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
- Connes’ integration and Weyl’s laws (arXiv). To appear in J. Noncomm. Geom., 29 pages.
- Dixmier trace formulas and negative eigenvalues of Schrödinger operators on curved noncommutative tori (with Edward McDonald)
(arXiv)
(doi).
Adv. Math. 412 (2023), 108815 (57 pages).
- Cwikel estimates and negative eigenvalues of Schrödinger operators on noncommutative tori (with Edward McDonald) (arXiv)
(doi). J. Math. Phys. 61 (2022), 043503 (37 pages).
- Weyl’s laws and Connes’ integration formulas for matrix-valued LlogL-Orlicz potentials (arXiv) (doi). Math. Phys. Anal. Geom. 25, 10 (2022), 33 pages.
- Periodicity and cyclic homology. Para-S-modules and perturbation lemmas (arXiv) (doi).
J. Noncomm. Geom. 14 (2020), 1365–1444.
- Noncommutative residue and canonical trace on noncommutative tori. Uniqueness results
(arXiv)
(doi).
SIGMA (Symmetry Integrability Geom. Methods Appl.) 16 (2020), 061, 31 pages.
- Functional calculus for elliptic operators on noncommutative tori, I (with Gihyun Lee) (arXiv)
(doi). J. Pseudo-Differ. Oper. Appl. 11 (2020), 935–1004.
- Connes’s trace theorem for curved noncommutative tori. Application to scalar curvature (arXiv)
(doi).
J. Math. Phys. 61 (2020), 042301 (27 pages).
- Laplace-Beltrami operators on
noncommutative tori (with Hyunsu Ha) (arXiv)(doi). J. Geom. Phys. 150 (2020), 103594 (25 pages).
- 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).
- 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).
- Tangent maps and tangent
groupoid for Carnot manifolds (with Woocheol Choi)
(arXiv)
(doi).
Differential Geom. Appl. 62 (2019), 136–183.
- Privileged coordinates and nilpotent
approximation for Carnot manifolds, II. Carnot coordinates (with Woocheol Choi)
(arXiv) (doi).
J. Dyn. Control Syst. 25 (2019), 631–670.
- Privileged coordinates and nilpotent
approximation for Carnot manifolds, I. General results (with Woocheol Choi)
(arXiv)
(doi).
J. Dyn. Control Syst. 25 (2019), 109–157.
- Cyclic homology and group actions
(arXiv)
(doi). J.
Geom. Phys. 123 (2018), 30–52.
- Noncommutative geometry and conformal
geometry. I. Local index formula and conformal invariants (with Hang
Wang)
(arXiv) (doi).
J. Noncomm. Geom. 12 (2018), 1573–1639.
- The cyclic homology of crossed-product algebras, I
(pdf)
(doi).
C. R. Acad. Sci. Paris, sér. I, 355 (2017), 618–622.
- The cyclic homology of crossed-product algebras, II
(pdf)
(doi).
C. R. Acad. Sci. Paris, sér. I,
355 (2017), 623–627.
- 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.
- 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.
- Noncommutative geometry and conformal
geometry. III. Vafa-Witten inequality and Poincaré duality (with Hang Wang)
(arXiv)
(doi).
Adv. Math. 272 (2015), 761–819.
- The Logarithmic singularities of the Green functions of the conformal powers of the Laplacian
(arXiv).
Contemp. Math. Vol. 630 (2014), pp. 247–273.
- 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.
- 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.
- 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.
- A New hypoelliptic operator on almost CR manifolds
(arXiv)
(doi).
Rev. Mat. Iberoamericana 27 (2011), no. 2, 393–414.
- 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.
- Traces on pseudodifferential operators and sums of commutators
(arXiv)
(doi).
J. Anal. Math. 110 (2010), 1–30.
- Logarithmic singularities of Schwartz
kernels and local invariants of conformal and CR structures
operators
(arXiv). Preprint, 44 pages.
- 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.
- Noncommutative geometry and lower dimensional volumes in Riemannian geometry.
(arXiv)
(doi).
Lett. Math. Phys. 83 (2008), 19–32.
- Noncommutative residue invariants for CR and contact
manifolds
(arXiv)
(doi).
J. Reine Angew. Math. (Crelle’s Journal) 614 (2008), 117–151.
- Heisenberg calculus and spectral theory of
hypoelliptic operators on Heisenberg manifolds
(arXiv)
(AMS Bookstore)
Mem. Amer. Math. Soc. 194 (2008) no. 906, 140 pages.
- Noncommutative residue for Heisenberg
manifolds and applications in CR and contact geometry
(arXiv)
(doi).
J. Funct. Anal. 252 (2007), 399–463.
- 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.
- Spectral asymmetry, zeta functions and the noncommutative residue
(arXiv).
(doi).
Int. J. Math. 17 (2006), 1065–1090.
- The tangent groupoid of a Heisenberg manifold
(arXiv)
(doi).
Pacific J. Math. 227 (2006), 151–175.
- 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.
- A new proof of the local regularity of the
eta invariant of a Dirac operator
(arXiv)
(doi).
J. Geom. Phys. 56 (2006), 1654–1665.
- 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.
- 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.
- On the asymptotic completeness of the
Volterra calculus (Appendix by Hayk Mikayelyan and myself)
(arXiv)
(doi).
J. Anal. Math. 94 (2004), 249–263.
- A new short proof of the local index formula and some of its
applications
(arXiv) (doi).
Comm. Math. Phys. 241 (2003), 215–234.
- 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.
- 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.