A few weeks ago I discovered Robert Ghrist via his web site. Robert is a professor of mathematics and electrical engineering. He describes his research as applied topology, something I’d never heard of. (Topology has countless applications to other areas of mathematics, but I’d not heard of much work directly applying topology to practical physical problems.) In addition to his work in applied topology, I was intrigued by Robert’s interest in old books.
The following is a lightly-edited transcript of a phone conversation Robert and I had September 9, 2010.
* * *
JC: When I ran across your web site one thing that grabbed my attention was your research in applied topology. I’ve studied applied math and I’ve studied topology, but the two are very separate in my mind. I was intrigued to hear you combine them.
RG: Those two are separate in a lot of people’s minds, but not for long. It’s one of those things that the time has come and it’s clear that the tools that were developed for very abstract, esoteric problems have really concrete value with respect to modern challenges in data, or systems analysis.
JC: Could you give some examples of that?
RG: Certainly. One of the first groups of people who do full-scale applied algebraic topology were Gunnar Carlsson’s group at Stanford doing applications to data analysis. The setup is you have a collection of data points in a space, a point cloud, that is a discrete representation of some interesting structure. For example, you might want to know how many connected components does this data set have. That might correspond to different features. For example, this might come from customer surveys that a corporation has out. It’s trying to cluster these customers. Or if it is medical data, say they are trying to discern different types of cancers. Then you might look at what statisticians call clustering, grouping data sets into connected components.
Well, topologists know that’s just the first step in a larger program of finding global structure. Besides having connectivity properties, spaces can have holes in them of various types. There are formal algebraic methods for finding and classifying those holes. That’s homology, cohomology, homotopy theory. And applying that to data turns out to give some really revolutionary techniques that don’t rely on projecting the data set down to a two-dimensional picture and trying to visualize what’s happening. It’s automatic.
There are similarly themed application in the work I do in engineering systems where you take data that comes from, say, a network of sensors or a communications network or a networked connection on computers or a networked collection of autonomous robots. And you try and take all that local information, say in the context of sensor networks, you take collections of local data and try to patch it together to give you global understanding of an environment. That kind of local-to-global transition is what the techniques of topology were built to do. And they are surprisingly efficacious in these very applied problems.
JC: I’m familiar with Van Kampen’s theorem in homotopy. Is that the sort of thing you’re talking about?
RG: Yeah, homotopy tends to be less computable than homology. Homology is much more natural in these contexts. The corresponding principle is the Mayer-Vietoris principle, the homological analog of Van Kampen. And the Mayer-Vietoris principle is really telling you something about integration, how you stitch together local bits of data, how you integrate it into a global understanding of your network. That is a very deep idea that is very important to transitioning from local to global data.
JC: One mental block I have when I’m thinking about these sort of things is that in my mind topology seems sorta fragile in the sense that one random connection can change something. Connect just one pair of dots and suddenly a disconnected space becomes a connected space. It seems that would be a problem in these settings where you make some sort of topological statement, but any missing data or erroneous data invalidates your conclusions. But I remember you said something that was sort of the opposite on your web site, that topological methods can be more robust. I could see that, but I’m having trouble resolving these two views.
RG: There are different types of robustness that are critical in different types of applications. Because the constructs of algebraic topology are invariant under homotopy or deformation, it turns out to be very robust with respect to, for example, coordinate changes. That is extremely useful when you’re dealing with data that has some anchor to the physical world.
Let’s say data that’s being collected by our cell phones. Put a couple sensors on a cell phone and we have lots and lots of interesting data streams. And that data is tied to physical locations. But you might not know exactly where it is. GPS doesn’t work so well when you’re inside a building, for example. In contexts like that, you want the kind of robustness that doesn’t depend on having a carefully laid-out coordinate system.
Now robustness with respect to noise, especially robustness with respect to errors, is a much more difficult problem to solve in general. But even in that case there are some topological tools that in specific examples can be deployed. This gets into slightly more technical stuff involving persistent homology and topological properties that persist over a range of samplings.
