Steve Trettel

Abstract
Created over 150 years ago, the periodic table of elements reveals many interesting similarities, and periodic behaviors in the elementary constituents of nature. Elements with similar chemical properties are placed on top of one another into columns, and the number of columns populated increases as one moves down the table. But inside this organizational tool one can find many mathematical patterns, which historically provided welcome hints to the nature of quantum theory itself.
One such pattern is simply noting that the number of elements in any given row of the table is always twice a perfect square. Why is that? Amazingly, this feature of nature is forced upon us by symmetry and mathematics, and is relatively insensitive to the precise form of quantum theory.
In this talk, we use the periodic table as a light house, illuminating our way on a journey through infinte dimensional linear algebra, symmetries, eigenvalues, eigenfunctions, and separable PDEs, giving students a taste of all the mathematics that now lays in front of them, after mastering Linear Algebra.
Occurrences
This talk was written as the final lecture to my undergraduate Linear Algebra course, to try and do something interesting among the chaos of moving online for the pandemic in the final days of the quarter. Here’s the recording I made: I will replace this next time I give a version of this talk that is recorded, as this particular recording is certainly tailored to our specific class:
Slides
Here are the slides from this lecture!