One of the major targets for next-generation cosmic microwave background (CMB) experiments is the precision mapping of CMB distortions due to the gravitational lensing effect of dark matter. Estimating this lensing is important for two reasons. First, lensing probes the nature of dark matter fluctuations in the sky. Second, lensing estimates can be used, in principle, to delense the observed CMB for reconstructing the primordial CMB fluctuations. Both of these problems are crucial for next-generation CMB experiments.

In the first part of the talk I will discuss the lensing estimation problem. This will center around an analysis of the current state-of-the-art tool, the quadratic estimate, developed by Hu and Okamoto more than a decade ago. This estimate is remarkable in many ways and has surprisingly attractive sampling properties. We will argue that these properties arise from distributional symmetries which occur when perturbative methods are applied to realizations of stationary random fields. This observation can be used to predict when similar estimates, derived for more general forms of non-stationarity, will share the same attractive sampling properties as the quadratic estimate.

If time permits, we will discuss ongoing work for the de-lensing problem. In many ways, the de-lensing problem is more difficult given the extra sensitivity required to address the relevant scientific questions for the primordial CMB (in particular, detecting the primordial B mode fluctuations). We will discuss likelihood approaches that take advantage of a new dynamical system characterization of lensing for pixelized maps. Unlike the typical Taylor series expansion for lensing, our new characterization can perform exact inverse lensing at no additional cost and represents a linear operator with determinant equal to exactly one, both of which are crucial for our purposes.