September 9, 2008

Donal O'Shea first thought it might be a hoax when his assistant told him that someone named Franco left a message asking
him to deliver a lecture to an Italian national television audience this coming November. After all, O'Shea is a mathematician,
a practitioner in a field not necessarily known for offering up scintillating oratorical fodder.

This was the second unusual invitation the dean of faculty and vice president for academic affairs received for this fall.
The other came from the Johannes Gutenberg University in Mainz, Germany, which is assembling a roster of leading physicists,
historians, cosmologists, philosophers and mathematicians for a conference in late September called *Beyond Einstein*
aimed at examining developments in general relativity and gravitation. O'Shea will present "The Unexpected Resolution of the
Poincaré Conjecture."

The genesis of these invitations isn't hard to trace. O'Shea's 2007 book, *The Poincaré Conjecture: In Search of the
Shape of the Universe*, was translated into 11 languages, including German and Italian, within two months of publication.
The book has been praised for explaining deep mathematical concepts in terms accessible to the savvy generalist. It has also
turned O'Shea into a sought-after explainer of one of the most exciting events in recent math scholarship, the publication
of a proof for a problem that had stumped mathematicians for a century.

The word "publication" in this case is a relative term because Grigory Perelman, the reclusive and eccentric Russian scholar
who solved a puzzle at the heart of a field called topology, decided to forgo the normal route of submitting his findings
to a peer-reviewed journal, instead posting the fruits of his research directly to the Internet. After giving a series of
lectures on the proof he was offering of the Poincaré Conjecture, Perelman dropped out of sight, leaving it to the professional
community to hash out whether the proof was valid. The verdict turned out to be that it is, and Perelman was awarded the Field
Medal, the highest prize in mathematics. Adding to the intrigue surrounding his personality Perelman spurned the honor, failing
to show up at the ceremony in Madrid.

Topology, O'Shea explained, is the study of how things are connected. It is distinct from geometry because it doesn't concern
itself with distance, instead focusing on underlying theoretical questions about shape. Think of a deflated balloon; even
though it may droop or be stretched, the points on its surface still have the properties of a potential sphere. In 1904 French
mathematician Henri Poincaré conjectured that any form that did not have holes in it was equivalent to a sphere. This idea
has implications for the shape of the largest entity we can think of, the universe itself. It turned out that the verity of
this proposition was easier to imagine than to prove. In fact, almost a century after Poincaré made this conjecture, or informed
guess, the Clay Mathematics Institute in Cambridge, Massachusetts, posted a $1million reward for a proof or disproof that
withstood the vetting of the mathematics community. Whether or not Perelman will claim the prize is an open question.

O'Shea is as interested in the story of how the concerns that led up to the Poincaré Conjecture evolved, starting with
the Babylonians and then through the ancient Greeks, as he is with how Perelman arrived at a milestone in this history that,
by many accounts, rocked the math world. The *New York Times* called it "a landmark not just of mathematics, but of human
thought." O'Shea is struck by parallels between how Poincaré went about his research, which led to his famous conjecture,
and how Perelman arrived at his conclusions. Each built on work that had come before, but instead of pursuing incremental
gains in understanding fundamental questions they came at the complexities that fascinated them from entirely new angles,
bringing seemingly irrelevant branches of research to bear.

"Everybody thought this problem would have a purely topological proof and people tried for 100 years to find one. There
were hundreds of false proofs," O'Shea said. "And then this Perelman guy came along and said the proof goes through geometry
and calculus, which no one really suspected, and he made it work." Not only did he make it work, but he did so in a way that
O'Shea described using words such as "beautiful," "stunning," and "ironic." Just as Poincaré changed the course of twentieth-century
mathematics, Perelman's proof, and the advances he made in arriving at it, might well shape the field in the next century.
"Perelman's work will probably provide future breakthroughs in general relativity by linking general relativity with quantum
theory," O'Shea said.

That is one of the messages O'Shea will bring to a gathering of learned colleagues in Germany. Only starting to adjust
to his acquired vocation as a popularizer of mathematics, O'Shea said he was "befuddled" by the invitation to participate
in such a prestigious gathering as the *Beyond Einstein* conference. The invitation from Italian television left him
downright suspicious that he was being punked. The letter that followed the phone message from Franco Pastrone, president
of the Associazione Subalpina Mathesis, allayed that fear. It informed O'Shea that his book was selected for the Premio Peano,
"a prize awarded to the author of a book of 'readable mathematics'." Following the presentation in Torino, the winner is called
on to present a lecture to a live audience for later broadcast to the entire country.