Beginning with the work, of Jean Bourgain and later the Ph.D. thesis of Benoit Bossard, researchers have been using notions and methods of descriptive set theory to provide remarkable embedding results for classes of separable Banach spaces.
More recently by considering a natural coding of the class of bounded linear operators between separable Banach spaces, many of these embedding results have
been generalized to factorization results for operators; especially uniform factorization results for subsets of operator ideals.
In this talk, I will explain this framework, outline some recent achievements in the area, and give avenues for future research.
The work presented can be found in several papers written jointly with several coauthors including Ryan Causey, Dan Freeman, Bruno Braga, and Leandro Antunes.