Editor’s note: This article was initially published in The Daily Gazette, Swarthmore’s online, daily newspaper founded in Fall 1996. As of Fall 2018, the DG has merged with The Phoenix. See the about page to read more about the DG.
As David Mumford, Professor of Applied Mathematics at Brown University and Fields Medalist, was setting up his laptop before Tuesday afternoon’s lecture entitled “What’s an Infinite Dimensional Manifold, and How Can It be Used in Hospitals?”, Professor Walter Stromquist remarked that “infinite dimensional manifold” is a phrase that scares mathematicians, and that was why Mumford added the bit about hospitals, to soften it. Nearby mathematicians agreed that this was so.
But despite the threat of scary mathematical concepts, Mumford’s lecture presented a straightforward and solid introduction to the benefits and difficulties of using mappings of infinite dimensional manifolds – highly precise mathematical analyses of non-linear spaces – in the medical world.
A manifold, essentially, is a nonlinear geometric “thing” which can be described piecemeal by a series of charts which assign coordinates to points on the manifold, creating a one-to-one linear mapping. For example, a globe is a manifold which can be precisely mapped by an atlas.
Two-dimensional manifolds were described by Gauss, but later mathematicians such as Riemann postulated that there might be manifolds for which one set of coordinates would not be enough; rather, to accurately describe the location of a point on the manifold you would need a function that mapped the point continuously over time.
One application of such a process is to chart a path between two similar shapes that shows how they are related. By finding a series of functions that describes the distance between two curves at various points, it is possible to describe the structure of any nonlinear space (such as the difference between two curves) mathematically.
What are the medical applications of these ideas?
Two shapes or forms that can be morphed into one another are said to be “diffeomorphic”, and essentially a healthy person is a diffeomorph of some ideal model human. Therefore, it should be possible to compare an MRI scan, for example, with an ideal model to check for differences. In order to perform such a comparison data must be collected which shows how the patient’s scan relates to the ideal model, and this collection constitutes a multi-dimensional manifold which can be studied.
Using this technique, doctors could study the shape of the hippocampus to diagnose schizophrenia or Alzheimers, or look at the shape of a patient’s heart or prostrate, and so forth.
For doctors who want to analyze this data to find characteristics of patients with diseases, traditional analytical tools will not be enough, since they are intended for use in linear space. Mumford illustrated a variety of analytical procedures that could be performed on manifolds. Chief among these were the use of geodesics (the shortest path from point A to point B on a manifold) to “linearize” these nonlinear spaces.
Geodesics, however, are not without their limitations. One obstacle for people who want to use these techniques both in medicine and elsewhere is the tendency of geodesics to curve unpredictably, leading to distortions and exaggerations. To illustrate how this happens in reality, Mumford showed a slide of the light from a distant galaxy curving around an intervening galaxy, resulting in an image that showed the more distant objects smeared around the outside edge of the picture.
In addition to the problem of curvature, Mumford is working on finding ways to apply manifolds to the field of pattern recognition, and finding probability ratios that make it possible to determine the nature of a shape mathematically.