When a top-tier mathematician announced in August that he had proved one of the greatest problems in mathematics, the claim was trumpeted in the New York Times, Nature, Science and the Boston Globe. But that didn’t make the abc conjecture proven. People often think of mathematics as a solitary pursuit, with a written proof as final product. In fact, it’s an unavoidably social activity, even for mathematicians who prefer to work alone. A theorem isn’t proven until the mathematical community is persuaded that it’s proven. And proofs today are often so complex that that persuasion mus...

Fermat’s Last Theorem is so simple to state, but so hard to prove. Though the 350-year-old claim is a straightforward one about integers, the proof that University of Oxford mathematician Andrew Wiles finally created for it nearly two decades ago required almost unimaginably complex theoretical machinery. The proof was a dazzling demonstration of that machinery’s value, but one aspect of it troubled mathematicians: It relied on stronger axioms than mathematics normally requires, and ones far more complex than are needed to state the problem. Surely, many mathematicians thought, it was poss...

Medicare could waste billions of dollars, bankrupt small businesses and leave seniors without crucial equipment like oxygen tanks and wheelchairs, some economists warn, with a new auction-based purchasing plan that ignores mathematical principles of competitive bidding. The new plan began with an apparently good idea: Medicare has started buying durable medical equipment through a competitive bidding process rather than through fixed prices influenced by industry lobbying. In theory, this could save the government a big chunk of the $14 billion it has spent on such equipment in a year. But in...

Climate treaty negotiators might be wise to have a conversation with a game theorist. So far, negotiators’ promises to reduce greenhouse gas production have been paltry and results paltrier, as both emissions and global temperatures have risen. A new game theoretic analysis published in the Oct. 23 Proceedings of the National Academy of Sciences both pinpoints why negotiations have accomplished so little and suggests how the parties might achieve better results. Since 2009, climate treaty negotiations have focused on one value: 2 degrees Celsius. If the planet warms more than this, scien...

The rate of change of bubble volume. If this quantity is positive, the bubble will grow; if it’s negative, it will shrink. A constant that depends on the temperature and the specific gas in the foam. (The foam on top of a glass of Guinness lasts unusually long because Guinness uses nitrogen in addition to carbon dioxide in its beer. K is smaller for nitrogen, so the bubbles change size more slowly.) The sum of the lengths of the edges of the surfaces where the bubble intersects other bubbles. For an isolated bub...

MATH TREK: A claimed proof that P≠NP spurs a massive collaborative research effort.