The Aleph Zero Categorical: There can only be one blog is written by Canadian mathematician Dr. Jason Polak. The blog started back in 2011, when Polak began his Ph.D. as a way to “showcase abstraction and its beauty in the realm of pure mathematics, especially in algebra.”
Its tagline was inspired by the show “Highlander” and relates to the blog’s title since “there can be only one countable aleph zero categorical model up to isomorphism.” His research interests began in ring theory, module theory, p-adic groups, automorphic representations, logic, and combinatorics and recently have shifted towards ecology and conservation. It was hard to pick which post to write about! I love that the blog features a content tab that lets you see all the posts in alphabetical order and those whose topics involve multiple posts.
For this tour, I will summarize some of the most recent posts and hopefully give you a glimpse of the blog’s style and content. What has been most amazing to me is the combination between older and more recent research interests. I definitely got pulled in by a post about ecology and ended up reading more about spectral sequences. You
can also see some fantastic bird pictures on Bad Birding, a joint blog he created with Emily Polak.
In this post, Polak discusses some of the ways people can misuse science (i.e. using science to harm society, individuals, or the environment) and our responsibility to recognize and address them. He provides different examples like to support science out of complacency, science to support ideologies and beliefs, and science that harms living organisms.
“Anyone who is trained in science has the knowledge and ability to recognize many of these issues. Such knowledge entails a responsibility to make decisions based on our understanding. These decisions, both personal and professional, do not have to be predicated on what is right, since we may not know what that is. But they do have to be made with the desire to understand and discover the truth even if that truth is uncomfortable, and connect with others in order to share our limited understanding in the hopes of creating a better environment for all living organisms on this planet.”
As a math ecologist, I was really eager to read this post which discusses how we measure biodiversity. Is it the number of species that count? If so, how do we decide how to count them? Should we consider other information such as order, family, and genus level classifications? For example, using the number of species in two lakes does not take into account relative differences among the species. He shares two great pictures of an Australian White Ibis (Threskiornis molucca) and Straw-necked Ibis (Threskiornis spinicollis) observed by the author in Darwin’s George Brown Botanical Garden. Both appear very similar even though they are different species from the same family and genera. An approach, for example, to compare the diversity in two lakes, is making a table that lists the last sighting of birds species into genii, families, and orders. While there are many ways to summarize this table, one of them called the Shannon entropy, measures how much information is stored in the probability distribution for each lake. As he explains,
“The higher the entropy, the more evenly the vector is distributed. The maximum entropy is obtained when the distribution is uniform in which case the Shannon entropy of that vector is log(n). The idea is that the more evenly distributed the probability distribution is, the more diverse the area with respect to the subset of organisms you are studying.”
Polak concludes that a way to improve biodiversity measures one needs better data which he hopes to analyze in more detail in futures posts.
Wild Spectral Sequences Series
In these multi-post expositions of proofs that use spectral sequences, Polak illustrates that these sequences are ‘safe’ and in fact, can be used in a variety of examples. The post assumes the reader is familiar with spectral sequences so I dived to find a ‘big picture’ idea of what these are. In the notes, “Spectral Sequences: Friend or Foe?“, Ravi Vakil describes spectral sequences as “a powerful book-keeping tool for proving things involving complicated commutative diagrams.” Through six posts/episodes, Ep.1 Snake, Ep.2 Five, Isomorphism!, Ep.3 Cohomological Dimension, Ep.4 Schanuel’s Lemma, Ep.5 Lyndon-Hochschild-Serre, and Ep.6 The 3×3 Lemma he gives examples of spectral sequences being used in ‘toy examples’. While the concept of spectral sequences was new to me, I appreciated seeing the main ideas behind how these sequences allow us to “clean-up” messy proofs. As he concludes in Ep.4,
“Notice that once we get used to spectral sequences, they can help remove a lot of the clutter that comes with ridiculous proofs that contain sentences of the form ‘let $x\in X$, then $f(x)\in Y$ is in the kernel of…’, which are exceptionally hard to read.”
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