Let $(\hat X,T^{1,0}\hat X)$ be a compact orientable CR embeddable three dimensional strongly pseudoconvex CR manifold, where $T^{1,0}\hat X$ is a CR structure on $\hat X$. Fix a point $p\in\hat X$ and take a global contact form $\hat\theta$ so that $\hat\theta$ is asymptotically flat near $p$. Then $(\hat X,T^{1,0}\hat X,\hat\theta )$ is a pseudohermitian $3$-manifold. Let $G_p\in C^\infty(\hat X\setminus\set{p})$, $G_p > 0$, with $G_p(x)\sim\vartheta(x,p)^{-2}$ near $p$, where $\vartheta(x,y)$ denotes the natural pseudohermitian distance on $\hat X$. Consider the new pseudohermitian $3$-manifold with a blow-up of contact form $(\hat X\setminus\set{p},T^{1,0}\hat X,G^2_p\hat\theta)$ and let $\Box_{b}$ denote the corresponding Kohn Laplacian on $\hat X\setminus\set{p}$. In this paper, we prove that the weighted Kohn Laplacian $G^2_p\Box_b$ has closed range in $L^2$ with respect to the weighted volume form $G^2_p\hat\theta\wedge d\hat\theta$, and that the associated partial inverse and the Szeg\"{o} projection enjoy some regularity properties near $p$. As an application, we prove the existence of some special functions in the kernel of $\Box_{b}$ that grow at a specific rate at $p$. The existence of such functions provides an important ingredient for the proof of a positive mass theorem in 3-dimensional CR geometry by Cheng-Malchiodi-Yang \cite{CMY}.

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Three dimensional Einstein gravity with negative cosmological constant -1/\ell^2 deformed by a gravitational Chern-Simons action with coefficient 1/\mu is studied in an asymptotically AdS_3 spacetime. It is argued to violate unitary or positivity for generic \mu due to negative-energy massive gravitons. However at the critical value \mu\ell=1, the massive gravitons disappear and BTZ black holes all have mass and angular momentum related by \ell M=J. The corresponding chiral quantum theory of gravity is conjectured to exist and be dual to a purely right-moving boundary CFT with central charges (c_L,c_R)=(0,3\ell /G).

We improve upon the local bound in the depth aspect for sup-norms of newforms on D^×, where D is an indefinite quaternion division algebra over Q. Our sup-norm bound implies a depth-aspect subconvexity bound for L(1/2,f×θ_χ), where f is a (varying) newform on D^× of level p^n, and θ_χ is an (essentially fixed) automorphic form on GL_2 obtained as the theta lift of a Hecke character χ on a quadratic field. For the proof, we augment the amplification method with a novel filtration argument and a recent counting result proved by the second-named author to reduce to showing strong quantitative decay of matrix coefficients of local newvectors along compact subsets, which we establish via p-adic stationary phase analysis. Furthermore, we prove a general upper bound in the level aspect for sup-norms of automorphic forms belonging to any family whose associated matrix coefficients have such a decay property.

The diffusive transport of passive tracers or particles can be enhanced by incompressible, turbulent flow fields. Analyzing the effective behavior is a challenging problem with theoretical and practical importance in many areas of science and engineering, ranging from the transport of mass, heat, and pollutants in geophysical flows to sea ice dynamics and turbulent combustion. The long time, large scale behavior of such systems is equivalent to an enhanced diffusion process with an effective diffusivity tensor D*. Two different formulations of integral representations for D* were developed for the case of time-independent fluid velocity fields, involving spectral measures of bounded self-adjoint operators acting on vector fields and scalar fields, respectively. Here, we extend both of these approaches to the case of space-time periodic velocity fields, allowing for chaotic dynamics, providing rigorous integral representations for D* involving spectral measures of unbounded self-adjoint operators.We prove the different formulations are equivalent. Their correspondence follows from a one-to-one isometry between the underlying Hilbert spaces. We also develop a Fourier method for computing D*, which captures the phenomenon of residual diffusion related to Lagrangian chaos of a model flow. This is reflected in the spectral measure by a concentration of mass near the spectral origin.