First band of Ruelle resonances for contact Anosov flows in dimension 3 (with Colin Guillarmou)
accepted in Commun. Math. Phys. (2021).

PDF 
arXiv 
We show, using semiclassical measures and unstable derivatives, that a smooth vector field X generating a contact Anosov flow on a 3dimensional manifold M has only finitely many Ruelle resonances in the vertical strips {s∈ℂ  Re(s)∈[−ν_min+ϵ,−1/2ν_max−ϵ]∪[−1/2ν_min+ϵ,0]} for all ϵ>0, where 0<ν_min≤ν_max are the minimal and maximal expansion rates of the flow (the first strip only makes sense if ν_min>ν_max/2). We also show polynomial bounds in s for the resolvent (−X−s)^{−1} as Im(s)→∞ in Sobolev spaces, and obtain similar results for cases with a potential. This gives a short microlocal proof of a particular case of the results announced by FaureTsujii in \cite{FaTs1}, using that dim E_u = dim E_s = 1.


PDF 
arXiv 
We show that for a generic conformal metric perturbation of a hyperbolic 3
manifold $\Sigma$, the order of vanishing of the Ruelle zeta function at zero equals $4 − b_1(\Sigma)$,
contrary to the hyperbolic case where it is equal to $4 − 2b_1(\Sigma)$. The result is proved by
developing a suitable perturbation theory that exploits the natural pairing between resonant
and coresonant differential forms. To obtain a metric conformal perturbation we need to
establish the nonvanishing of the pushforward of a certain product of resonant and coresonant states and we achieve this by a suitable regularisation argument. Along the way
we describe geometrically all resonant differential forms (at zero) for a closed hyperbolic
$3$manifold and study the semisimplicity of the Lie derivative.

Generic dynamical properties of connections on vector bundles (with Thibault Lefeuvre)
accepted in IMRN (2021).

PDF 
arXiv 
Given a smooth Hermitian vector bundle $\mathcal{E}$ over a closed Riemannian manifold $(M,g)$, we study generic properties of unitary connections $\nabla^{\mathcal{E}}$ on the vector bundle \mathcal{E}. First of all, we show that twisted Conformal Killing Tensors (CKTs) are generically trivial when dim(M) ≥ 3, answering an open question of GuillarmouPaternainSaloUhlmann. In negative curvature, it is known that the existence of twisted CKTs is the only obstruction to solving exactly the twisted cohomological equations which may appear in various geometric problems such as the study of transparent connections. The main result of this paper says that these equations can be generically solved. As a byproduct, we also obtain that the induced connection $\nabla^{\mathrm{End} \mathcal{E}}$ on the endomorphism bundle $\mathrm{End}\mathcal{E}$ has generically trivial CKTs as long as $(M,g)$ has no nontrivial CKTs on its trivial line bundle. Eventually, we show that, under the additional assumption that $(M,g)$ is Anosov (i.e. the geodesic flow is Anosov on the unit tangent bundle), the connections are generically opaque, namely there are no nontrivial subbundles of \mathcal{E} which are generically preserved by parallel transport along geodesics. The proofs rely on the introduction of a new microlocal property for (pseudo)differential operators called operators of uniform divergence type, and on perturbative arguments from spectral theory (especially on the theory of PollicottRuelle resonances in the Anosov case).

Resonant spaces for volume preserving Anosov flows (with Gabriel P. Paternain)
Pure and Applied Analysis 24 (2020), 795840.

PDF 
arXiv 
We consider Anosov flows on closed 3manifolds preserving a volume form $\Omega$.
Following \cite{DyZw17} we study spaces of invariant distributions with values in the bundle of exterior forms whose wavefront set is contained in the dual of the unstable bundle. Our first result computes the dimension of these spaces in terms of the first Betti number of the manifold, the cohomology class $[\iota_{X}\Omega]$ (where $X$ is the infinitesimal generator of the flow) and the helicity. These dimensions coincide with the PollicottRuelle
resonance multiplicities under the assumption of {\it semisimplicity}. We prove various results regarding semisimplicity on 1forms, including an example showing that it may fail for time changes of hyperbolic geodesic flows.
We also study non nullhomologous deformations of contact Anosov flows and we show that there is always
a splitting PollicottRuelle resonance on 1forms and that semisimplicity persists in this instance.
These results have consequences for the order of vanishing at zero of the Ruelle zeta function. Finally our analysis also incorporates a flat unitary twist in both, the resonant spaces and the Ruelle zeta function.

