Here is the original paper on total variation for denoising.

Here is a talk from 2003 where Stan describes and shows images from the attack on the truck driver Reginald Denny during the riots in LA (skip to 11:00 for the story).

Here is the paper on the level set method.

The company Stan cofounded, Luminescent Technologies, Inc, used the level set method for inverse lithography technology.

Here is a paper by Candes, Romberg and Tao on compressed sensing, providing rigorous theory for use of the L1 norm.

An example of "thinking continuously rather than discretely" is the analysis of Su, Boyd, and Candes in providing a short and simple proof for Nesterov acceleration in the continuous setting via a continuous ODE (see Theorem 3 in this ⁠paper⁠).