JC: Could you give an example of how knowing the homology of a data set might tell you something about a physical phenomena?
RG: One example that I’ve worked with a lot has to do with coverage problems in sensor networks. Let’s say we’re talking about cell phone coverage, because everyone’s familiar with that. If you get into a hole in the cell phone network where you’ve dropped coverage, that’s frustrating. You’d like to know whether you have full coverage over an area or not, whether you have holes.
This gets much more critical when you’re talking about a security setting. You have video cameras or satellites that cover a region and you want whether you’ve covered everything, or whether there are holes where you are missing information. One of the things I’ve used homology theory for is to give criteria that guarantee coverage, that guarantee no holes in your sensor network based exclusive on coordinate-free data. So even though the sensors may not know where they are, and only know the identities of the sensors near by, it’s still possible to verify coverage based on homological criteria.
JC: Interesting. I suppose especially if you’re looking at higher-dimensional data you can’t just draw a lot of circles on a map and see whether they overlap. You have to do something more computational.
RG: Exactly. Especially if those circles are in motion and you want to know what’s happening over time. Especially if there are no coordinates and you don’t know where to draw the circles to see how things overlap.
One thing I want to get across when I’m talking with people is that I view a mathematics library the same way an archaeologist views a prime digging site. There are all these wonderful treasures that are buried there and hidden from the rest of the world. If you pick up a typical book on sheaf theory, for example, it’s unreadable. But it’s full of stuff that is very, very important to solving really difficult problems. And I have this vision of digging through the obscure text and finding these gems and exporting them over to the engineering college and other domains where these tools can find utility.
Now, a lot of esoteric mathematics has already crossed the fence. No one will claim any more that number theory is useless. But topology is the place where you have the most-useful, least-used tools. So that’s my vision for what I want to see happen in mathematics and what I’m trying to accomplish.
JC: Switching gears a little bit, another thing on your site that caught my eye was your quote about old books.
Reading anything less than 50 years old is like drinking new wine: permissible once or twice a year and usually followed by regret and a headache.
I thought about C. S. Lewis’ exhortation to mix in old books with your reading of new books because each age has its own blind spots. Old authors have their own blind spots, but maybe they’re complementary to the ones we have.
RG: Exactly. I definitely have followed that dictum. Maybe a little too much so, in that I rarely read anything modern at all. When it comes to books. I don’t follow that rule when it comes to music or movies or blogs. But on the level of books, there is so much good stuff out there that has stood the test of time, I don’t run out of interesting things to read. I had the wonderful experience as a college student to take a great books-type course that involved a lot of reading, a lot of discussion. Really changed my outlook and got me loving the classics and really living inside a lot of those books.
JC: Were you a math major when you took this great books class?
RG: No, I was an engineering major. My undergraduate degree was in engineering. I came to math a little late in life.
JC: You said you were particularly fond of Dante.
RG: That’s correct. Yeah, I’ve lived in that book [The Divine Comedy] a long time and I still find new and very engaging ideas in it every time I crack it open. Most people don’t get very far past the Inferno. The Inferno is the exciting, action-movie part of the story. But the later parts of the story — purgatory, paradise — those are really nice places to live.
JC: Have you had to do a lot of historical research to be able to read Dante?
RG: If you get a good translation with a good set of notes, that makes it much easier. I find the translation by Dorothy Sayers has excellent notes. She got turned on to Dante late in life. There’s an interesting story. While they were having bombings in the early 1940’s, she was going down to the bomb shelter to spend a few hours, she decided to grab a book off the shelf on the way down. She saw a copy of Dante. She said “You know, I’ve never really read Dante.” Pulled it off the shelf. For the next two days she didn’t eat or sleep. She was engrossed with the story and how masterful it was. And she wound up devoting the rest of her life to mastering the Italian and producing her own translation.
* * *
For daily tweets on topology and geometry, follow @TopologyFact on Twitter.