COUNCIL: State of the University 

November 3, 2009,
Volume 56, No. 10

Robert W. Ghrist, Andrea Mitchell University Professor
Thank you, President Gutmann. Let me say to you all how grateful I am for your courtesy in listening to a talk about, essentially, mathematics. Let me take away your suspense: no formulae, and no quiz. Promise!
Penn’s Integrating Knowledge program holds a special appeal to me, as a mathematician. You see, I love integration. Of course, I’m fond of differentiation as well, and am perfectly at peace with limits, linearity, sequences, series, contours, cohomologies, and all kinds of calculus.
Calculus, in particular, forms the focus about which Mathematics research and pedagogy orbits. The calculus of Newton and Leibnitz is a zenith of human thought. It is a true pleasure to train our students in the art and implementation of the calculus, in no small part because it is so easy to connect them to the cutting edge research that Penn conducts.
Much of my current research is focused on building new calculi for sensor networks. If you look about the room, you are sure to see a whole host of sensing devices: audio sensors, thermostats, motion detectors, etc. Penn’s campus has over 180,000 sensors as part of its heating/cooling system.
However, the most sophisticated and ubiquitous sensors in the room are the ones you are using to listen and look.
Many of the emerging problems in this domain concern, precisely, a problem of integration, of turning local signals into global meaning. For example, could the collection of cell phones in this room collaborate to determine the size and shape of the room? Whether and where there might be H1N1 present? Or a radiological substance?
From monitoring the environment, to monitoring your heart rate and blood circulation, ubiquitous sensor networks are poised to impact society in dramatic ways. But in order to help the “walls wake up” we need the mathematics to complement the machines.
I am leading a team of researchers in a 4year $8 million DARPAfunded project on Sensors to develop the mathematics crucial to ubiquitous sensing. One important product of this project has been the creation of a new calculus, built specifically for sensor networks. This integral calculus comes from a branch of Mathematics—sheaf theory—that, even among mathematicians, has a reputation for being abstract, unapplied, and perhaps inapplicable. Not so!
I would love to tell you some details, but I’m afraid there’s not quite enough room at the margin of my talk for precise theorems and proofs. Suffice to say we are developing a new calculus for data with the same language of integration that we teach to our youngest scholars, while leveraging the depths of Mathematics generated by its eldest luminaries. This is what the Penn’s PIK program is accomplishing.
Now, that there are many kinds of calculus in the world comes as a surprise to most people, yet mathematics is constantly evolving new species of structures. If, donning an engineer’s cap, I tried to convince you of progress, I would find ready assent: consider your cell phone, your iPod, your Kindle, your tablet.
You may know Moore’s Law, which states the number of transistors on an integrated circuit is exponential: it predicts, accurately, dramatic improvement in processor performance, with corollaries to memory, storage, speed, and more. Hardware improves.
What about mathematics? It is a common misconception that mathematics, like Shakespeare’s Folio, is a complete work, with nothing more to do than work out little bits at the margin. Nothing could be further from truth.
Indeed, mathematics has grown at an exponential clip not unlike the semiconductor industry.
Margaret Wright, a mathematician at NYU, did the following thought experiment. Consider Linear Programming, or “LP” problems, a class of optimization problems important in scheduling air traffic, communications, manufacturing, and design. Every trip you plan on Orbitz, every Amazon purchase, every GPS routing, requires the solution of one or more LP problems.
Imagine it’s the 1970s and, in a burst of lights and disco music, a human from the future appeared in a time machine, Starbucks in hand, to make you the following offer: you can have the top supercomputer from the year 2001 OR you can have the top theorems & algorithms from 2001, but you have to run them on your disco1000 IBM mainframe. Which would you choose? Machine? Or mathematics?
Knowing Moore’s Law, you would clearly choose the 21st century machine! And you would have chosen poorly. Not only would the algorithms of the future weigh less (and cost less!), they would be superior. Mathematics, too, is exponential.
This example works because there is clear communication between mathematicians and scientists in the realm of LP problems. But the sciences abound with difficult problems, harder to communicate, the mathematical tools for which may already exist, but lie buried under layers of abstraction until unearthed and recognized as such.
In order to take advantage of and direct the exponential growth in mathematical tools, we need, desperately, a better transfer of technology between the mathematical arts and the engineering, physical, life, and social sciences, with ideas flowing in both directions. It is to that goal that Penn’s PIK program can have immediate and dramatic impact. 