Out from Obscurity
Old math theories take on new life in engineering.
By Mark Matthews
Not long ago, algebraic topology was so abstract as to belong in “the stratosphere of human thought,” or so Gleb Nerzhin mused in Aleksandr Solzhenitsyn’s The First Circle. Now, topologists, who study surfaces and relationships between proximate and global spaces, are mastering a growing number of real-world applications, from preventing drone collisions to guiding six-legged robots and managing vast amounts of sensor data.
All this puts Robert Ghrist in the right place at the right time as a professor of both mathematics and electrical and systems engineering at the University of Pennsylvania, where he and research colleagues give once obscure theories new life as power tools for the digital age.
Concepts from decades or even centuries past, like the “characteristic” named for Enlightenment mathematician Leonhard Euler, can simplify and cut the cost of solving 21stcentury problems in such fields as signals processing, data aggregation, and network coding, Ghrist contends. In a vivid application of topology, his students mapped an airport’s interior space, revealing a hole-like atrium, just by using Apple laptops to capture the signal strength of various Wi-Fi hotspots. Topology also can help organize swarms of drones or figure out the behavior of nanoscale materials. “It’s custom-built for dealing with so many modern problems,” he says.
Just as Moore’s law predicted the exponential growth of computing power, Ghrist says, we’re witnessing a comparable expansion of math knowledge and practice. In fields such as neuroscience or gene mapping, for example, math could outpace machines in hastening discoveries. “If you’re interested in protein folding, you’ll wait a long time for supercomputers to catch up,” he asserts.
Ghrist’s work is particularly suited to sensor networks, fielding robots in uncertain terrain, and pursuit-and-evasion strategy. As such, it fits the Pentagon’s aim of relying more on smart technology to maintain America’s military edge. Along with directing other Defense Department-funded research, Ghrist led more than a dozen researchers from nine institutions in an $8 million, four-year project called Sensor Topology and Minimal Planning (SToMP).
A mechanical engineer with a Ph.D. in applied mathematics, Ghrist, 45, has enjoyed a rapid, if frenetic, academic rise, winning a National Science Foundation CAREER grant in 2002 and a President’s Early Career Award for Scientists and Engineers (PECASE) two years later. Following faculty appointments at Georgia Tech and the University of Illinois, Urbana-Champaign, he became Penn’s first Andrea Mitchell PIK University Professor in 2008. Endowed by the veteran NBC correspondent and her husband, retired Federal Reserve Chairman Alan Greenspan, the post is a dual appointment in arts and sciences and engineering.
Part of Ghrist’s success undoubtedly comes from a compelling lecture style, which makes him a sought-after speaker at events ranging from a Technion symposium to a recent Capitol Hill presentation arranged by the American Mathematical Society. Videos in which he strides across stage, mic in hand, spreading the applied-math gospel, represent “a key step in increasing bandwidth – in mathematics in particular,” he says. Another boon is massive open online courses, or MOOCs. Ghrist spent close to 1.5 years developing his own calculus MOOC for Coursera, complete with 15 hours of painstaking animation and the self-published FLCT: Funny Little Calculus Text. Some 5,000 students worldwide have completed it, including a 10-year-old American “genius” and a Mumbai teenager who went on to create videos to explain calculus to high school classmates.
If students could enter college having completed prerequisites from MOOCs, such as introductions to math, psychology, and economics, they could graduate in three years and save $60,000, Ghrist contends. He also sees MOOCs that feature expert lecturers as a preferred alternative to Advanced Placement tests – at least in math. Ghrist, who says he must spend time undoing what students were taught in AP Calculus, looks aghast when told of a campaign to introduce an AP engineering course (Prism cover story, April 2014.) “Oh God – really?” he asks. The result, he fears, could be “students who are good at the AP engineering test [but] don’t know a blasted thing about engineering.”
Although America is second to none at postgraduate teaching, the nation performs less well at the precollege and undergraduate levels. One reason Ghrist, a fan of Dante and other classic authors, hopes to change that is the flashes of mathematical insight often found only in old books. “Branches of math have been lost because they were not taught well,” he laments.
Mark Matthews is editor of Prism.
Image courtesy of Tommy Leonardi/Thinkstock