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

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.

(submitted)

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.

(submitted)

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.

(submitted)

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.

(submitted)

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)