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The Remarkable Interaction between Mathematics and the Computer: Examples Old and New

The image of the lonely genius locking the office door working away on a problem is something of the past. And it’s not an accurate representation of the majority of advances in mathematics,” said Jill Pipher, director of the Institute for Computational and Experimental Research in Mathematics (ICERM) at Brown University. On the last day of Mathematics Awareness Month 2012, Pipher presented “The Remarkable Interaction of Mathematics and the Computer” as part of MAA’s Distinguished Lecture Series. 

The intersection of mathematics with computation and experimentation, Pipher argued, has changed mathematical practice in exciting ways that challenge stereotypes of the discipline (and its practitioners!).

Acknowledging that a talk about her chosen subject matter “could really go on forever,” Pipher mastered her obvious enthusiasm and confined her remarks to the hour allotted. She organized the “panorama of examples” she’d chosen into three categories: personal, classical, and contemporary.

Although tight-lipped about how to get rich in cryptography, Pipher counts the science of code making and breaking among her research interests and jointly holds four patents for the NTRU encryption and digital signature algorithms. She treated her audience to a tantalizing overview of the history of the public key cryptosystems (PKC) on which Internet commerce relies.

Pipher quoted Whitfield Diffie’s declaration that he and his colleagues stood “on the brink of a revolution in cryptography” and showed a picture of a historic garment. Printed with the RSA algorithm, the 1980s T-shirt represents to Pipher the “mathematical and political impact” of PKC. “The printing of that algorithm on that T-shirt at that moment in time . . . turned that T-shirt into a munition,” Pipher explained. “It made it something that was not permissible to export under U.S. export law or even to show to a foreign national.”

For the classical segment of her talk, Pipher highlighted several questions that, while stated in centuries past, were “recently answered—by geological standards—with the help of a computer.” They included the four-color theorem and the Kepler conjecture.

The attacks leveled against these problems by mathematicians armed with unprecedented computing power, Pipher noted, forced the mathematical community to grapple with what constitutes proof in mathematics. In 1976, to prove that four colors suffice to color any map, Kenneth Appel and Wolfgang Haken used an approach that relied on a computer program verifying what happened in a little more than a billion cases.

“Are we any more or less certain if a computer checks all these cases than if a mathematician sits down and checks them by hand?” Pipher asked.

Outsourcing key components of their work to computers may make mathematicians uneasy, but that disquiet hasn’t stopped them from doing it. Pipher’s final set of examples illustrated the fascinating and useful results achieved by those open-minded enough to reconceive of how math is done.

Jill Pipher presenting an MAA Distinguished Lecture in MAA Carriage House.

A numerical experiment suggested a surprising solution to the three-body problem: an orbit in the shape of a figure eight. David Gracias andGovind Menon work at what Pipher called the “intersection of mathematics and materials,” developing self-assembling, 3D nanostructures. And the Internet facilitates mathematical collaboration on a massive scale in Tim Gowers’s Polymath Project. Thousands of contributors making incremental progress on unsolved problems demonstrate the power of “many mathematicians connecting their brains efficiently” in cyberspace.

“That’s mathematicians imitating computers, rather than computers imitating mathematicians,” Pipher said.

Pipher recognized that she was just scratching the surface, merely whetting her audience’s appetite for more information on the ways computers and experimentation have broadened what mathematics and mathematicians can do. She referred interested listeners to other Distinguished Lectures. Those keen to learn more about cryptography should look up Alice Silverberg’s Distinguished Lecture, for instance.

And for the dirt on the Riemann zeta function? “I’m not going to be able to tell you everything that’s beautiful about this,” Pipher mourned—and recommended Brian Conrey’s lecture on the topic. —Katharine Merow


 Listen to the full lecture (mp3)

 

This MAA Distinguished Lecture was funded by the National Security Agency.

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