I am going to talk on a work aimed at formalising approximation procedures in physics.  The main new model-theoretic tool in this work is the notion of the ultraproduct in the classes of emerging metric structures which generalises the ultraproduct  of general structures developed by J.Kiesler. In particular, the structure of Minkowski spacetime with the action of the Lorentz group is an emerging metric ultraproduct of certain finite structures invariant under the action of appropriate finite groups. Also, it is shown that any compact simple Lie group is representable as emerging metric ultraproduct of finite groups.
 

Choosing where to stop an iteration or how far to increase a model complexity parameter is a recurring problem in numerical computation and data analysis. Typical symptoms are diminishing returns, a noise-dominated floor, and overfitting---accordingly, many heuristics seek an elbow or plateau beyond which further effort is not worthwhile.  AAA rational approximation provides a sharp instance of this difficulty when constructing rational approximations from noisy data, where the error often decreases rapidly at first and then fluctuates in a noisy band.  Standard AAA has no mechanism to recognize this regime and may continue iterating until a preset degree cap is reached. We thus propose noiseChop, a noise-aware stopping rule designed to run online alongside AAA. The method is inspired by Chebfun's standardChop but is tailored to AAA by using quantities already available during the iteration---a monotone envelope of the $\infty$-norm nonlinear error and the linearized error from the Loewner least squares step.  
The method first detects evidence of stagnation and then selects an early cutoff degree that achieves good accuracy without chasing noise. Numerical tests illustrate robust behavior across several functions, sample sets, and noise levels. The method is soon to be available as an optional feature in Chebfun's AAA code.