# Precision Problem Solving: Topological Data Analysis Driving Advances in Medicine and Biology

By MAA FOCUS, news magazine

Mathematician Chad Giusti spoke with MAA FOCUS, the news magazine of the Mathematical Association of America.

Chad Giusti is an assistant professor of mathematics at Oregon State University. He works in pure and applied topology, with applications principally in neuroscience and complex systems. His work has appeared in journals such as PNAS and Crelle’s and has been supported by the NSF, AFOSR, and AFRL. Here, we learn about the fascinating work Chad has done in applying the tools of topological data analysis to problems in medicine and biology.

1. You are an expert in topological data analysis (TDA), a field that many people in our community are unfamiliar with. How would you describe TDA to someone who just finished the calculus sequence? How would you describe TDA to someone who has taken a standard introductory course in topology?

The usual quip is that topological data analysis characterizes complex systems or data in terms of qualitative notions of “shape.” I think this is at the same time too vague and too specific.

Calculus students are adept at describing shape in qualitative ways. A common exercise is to read off various information about a polynomial by looking at its graph or the graph of its derivative. By counting extrema and roots, examining behavior “at the ends,” and so on, we can determine things like the minimum possible degree, sign of the leading coefficient, and so on. While these are, in principle, numeric answers, they aren’t exact measurements—they’re bounds and ranges of possible values. Even if I only provide a scattering of points on the graph of the polynomial, it’s not much harder to provide the same data about the underlying polynomial.

For students, I would say that topology, particularly algebraic topology, provides a set of mathematical tools for a similarly qualitative characterization of more complex shapes: surfaces and higher dimensional analogues called manifolds, and more abstract structures like graphs. We most commonly formalize “qualitative” as meaning “up to continuous deformation” – stretching or compressing, without cutting or gluing. A circle remains a circle, topologically, even if we stretch it into a wiggly mess as we might do with a rope, so long as we don’t cut it open into a long strand or glue distant points together. This flexibility reduces the specificity of what we can say about systems, but it makes these descriptors more applicable in the presence of noise or incomplete data, both of which are particularly pernicious in biological and medical applications.

Students in an undergraduate topology course might not recognize much of what we do in TDA immediately. However, many will have seen the fundamental group of a topological space, or the topological classification of smooth surfaces, which are cousins of the kind of measurements and classifications we employ when studying “shape” in applications. However, data is rarely given to us in the form of a topological space—we must build approximations of our spaces from things like finite collections of points sampled on (or noisily near) a surface we want to study.

Currently, the most common tool used in TDA is called persistent homology, which characterizes how qualitative features of a shape evolve as some parameter changes. The parameter can be a measure of size (“how big are the features”), time (“when do the features appear”), or something more esoteric and domain dependent. Persistent homology gives us a collection of vector spaces associated to the space, much like the fundamental group gives us a group. By comparing these vector spaces across different data sets—results of some experiment under different conditions, for example—we can use the similarity or differences between the evolution of features to reason about how the underlying systems compare. Differences in shape can point to differences in organization in a complex system. For example, neural activity that encodes the head direction of a mouse is well-described by a circle, but that which describes the head direction of a bat generally requires a shape that can encode three dimensions of motion. (In fact, experimentally it appears to be a torus, not a sphere!)

Image of Chad Giusti's MAA FOCUS magazine article.

2. You apply TDA to current problems and systems that arise in biology and medicine. Can you elaborate more on those applications and what got you interested in pursuing them?

When I think about my applied work, I usually place it in the field of theoretical neuroscience, in the context of developing a theory of how neural populations encode information and perform computations. It turns out that many of the models that neuroscientists have developed to describe these phenomena “look” topological in the sense that it’s easy (for an applied topologist like me) to imagine formalizing them using language from TDA.

In fact, this is how I first got started in the area. As a graduate student, I worked in pure algebraic and geometric topology studying spaces of knots, though my projects always had a computational bent. One year on the job market, I had two offers: one to go to Belgium and work on this very theoretical type of mathematics, and another to go to Lincoln, Nebraska and try to apply topology to the study of neural codes. The PIs on that project, Vladimir Itskov and Carina Curto, showed me some pictures of place fields, which diagram how individual neurons in the hippocampus respond to an animal’s location in its environment.

These look a great deal like the topological notion of a “cover” of a space, which is one of our fundamental tools for studying shape. Their notion, which turned out to be an excellent one, was that we should be able to use tools from TDA to study this structure in neural activity, providing a platform for mathematically formalizing some of these informal models. The idea of developing an entirely new way of studying how the brain works—and doing it using all of the abstract math I’d fallen in love with in graduate school—was a very compelling offer.

I think it’s important to note that, as compelling as the offer was, pursuing this route was a risky decision. Novel applications of mathematics, particularly areas of math that aren’t well established for applications, very often don’t gain traction or take many years to do so, and a postdoc project that doesn’t go anywhere usually doesn’t lead to further employment. I had the privilege to be able to take that risk in large part because I had a strong economic and personal support system, including skills that would allow me to seek alternative employment if the project didn’t work out. It would behoove us to provide more support to early career academics so it’s easier to take these big risks.

Lastly, I should note that my own narrow conception of my work is not exactly accurate: I’ve done or supervised projects in human neuroscience/neurology, physics of granular media, plant/pollinator networks, collective behavior of swarms, and elsewhere. I’m currently working with researchers on problems in climate science and cancer genetics. I suppose the point is that it doesn’t take a lot of persuasion to get me interested in a good problem.