative geometry and conformal geometry. I. Local index formula and conformal invariants (with Hang Wang) (arXiv) (doi). J. Noncomm. Geom. 12 (2018), 1573–1639.
  15. The cyclic homology of crossed-product algebras, I (pdf) (doi). C. R. Acad. Sci. Paris, sér. I, 355 (2017), 618–622.
  16. The cyclic homology of crossed-product algebras, II (pdf) (doi). C. R. Acad. Sci. Paris, sér. I, 355 (2017), 623–627.
  17. Noncommutative geometry and conformal geometry. II. Connes-Chern character and the local equivariant index theorem (with Hang Wang) (arXiv) (doi). J. Noncomm. Geom. 10 (2016), 307–378.
  18. Index map, σ-connections, and Connes-Chern character in the setting of twisted spectral triples (with Hang Wang) (arXiv) (doi). Kyoto J. Math. 56 (2016), 347–399.
  19. Noncommutative geometry and conformal geometry. III. Vafa-Witten inequality and Poincaré duality (with Hang Wang) (arXiv) (doi). Adv. Math. 272 (2015), 761–819.
  20. The Logarithmic singularities of the Green functions of the conformal powers of the Laplacian (arXiv). Contemp. Math. Vol. 630 (2014), pp. 247–273.
  21. Conformal invariants from nodal sets (with Yaiza Canzani, Rod Gover and Dmitry Jakobson) (Appendix by Rod Gover and Andrea Malchiodi) (arXiv) (doi). Int. Math. Res. Notices 2014 (2014), 2356–2400.
  22. Nullspaces of conformally invariant operators. Applications to Q-curvature (with Yaiza Canzani, Rod Gover and Dmitry Jakobson) (arXiv) (doi). Electron. Res. Announc. Math. Sci. 20 (2013), 43–50.
  23. On the singularities of the zeta and eta functions of an elliptic operator (with Paul Loya and Sergiu Moroianu) (arXiv) (doi). Int. J. Math. 23 (2012), no. 6, 1–26.
  24. A New hypoelliptic operator on almost CR manifolds (arXiv) (doi). Rev. Mat. Iberoamericana 27 (2011), no. 2, 393–414.
  25. A microlocal approach to Fefferman’s program in conformal and CR geometry (full volume). Clay Math. Proc., 12, Amer. Math. Soc., Providence, RI, 2011, pp. 219–242.
  26. Traces on pseudodifferential operators and sums of commutators (arXiv) (doi). J. Anal. Math. 110 (2010), 1–30.
  27. Logarithmic singularities of Schwartz kernels and local invariants of conformal and CR structures operators (arXiv). Preprint, 44 pages.
  28. Noncommutative residue and new invariants for CR and contact manifolds (pdf). Traces in Geometry, Number Theory and Quantum Fields (Max Planck Institute, Bonn, Germany, Oct. 24-28, 2005), pp. 179–186. S. Albeverio, M. Marcolli and S. Paycha eds., Aspects of Mathematics, Vieweg Verlag, 2008.
  29. Noncommutative geometry and lower dimensional volumes in Riemannian geometry. (arXiv) (doi). Lett. Math. Phys. 83 (2008), 19–32.
  30. Noncommutative residue invariants for CR and contact manifolds (arXiv) (doi). J. Reine Angew. Math. (Crelle’s Journal) 614 (2008), 117–151.
  31. Heisenberg calculus and spectral theory of hypoelliptic operators on Heisenberg manifolds (arXiv) (AMS Bookstore) Mem. Amer. Math. Soc. 194 (2008) no. 906, 140 pages.
  32. Noncommutative residue for Heisenberg manifolds and applications in CR and contact geometry (arXiv) (doi). J. Funct. Anal. 252 (2007), 399–463.
  33. A new short proof of the Atiyah-Singer index formula (doi). Noncommutative Geometry and Number Theory (Max Planck Institute, Bonn, Aug. 17-22, 2003), 361–372, Aspects of Mathematics, Vieweg Verlag, 2006.
  34. Spectral asymmetry, zeta functions and the noncommutative residue (arXiv). (doi). Int. J. Math. 17 (2006), 1065–1090.
  35. The tangent groupoid of a Heisenberg manifold (arXiv) (doi). Pacific J. Math. 227 (2006), 151–175.
  36. Comments on: “Operator K-theory for the group SU(n,1)” by P. Julg and G. Kasparov (arXiv). To be included in another paper, 7 pages.
  37. A new proof of the local regularity of the eta invariant of a Dirac operator (arXiv) (doi). J. Geom. Phys. 56 (2006), 1654–1665.
  38. Analogues of the holomorphic Morse inequalities in CR geometry (pdf). Proceedings of the Hayama symposium in several complex variables, Hayama, Japan, Dec. 18-21, 2005, T. Oshawa et al. eds., 6 pages.
  39. Szegö projections and new invariants for CR and contact manifolds (pdf). Proceedings of the workshop “Analytic geometry of the Bergman kernel and related topics”, RIMS, Kyoto University, Japan, Dec. 12-16, 2005, T. Oshawa et al. eds., 14 pages.
  40. On the asymptotic completeness of the Volterra calculus (Appendix by Hayk Mikayelyan and myself) (arXiv) (doi). J. Anal. Math. 94 (2004), 249–263.
  41. A new short proof of the local index formula and some of its applications (arXiv) (doi). Comm. Math. Phys. 241 (2003), 215–234.
  42. Géométrie spectrale et formules d’indices locales pour les variétés CR et contact (in French) (pdf) (doi). C. R. Acad. Sci. Paris, sér. I, 332 (2001), 735–738.
  43. Calcul fonctionnel sous elliptique et résidu non commutatif pour les variétés de Heisenberg (in French) (pdf) (doi). C. R. Acad. Sci. Paris, sér. I, 332 (2001), 611–614.

Theses

  • Calcul hypoelliptique sur les variétés de Heisenberg, résidu non commutatif et géométrie pseudo-hermitienne (in English) (pdf). PhD thesis, University of Paris-Sud (Orsay, France), December 2000 (Advisor: Alain Connes).
  • Résidu de Wodzicki et géométrie non commutative (in French) (pdf). Mémoire de magistère (Master’s thesis), Ecole normale supérieure, Paris, Dec. 96.
  • Trace de Dixmier (in French) (pdf). Mémoire de DEA (Master’s thesis), University of Paris-Sud, Orsay, November 1995.
  • Formule de Feynman-Kac (in French) . Senior thesis, Ecole normale supérieure, Paris, September 1994.