Polyhedral billiards, eigenfunction concentration and almost periodic control (with Bogdan Georgiev and Mayukh Mukherjee)
Commun. Math. Phys. 377 (2020), 24512487.

PDF 
arXiv 
We study dynamical properties of the billiard flow on convex polyhedra away from a neighbourhood of the nonsmooth part of the boundary, called "pockets". We prove there are only finitely many periodic immersed tubes missing the pockets and moreover establish a new quantitative estimate for lengths of such periodic tubes. This extends wellknown results in dimension 2. We then apply these dynamical results to prove a quantitative Laplace eigenfunction mass concentration near the pockets of convex polyhedral billiards. As a technical tool for proving our concentration results on irrational polyhedra, we establish a controltheoretic estimate on a product space with an almostperiodic boundary condition. This extends previously known control estimates for periodic boundary conditions, and seems to be of independent interest.

The Calderón problem for the fractional Schrödinger equation with drift (with YiHsuan Lin and Angkana Rüland)
Calc. Var. Partial Differential Equations 59 (2020), no. 3, Paper No. 91, 46 pp.

PDF 
arXiv 
We investigate the Calderón problem for the fractional Schrödinger equation with drift, proving that the unknown drift and potential in a bounded domain can be determined simultaneously and uniquely by an infinite number of exterior measurements. In particular, in contrast to its local analogue, this nonlocal problem does \emph{not} enjoy a gauge invariance. The uniqueness result is complemented by an associated logarithmic stability estimate under suitable apriori assumptions. Also uniqueness under finitely many measurements is discussed. The inverse problem is formulated as a partial data type nonlocal problem; it is considered in any dimensions n≥1.

Harmonic determinants and unique continuation
preprint (2018), arXiv:1803.09182.

PDF 
arXiv 
We give partial answers to the following question: if F is an m by m matrix on R^n satisfying a second order linear elliptic equation, does det F satisfy the strong unique
continuation property? We give counterexamples in the case when the operator is a general
nondiagonal operator and also for some diagonal operators. Positive results are obtained
when n = 1 and any m, when n = 2 for the LaplaceBeltrami operator and also twisted with
a YangMills connection. Reductions to special cases when n = 2 are obtained. The last
section considers an application to the Calderón problem in 2D based on recent techniques.

Calderón problem for YangMills connections
J. Spectr. Theory 10 (2020), 463513.

PDF 
arXiv 
We consider the problem of identifying a unitary YangMills connection ∇ on a Hermitian vector bundle from the DirichlettoNeumann (DN) map of the connection Laplacian ∇∗∇ over compact Riemannian manifolds with boundary. We establish uniqueness of the connection up to a gauge equivalence in the case of trivial line bundles in the smooth category and for the higher rank case in the analytic category, by using geometric analysis methods and essentially only one measurement.
Moreover, by using a Rungetype approximation argument along curves to re cover holonomy, we are able to uniquely determine both the bundle structure and the connection, but at the cost of having more measurements. Also, we prove that the DN map is an elliptic pseudodifferential operator of order one on the restriction of the vector bundle to the boundary, whose full symbol determines the complete Taylor series of an arbitrary connection, metric and an associated potential at the boundary.

Calderón problem for connections
Comm. Partial Differential Equations 42 (2017), no. 11, 17811836.

PDF 
arXiv 
In this paper we consider the problem of identifying a connection ∇ on a vector
bundle up to gauge equivalence from the DirichlettoNeumann map of the connection
Laplacian ∇∗∇ over conformally transversally anisotropic (CTA) manifolds.
This was proved in [ 1] for line bundles in the case of the transversal manifold
being simple – we generalise this result to the case where the transversal manifold
only has an injective ray transform. Moreover, the construction of suitable
Gaussian beam solutions on vector bundles is given for the case of the connection
Laplacian and a potential, following the works of [ 2]. Consequently, this enables us
to construct the Complex Geometrical Optics (CGO) solutions and prove our main
uniqueness result. Finally, we prove the recovery of a flat connection in general from
the DN map, up to gauge equivalence, using an argument relating the Cauchy data
of the connection Laplacian and the holonomy.
