Schwartz Functions on Open Subsets of Rn

Schwartz Functions on Open Subsets of Rn

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Ary Shaviv, Princeton University
Fine Hall 214

Please note only Princeton University ID holders will be admitted. All attendees must be masked and sign in upon entry.

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 (which we will call the Schwartz space) was first introduced by Laurent Schwartz in the first half of the 20th century, in an attempt to give a rigorous framework to quantum mechanics and in particular to Dirac’s Delta function. Since then, Schwartz spaces were defined and studied on various objects, and were used in various fields, e.g., Representation Theory, Number Theory, and maybe even Machine Learning.

In this talk I will first briefly explain the historical motivation described above. Then, I will explain how one can attach a Schwartz space to an arbitrary open subset of Rn, and 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 under reasonable model theoretic assumptions (namely, in the polynomially bounded o-minimal definable case - I will explain what that means) 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 Eden Prywes) we will explore this question in the quasiconformal setting.

This talk will start with very little Physics, and develop to a fun blend of Analysis, Model Theory, and Quasiconformal Geometry. Lots of easy examples will be presented, and no prior knowledge in any of these ingredients will be assumed.