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The best way of checking you understand something is to write it down. This is a collection of short notes that are aimed at explaining one small piece of mathematics, mostly geared toward topics in geometric analysis and PDE as that is the field I'm trying to learn as part of my research.
Title: On the asymptotic behavior of the magnitude function for odd-dimensional Euclidean balls.
Abstract: Magnitude is a numerical invariant of metric spaces with origins in the notion of the Euler characteristic of a category. To this day the only convex sets in Euclidean space for which we can compute magnitude are the odd-dimensional Euclidean balls. Recent results have shed light on the asymptotic behavior of the magnitude function for these Euclidean balls, and in particular recent work by Meckes showed that the first order small-$t$ asymptotics of the magnitude function recovers its first intrinsic volume. The aim of this thesis is to survey work done to understand the magnitude function for odd-dimensional Euclidean balls and to compute its second order small-$t$ asymptotics.
Note: I'm posting these mostly for archival purposes. Most of these notes were written while I was a freshman / sophomore at university and therefore I can't attest to their quality. Sadly I don't have the LaTeX sources for these anymore so PDFs they will remain.
All the notes relating to Category Theory were written when I was learning some basic Category Theory with Professor Nick Gurski in the summer of my sophomore year.