Research
I am broadly interested in global analysis. My research involves spectral theory and microlocal/semiclassical analysis in geometric contexts such as gauge theory, spin geometry (Dirac operators), CR/complex geometry, contact and sub-Riemannian geometry.
Articles
Geometric quantization results for semi-positive line bundles on a Riemann surface.
with
G. Marinescu
In earlier work the authors proved the Bergman kernel expansion for semipositive line bundles over a Riemann surface whose curvature vanishes to atmost finite order at each point. Here we explore the related results and consequences of the expansion in the semipositive case including: Tian's approximation theorem for induced Fubini-Study metrics, leading order asymptotics and composition for Toeplitz operators, asymptotics of zeroes for random sections and the asymptotics of holomorphic torsion.
J. Geom. Anal. 34 (2024), no. 5, Paper No. 138, 41 pp.
arxiv
Kähler-Einstein Bergman metrics on pseudoconvex domains of dimension two
with
M. Xiao
We prove that a two dimensional pseudoconvex domain of finite type with a Kähler-Einstein Bergman metric is biholomorphic to the unit ball. This answers an old question of Yau for such domains. The proof relies on asymptotics of derivatives of the Bergman kernel along critically tangent paths approaching the boundary, where the order of tangency equals the type of the boundary point being approached.
Duke Math. J. (to appear)
arxiv
Quantitative version of Weyl's law.
We prove a general estimate for the Weyl remainder of an elliptic, semiclassical pseudodifferential operator in terms of volumes of recurrence sets for the Hamilton flow of its principal symbol. This quantifies earlier results of Volovoy (Comm Partial Differential Equations 15:1509–1563, 1990; Ann Global Anal Geom 8:127–136, 1990). Our result particularly improves Weyl remainder exponents for compact Lie groups and surfaces of revolution. And gives a quantitative estimate for Bérard’s Weyl remainder in terms of the maximal expansion rate and topological entropy of the geodesic flow.
Ann. Global Anal. Geom. 64 (2023), no. 2, Paper No. 14, 24 pp.
arxiv
Bergman-Szegő kernel asymptotics in weakly pseudoconvex finite type cases
with
C-Y. Hsiao
We construct a pointwise Boutet de Monvel-Sjöstrand parametrix for the Szegő kernel of a weakly pseudoconvex three dimensional CR manifold of finite type assuming the range of its tangential CR operator to be closed; thereby extending the earlier analysis of Christ. This particularly extends Fefferman's boundary asymptotics of the Bergman kernel to weakly pseudoconvex domains in \mathbb{C}^{2}, in agreement with D'Angelo's example. Finally our results generalize a three dimensional CR embedding theorem of Lempert.
J. Reine Angew. Math. 791 (2022), 173–223.
arxiv
Hyperbolicity, irrationality exponents and the eta invariant
We consider the remainder term in the semiclassical limit formula for the eta invariant on a metric contact manifold, proving in general that it is controlled by volumes of recurrence sets of the Reeb flow. This particularly gives a logarithmic improvement of the remainder for Anosov Reeb flows, while for certain elliptic flows the improvement is in terms of irrationality measures of corresponding Floquet exponents.
Comm. Partial Differential Equations 47 (2022), no. 5, 989–1023.
arxiv
Spectrum and abnormals in sub-Riemannian geometry: the 4D quasi-contact case
We prove several relations between spectrum and dynamics including wave trace expansion, sharp/improved Weyl laws, propagation of singularities and quantum ergodicity for the sub-Riemannian (sR) Laplacian in the four dimensional quasi-contact case. A key role in all results is played by the presence of abnormal geodesics and represents the first such appearance of these in sub-Riemannian spectral geometry.
arxiv
Sub-leading asymptotics of ECH capacities
with
Daniel Cristofaro-Gardiner
In previous work, the first author and collaborators showed that the leading asymptotics of the embedded contact homology (ECH) spectrum recovers the contact volume. Our main theorem here is a new bound on the sub-leading asymptotics.
Selecta Math. (N.S.), 26 (2020), no. 65.
arxiv
Bochner Laplacian and Bergman kernel expansion of semi-positive line bundles on a Riemann surface
with
G. Marinescu
We generalize the results of Montgomery for the Bochner Laplacian on high tensor powers of a line bundle. When specialized to Riemann surfaces, this leads to the Bergman kernel expansion and geometric quantization results for semi-positive line bundles whose curvature vanishes at finite order. The proof exploits the relation of the Bochner Laplacian on tensor powers with the sub-Riemannian (sR) Laplacian.
Math. Ann. 389 (2024), no. 4, 4083–4124.
arxiv
A Gutzwiller type trace formula for the magnetic Dirac operator
For manifolds including metric-contact manifolds with non-resonant Reeb ow, we prove a Gutzwiller type trace formula for the associated magnetic Dirac operator involving contributions from Reeb orbits on the base. As an application, we prove a semiclassical limit formula for the eta invariant.
Geom. Funct. Anal., 28 (2018), pp. 1420-1486.
arxiv
Koszul complexes, Birkhoff normal form and the magnetic Dirac operator
We consider the semi-classical Dirac operator coupled to a magnetic potential on a large class of manifolds including all metric contact manifolds. We prove a sharp local Weyl law and a bound on its eta invariant. In the absence of a Fourier integral parametrix, the method relies on the use of almost analytic continuations combined with the Birkhoff normal form and local index theory.
Anal. PDE, 10 (2017), pp. 1793-1844.
arxiv
The Gauss-Bonnet-Chern theorem: a probabilistic perspective
with
L. Nicolaescu
We prove that the Euler form of a metric connection on real oriented vector bundle E over a compact oriented manifold M can be identified, as a current, with the expectation of the random current defined by the zero-locus of a certain random section of the bundle. We also explain how to reconstruct probabilistically the metric and the connection on E from the statistics of random sections of E.
Trans. Amer. Math. Soc. 369 (2017), no. 4, 2951–2986.
arxiv
Asymptotics of the eta invariant
We prove an asymptotic bound on the eta invariant of a family of coupled Dirac operators on an odd dimensional manifold. In the case when the manifold is the unit circle bundle of a positive line bundle over a complex manifold, we obtain precise formulas for the eta invariant.
Comm. Math. Phys., 332 (2014),
pp. 847-884
arxiv
Spectral Asymptotics for Coupled Dirac Operators
This is my thesis, where I study the problem of asymptotic spectral flow for a family of coupled Dirac operators. I exhibit a leading term followed by a non-sharp remainder. The thesis gives background material and performs certain computations for spectral flow not contained in the above articles.
MIT DSpace (2012)