The n-Category Café

• Lecture 1: Measuring diversity: the axiomatic approach

  Ecologists have been debating the best way to measure diversity for more than 50 years. The concept of diversity is relevant not only in ecology, but also in other fields such as genetics and economics, as well as being closely related to entropy. The question of how best to quantify diversity has surprising mathematical depth. I will argue that the best approach is axiomatic: to enable us to reason logically about diversity, the measures we use must satisfy certain mathematical conditions, and those conditions dramatically limit the choice of measures. This point will be illustrated with a theorem: using a simple model of ecosystems, the only diversity measures that behave logically are the Hill numbers, which are very closely related to the Rényi entropies of information theory.

  Friday 21 October, 16:00 Japan, 08:00 UK, 09:00 continental Europe

• Lecture 2: Measuring diversity: species similarity

  Traditional measures of the diversity of an ecological community depend only on how abundant the species are, not the similarities or differences between them. To better reflect biological reality, species similarity should be incorporated. Mathematically, this corresponds to moving from probability distributions on sets to probability distributions on metric spaces. I will explain how to do this and how it can change ecological judgements. Finally, I will describe a surprising theorem on maximum diversity (joint with Meckes and Roff), which reveals close connections between maximum diversity and invariants of geometric measure.

  Friday 28 October, 16:00 Japan, 08:00 UK, 09:00 continental Europe

Most of what I’ll say will be from my book Entropy and Diversity: The Axiomatic Approach, although the work with Emily Roff — about maximum diversity on a compact space — is recent enough that it barely made it into the book.