An undergraduate seminar during the fall of 2021.
How might one go about designing a useful measure of diversity for, say, an ecological system? One approach is to begin with a set of desiderata, a list of properties that any good measure of diversity ought to satisfy, and go about constructing all measures that satisfy these properties. With enough properties, it may be the case that the measure in question is essentially uniquely determined.
This is the approach of our text, in which we will discover that any measure of diversity satisfying a set of reasonable assumptions coming from ecological principles is amongst a family of well-known diversity measures, the Hill numbers. This in turn is a simple transformation of another well-known measure of variety, entropy. As such, I hope our journey through this book will give many different perspectives on the notion of entropy, and related concepts.
Each participant will give 2 talks over the course of the semester, during which I hope one enjoys some interesting mathematics, as well as meditates on the art of presenting and talking about mathematics. During the rest of the semester, I hope that those that are not presenting will help form a lively and friendly seminar environment, one in which all participants can feel comfortable to discuss and learn in.
The primary reference for this seminar is the book
Some related papers that might be fun to look at include:
Tai-Danae Bradley (2021), Entropy as a Topological Operad Derivation.
David Spivak and Timothy Hosgood (2021), Dirichlet polynomials and entropy.
We meet Monday and Wednesdays from 10 AM to 11 AM in Mathematics Room 622.