Bird's-eye views of Structure and Randomness (Series)

The first posts that I read from Fields Medalist Terence Tao's research blog "What's new" were pieces of advice to aspiring mathematicians, such as mathematical writing tips or what it takes to do math. His blog helped me decide to create my own blog to talk about my own math research, among other things. But his technical posts put me off reading his blog until I recently read a compilation of some of his posts in his book Structure and Randomness (Tao, 2008) (See cover at right).

His expository articles on math and science were surprisingly nontechnical, as well as fun and enriching to read. They conveyed the big picture of the topics in question, imparting an intuition and wonder of the way that math and mathematicians work. I could understand only a few articles, but even so I decided to share my joy with other nontechnical readers like myself. I have extracted what I could understand, expanded on it and adapted it to minimize prerequisite math knowledge, into the following six-part article series:

Expository articles, lectures and open problems.

1. Tomb raider: an analogy for quantum weirdness (adapted from Quantum mechanics and Tomb Raider)
2. Image compression basics (adapted from Compressed sensing and single-pixel cameras)
3. Strengthening inequalities: a mathematical trick (adapted from Amplification, arbitrage, and the tensor power trick)
4. Drawing networks on a plane (adapted from The crossing number inequality)
5. Analogies between finitary and infinitary math (adapted from Soft analysis, hard analysis and the finite convergence principle)
6. Infinities as numbers: purging the epsilons and deltas from proofs (adapted from Ultrafilters, nonstandard analysis, and epsilon management)

(1) and (2) are non-technical discussions of math and science topics; (3), (4), and (6) are demonstrations of mathematical tools and tricks; and (5) is about a piece of "mathematical folklore" (see "Why blog about research?" below).

Tao's writing is concise enough that any further extracting, as I have done, misses the complete picture. He is careful to note down important caveats to the insights he presents. I have ignored those caveats, as worthy sacrifices to pique the interest of less technical readers. However, I hope my adaptations also serve as gateways to attract technical readers towards Tao's well-written and more comprehensive exposition. They'd probably understand more of what he writes than I did!

Besides the many expository articles I didn't understand enough to confidently present, Structure and Randomness included transcripts of some of his lecture series and some of his favourite open problems in math. The target audience of the former is more technical than I am, and the latter is (by definition) at the forefront of mathematical knowledge. However, I hope that able readers check those out as well for their equally illuminating focus on the big picture without the most technical details.

Tao's concise writing and excellent choice of words sometimes forces me to lift directly from him rather than cobble together my own contrived paraphrase. I hope he doesn't mind; this is testament to his writing skills. If you find my adaptations convincing and enjoyable, most of the credit goes to Tao.

Why blog about research?

Tao describes in the preface to Structure and Randomness the advantages of research blogging and its niche between traditional print media (journals, books) and informal communications (lectures, conferences). I will skip the benefits for the blogger himself in favor of those for the readers.

• Research blogs are informal, dynamic, and interactive (discussions in the comments), but with a permanent record, well-defined author, and links to further resources.
• The math community circulates pieces of "mathematical folklore", or common intuitions about a field and their interpretations (e.g. "discrete" vs. "continuous"; also, (5) is folklore about the field of analysis), too fuzzy for formal literature but suitable for blogging.

However, blog posts are not permanent enough to be cited. This pushed Tao to compile some of those posts, with corrections and further ideas from reader comments, into Structure and Randomness.

The phrase "Birds' eye views" was inspired by Freeman Dyson's characterization of mathematicians as "birds" or "frogs" (Dyson, 2009).

References

1. (Tao, 2008) Terence Tao. Structure and Randomness: Pages from Year One of a Mathematical Blog. American Mathematical Society (2008).
2. (Dyson, 2009) Freeman Dyson. Birds and Frogs. Notices of the American Mathematical Society 56, 212-222 (2009).