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Putting Topology to Work

Robert Ghrist has the upbeat inflection (and snazzy PowerPoint slides) of a practiced and confident presenter. The University of Pennsylvania professor also demonstrates a master educator’s ability to give cultural context and practical relevance to an abstruse field of mathematical study—algebraic topology, in Ghrist’s case.

Who knew, for instance, that John Milton pioneered our modern usage of the word “space” in his epic poem Paradise Lost or that a topology-based analysis could identify a subgroup of breast cancers?

Robert Ghrist

These facts—and many more—beguiled the crowd that packed the MAA Carriage House for the NSA-funded Distinguished Lecture Ghrist delivered on September 19. “Putting Topology to Work” offered a lay audience a surprisingly literary look at what topology is and how mathematicians are applying it to real-world problems.

Ghrist defined topology as “the study of abstract space,” and acknowledged 17th-century British poet John Milton for his preliminary pondering of the subject.

In composing Paradise Lost, Ghrist explained, in weaving his tale of God and man and the fallen angel Satan, Milton had a cosmologically thorny question to contend with, namely “where on earth to put hell.” While his literary forebears had situated the underworld underground, Milton could not site it there without stretching the credulity of his more earth-savvy readers.

Ghrist described the configuration Milton settled on: the empyrean; hell remote from it across a vast, chaotic void; the universe suspended from the empyrean as if by a golden chain.

“Now that is the very beginnings—within poetry—of what it took mathematicians a little bit longer to do more formally,” Ghrist said. “That is, come up with a notion of an abstract space.”

Ghrist then launched into a “gentle introduction” to a more mathematical take on this notion, though he would circle back to poetry soon enough.

As an example of a one-dimensional space, Ghrist cited the line. For two dimensions, he conjured up a video game from his childhood, one in which the airplanes that exited the rectangular screen on one side reappeared on the opposite side. It was as if, Ghrist observed, the opposite sides were associated, the left joined to the right and the top to the bottom, the screen deformed into a torus.

For his three-dimensional example, though, Ghrist appealed again to literature, this time to Dante’s Divine Comedy. Ghrist suggested that when, in Canto 28 of Paradiso, Dante turns around and sees the planetary spheres—their order reversed—rotating around a singular point, the effect is caused by the earth being at the bottom of the universe and heaven at the top, the whole comprised of nested two-dimensional spheres.  

Ghrist marveled at finding such a “a nice example of a non-trivial, three-dimensional topological space”—the three-sphere, the boundary of a four-dimensional ball—“in a poem hundreds of years before mathematicians got around to [conceiving of] it.”

Poets may be prescient in their way, but another take-home from Ghrist’s lecture was that today’s topologists are themselves arguably ahead of their time, at least by the estimation of the third literary luminary Ghrist invoked over the course of the evening.

Aleksandr Solzhenitsyn was not only trained as a mathematician, but also had a topologist for an advisor, so when a character in his 1968 novel The First Circle remarks of topology that “in the twenty-fourth century it might possibly be of use to someone,” the writer is expressing informed—and widely shared—doubts about topology finding application anytime soon.

According to Ghrist, though, a number of applications have “bubbled up” over the past 10 or 15 years.

“Over the past century we’ve developed manifold approaches, techniques, perspectives for answering topological questions, qualitative questions about spaces and mappings between them,” Ghrist said. “And they come in several flavors which are very, very relevant to problems associated with data, problems associated with engineering and science.”

Descriptive topological tools qualitatively describe spaces or transformations on spaces; integrative tools “take local data and turn it into something global”; obstructive tools “tell you whether you can or cannot solve a particular problem.”

After alluding to applications in signal processing, robotics, and genetics, Ghrist explained in more detail a few with which he has personal experience. An ad hoc wireless communication network comprised of sensors with a limited range and the ability to communicate with only their immediate neighbors, for example, presents a “problem of going from local sensor data to some sort of global understanding of your environment.”

“That is perfectly primed for topological methods,” Ghrist said.  

As many problems of interest may prove to be in the future. Displaying a slide reading “novel challenges necessitate novel math,” Ghrist concluded his talk by emphasizing the need to continually develop new tools to tackle what the modern world throws at us.

In the effort to meet the challenges of today and tomorrow, Ghrist said, “mathematics has a lot to offer.” —Katharine Merow


 Listen to an interview with Robert Ghrist and MAA Director of Publications Ivars Peterson (mp3)

 

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

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