# "Love and Math" vs "Flash Boys"

"Love and Math" is by Edward Frenkel. It tries to show the beauty of modern mathematics by taking the reader on an autobiographical tour of Frenkel's career, explaining the concepts and connections he's worked with over the years. Mathematical beauty, he says, ought to be accessible to non-mathematicians in the same way that beautiful paintings are accessible to non-painters.

"Flash Boys" is by Michael Lewis. It tries to show the workings of modern stock markets by taking the reader on a biographical tour of Brad Katsuyama's career and the founding of IEX. Wall Street, he says, is screwing the little guy (and big institutional investors) by delivering their trades into the hands of high-frequency traders.

Frenkel doesn't succeed in making the math simple. I had some undergraduate courses in "modern algebra" (not really modern at all), including Galois theory, and have maintained an amateur interest since then. I found myself flailing about halfway through the book, and never really developed a good sense what the Langlands Program actually was. The analogy he uses is a Rosetta stone tying together number theory, curves over finite fields (which he does a good job explaining), and Reimann surfaces (ditto.) Quantum theory comes in at the end too! But few of the conjectures that tie the different languages together are stated explicitly. Much of the mathematical meat is buried in the endnotes. This makes it harder for a somewhat-mathematically-sophisticated reader like me to follow the flow, but I can't see how somebody with less background than I comprehends the ideas and examples at all. There are a lot of concepts floating around that make me echo Enrico Fermi: "If I could remember the names of all these particles, I'd be a botanist."

Here's an example: I understand finite fields, and curves over finite fields. I kind of understand a manifold, and his glossary provides the definition "A smooth geoemetric shape such as a circle, a sphere, or the surface of a donut." He does a good job developing these ideas. Then later we get "manifold over a finite field" and I'm lost--- finite fields aren't smooth! Google or Wikipedia are no help, I just get dumped into reading about Calabi-Yau manifolds, whatever the heck they are.

That said, I applaud Frenkel for the attempt. I really appreciate having a book about what modern mathematicians are really doing. The sections on being a Jewish mathematician in the USSR are alone worth the price of admission. Antisemitism kept him out of Moscow University (and would have placed further roadblocks on graduate study) but the mathematicians of Moscow created informal networks that trained an entire generation of brilliant students.

Frenkel is also unabashedly Platonist which causes some hair-tearing. Perhaps pure mathematicians should be introduced to the concept of "confirmation bias."

Lewis's book succeeds where Frenkel does not. It not only tells a good story, but it also lays out very clearly how high-frequency trading works and affects institutional investors. This picture may not be accurate or complete--- but it's at least part of the story.

In brief, today's stock market is not a singular location. There are numerous exchanges each with their own rules, practices, and costs. When a broker gets a buy order he typically has to fill it from multiple exchanges (in fact a regulation requires him to go find the best prices!) But a high-frequency trader can monitor his activity on a subset of exchanges (or even one) and beat the broker to other exchanges, buy the available stock and re-sell it immediately at a slightly higher price. This is how high-frequency traders account for 50% of the shares traded--- they're not actually providing any "liquidity", they're just stepping in front of party-to-party sales that would otherwise happen anyway.

There are, of course, numerous criticisms of Lewis's account. Some claim this dynamic was already well known and published in books at the time his protagonist was puzzling it out. Or that the same was true before electronic exchanges. (This is not a positive.) Or that sophisticated investors were already deploying countermeasures before IEX, and IEX is a scam of some sort under the thumb of known insider traders. Or that Lewis is getting only the "buy-sider" picture of the world and HFT provides incalculable benefits in ways that only people who really know the stock exchanges can understand. That the whole book is nothing but an advertisement for IEX, who are less savvy and ethical than actually portrayed.

Some of these criticisms I think miss the mark. It's not important who was first to understand HFT or develop countermeasures; it's important to explain it in a clear and engaging fashion. Frankel likes to think that his mathematics is clear, elegant, and beautiful to an outsider--- but ends up, frankly, in a morass of jargon. Financial insiders like to cloud their basic operation in a cloud of strange terms and vague feel-good explanations--- but where their money comes from can usually be explained very simply.

The silliest criticism I hear is that individual investors ("the little guy") shouldn't care because their trades are so small that HFT can't exploit them. This is an out-and-out lie in two forms. The first is that some individual trades are certainly big enough to suffer. One of the telling anecdotes in the book is a trader with Bloomberg access performing a late-night trade on his personal account, and setting off a flurry of activity which he can observe. The second is that individual investors often put their money into mutual funds which are--- guess what--- big institutional investors who are affected.

Michael Lewis told Salon that IEX has been flooded with resumes and would-be whistleblowers since the publication of his book. Sadly, I think Frenkel's book is not likely to have a similar impact.
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