Mathematical Conversations

I will first give an outline of Lin-Friis-Rordam’s proof of the fact that almost commuting matrices are close to commuting matrices uniformly in the dimension. The proof is short and beautiful, but it involves an infinite-dimensional argument which makes it virtually impossible to recover a good qualitative distance estimate. I will discuss some "finite-dimensional" approaches to this and related problems, some of which are already known to work, some will most likely work, and some are just dreams.