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# Go figure

They love the Birch and Swinnerton-Dyer conjecture. Shouldn’t we?

Avner Ash and Robert Gross, who are old friends as well as faculty colleagues, specialize in number theory, or relationships among whole numbers. Once called arithmetic, it may well be the primal subdivision of mathematics, evidenced in equations scratched into Babylonian clay tablets nearly 4,000 years ago.

Until quite recently number theory was considered (sniffily, by some of its practitioners) part of *pure* mathematics, without practical implications; but cryptography and cybersecurity, which are dependent on number theory, changed that, and the area is one of three—along with algebraic geometry and topology—in which the Boston College mathematics department has invested since it was enlarged and transformed from a teaching department into a doctorate-granting unit in September 2010.

Gross, a scholarly-bearded associate professor and the kind of classical music lover who reads orchestral scores like they were stories, arrived at the Heights from MIT in 1984. Ash, with a doctorate from Harvard and an endearing habit of smiling encouragingly ahead of the punchline you’re about to offer (and which he’s already figured out), arrived as a full professor in 2000, having previously taught at Ohio State. Friends since the early 1990s, they share a high regard for serious reading and thought, a belief that jeans and sneakers are fashion enough for this world, and a habit of congenial dispute, like a couple of cousins who grew up next door to each other.

And both are distressed by the fact that mathematics, which they count a cultural and intellectual glory—”This wonderful subject to which we’ve devoted our lives,” says Ash without irony—is not viewed as glorious in their native land. “In France,” notes Gross, *with* intended irony, “great mathematicians are on postage stamps!”

And so in a modest attempt to fix American culture, soon after Ash came to Boston College, the two men, who had previously collaborated on scholarly papers (e.g., “Generalized non-abelian reciprocity laws: a context for Wiles’s proof”), began collaborating on a book that would explain number theory to those who cared to know. “We wanted to put out a book that was accessible but that was pure and theoretical,” says Gross—by which he means it would be longer on numbers and equations, and shorter on words—”because that’s the way we think about mathematics.” “A book that’s not about puzzles or the weird habits of mathematicians,” adds Ash, with a not terribly sympathetic nod toward popular movies such as *A Beautiful Mind* and *Good Will Hunting* that are focused on the troubled adventures of charismatic, dysfunctional math geniuses.

Beginning in 2006, and over the course of 10 years, Ash and Gross published three books for a Princeton University Press series designed to address readers who have a sophisticated—but not professional—interest in mathematics. The first was titled *Fearless Symmetry: Exposing the Hidden Patterns of Numbers*. “It’s for people who want to gain insight [into number theory] and are not afraid of algebra,” says Princeton editor Vickie Kearn. Their second book, published in 2012 and built for readers who have basic calculus as well as algebra, is *Elliptic Tales: Curves, Counting, and Number Theory*, which elucidates, among other worthy mysteries, the Birch and Swinnerton-Dyer conjecture, an open problem in number theory, and one of seven “Millennium Prize Problems” worth a million dollars in bounty to the person who arrives at a solution. (Ash and Gross, after making several efforts to jointly describe the book to me in one sentence, were finally satisfied with landing on this bit of a prose poem: “It’s about solving certain cubic equations where very surprising geometry intersects with the numerical properties of the solutions.”) Ash and Gross’s third book, published in 2016, is *Summing It Up: From One Plus One to Modern Number Theory*, and is the most challenging of the volumes. The first section requires college algebra, the second calculus, and the third “some knowledge of group theory and geometry,” says Kearn, whose main—and considerable—business is mathematics monographs, but who very deliberately publishes a handful of books each year that are aimed at “readers who are curious about contemporary mathematics.”

Sitting in Gross’s Coolidge Corner living room on a sunny late-winter afternoon, Ash and Gross talked about how they came to be academic mathematicians. For Ash, it was the influence of his father, an engineer, who first taught him math; for Gross it was alert teachers who provided what he needed. Ash talked about the methodical sensibility that mathematics requires. “Math isn’t like literature,” he says, “where if you don’t understand *Moby-Dick*, you can still understand *Northanger Abbey*.” Math, he continues, “is always building on itself, and if you can’t collect and retain successive layers of information, you get lost. And that may be why a lot of people drop out of the game. They missed a step.” Gross recalls learning how to “cast out nines”—a way of checking arithmetic solutions—when he was a young child, and going home and writing “lists of numbers and adding them on a calculator and then casting out the nines and seeing that it always worked!” Ash talks about the pleasure of wrestling with seemingly intractable theorems. “It’s like romance,” he says. “You start with a feeling toward someone [and] all kinds of experiences crystalize around that; all kinds of efforts, some successful, some unsuccessful—a history.” Ash says he believes it’s the concept of proof that is one of math’s central attractions. In the fading winter light, he and Gross trade names of mathematicians who made famous discoveries while failing to solve the problem they were chasing.

As to the books they’ve authored, none is a bestseller, but they do sell, to people who email their critiques and gratitude and requests for tutorial assistance. Judging from responses, Ash and Gross say, their typical reader is an engineer, a retired scientist or physician, or a high school hotshot who wants to prove he can tangle with a couple of university faculty. They respond to every communication, and they’re considering a fourth book.

Read more by Ben Birnbaum