July 16
Monroe Eskew,
University of Vienna
Integration with filters
In a recent Quanta Magazine article discussing difficulties and progress related to Feynman path integrals, Charlie Wood writes, 'No known mathematical procedure can meaningfully average an infinite number of objects covering an infinite expanse of space in general.' This statement is arguably refuted by Nonstandard Analysis, but what is perhaps lacking is a constructive approach. We present such an approach based on reduced powers and a class of algebraic structures we call comparison rings. This construction has a nice iteration theory and is able to represent classical integrals via standard parts. We discuss an example of a filter on $\mathbb R^{\lt\omega}$, the direct limit of the $\mathbb R^n$, that respects classical volumes in different dimensions simultaneously, with lower dimensional surfaces being infinitesimal relative to higher dimensional ones. This suggests a corresponding generalization of dimension, which we show under some set-theoretic assumptions may constitute a dense linear order without $(c,c)$-gaps. This is joint work with Emanuele Bottazzi.