Schwartz functions are classically defined on Rn as C∞-smooth functions such that they, and all their partial derivatives, decay at infinity even when being multiplied by any polynomial (think of a Gaussian on the real line). The space of Schwartz functions is a Fr´echet space, and its continuous dual is called the space of tempered distributions.

In this talk I will first explain how one can attach a Schwartz space (a space of Schwartz functions) to an arbitrary open subset of Rn, and mainly focus on the question under what conditions two open subsets of Rn have isomorphic Schwartz spaces? Such sets are said to be Schwartz equivalent.

We will see that in the polynomially bounded o-minimal definable case things behave very well, and briefly mention how in this case we can construct the entire theory on general smooth manifolds as well (not just on open subsets of Rn). In the second part of the talk (that is based on a joint work with Ededn Prywes) we will explore this question in the quasiconformal setting,and might find clues for “almost definable behaviour” there as well. No advance prior knowledge in analysis is assumed, and in particular I hope you will be able to follow everything even if you never heard of Schwartz functions, Fr´echet spaces and tempered distributions, and have no idea what is quasiconformal geometry