Books to Read While the Algae Grow in Your Fur, September 2014

Attention conservation notice: I have no taste.

Lauren Beukes, Broken Monsters

In which Detroit, which evidently hasn't suffered enough, must deal with an outbreak from the dungeon dimensions, cleverly disguised as a mere psycho killer loose in its art scene. But also, remarkably, a lot of humanity and sympathy for all the characters, even the ones it would have been easy to make into mere caricatures (like the hipster failed writer), even for the monster. I read it as quickly as life would let me, and wished there were more.

— Further comments outsourced to Steph Cha at the LA Review of Books.

J. F. Traub and A. G. Werschulz, Complexity and Information

A survey of information-based complexity, as it appeared in the late 1980s. This is a branch of computational complexity theory, but it takes continuous real numbers (indeed, continuous function spaces like Hilbert or Banach spaces) as the primitive objects. A typical problem might be calculating the action of a path, i.e., to approximate something like \( A(f) = \int_{0}^{1}{L(f(t), f^{\prime}(t)) dt} \) where the form \( L \) is known and fixed, but \( f \) is allowed to vary over some class of functions \( \mathcal{F} \). What distinguishes information-based complexity from plain numerical analysis is that we are not supposed to have \( f \) in some explicit form, but are merely able to evaluate it at some limited number of points, say \( f(t_1), \ldots f(t_n) \), or more generally evaluate \( n \) functionals of \( f \). It is this set of \( n \) functionals that are meant by "information" in this context; they constitute the information we have on \( f \). One then wants to know how closely \( A(f) \) can be approximated in terms of some measurable, or even linear, function of the \( f(t_i) \). Turned around, one asks how many functionals of what kind are required to get an \( \epsilon \) approximation to \( A(f) \).

Complexity and Information is just an introductory sketch, without any proofs or details, and in many places I wish it had made deeper connections with related subjects (e.g., conventional information theory, or minimax bounds on optimization). But it covers an awful lot in just 100 pages, and made me want to learn more, and know about what's happened in the last 25 years. Also, I imagine that if the book were being written today, the pricing of collateralized mortgage obligations might not receive quite so much attention as a success story.

(The Wikipedia page on IBC is, if not necessarily descended from the text, is at least a very close relative.)

Disclaimer: I have a slight acquaintance with Prof. Traub, professionally and socially, since we're both on the external faculty at the Santa Fe Institute.

Jordan Ellenberg, How Not to Be Wrong: The Power of Mathematical Thinking

It would be unkind to gloat how much of this book about mathematical thinking is specifically about statistics. (After all, it's almost an act of filial piety on the part of the author.) It is however quite fair to point out that there is a deep reason for this: statistics is the branch of mathematical engineering which is concerned with designing reliable ways of drawing inferences from imperfect information. (I realize "applied mathematics" is more usual than "mathematical engineering", but too often "applied math" translates to "solving partial differential equations".) As such it's one of the places which has had to realize that "not being wrong" is a distinct goal in its own right, separate from "being right". Less portentiously, it's one of the places which has had to think very hard about how to avoid fooling yourself. There are however other, quite distinct, branches of mathematics, which have very different purposes. These are, it seems to me, articulations of, first, extrapolating from assumptions ("If I'm right, then...") and sheer delight in solving puzzles. Since Ellenberg's own mathematics is very much of the latter varieties, these get touched on somewhat, but the focus is very much on statistical conc