An MRI scan (top and center) reveals the medial prefrontal cortex, a region of the brain that has been implicated in depression and bipolar disease. Because the convolutions of the cortex are as individual as a fingerprint, researchers have used a mathematical transformation (bottom) to map this region onto a flat surface. This allows them to make much more precise comparisons between diseased and non-diseased subjects, and to pinpoint where the differences lie. Credit: Monica K. Hurdal, Department of Mathematics, Florida State University

Many mathematical and statistical problems demand innovative ways of integrating numerous small pieces of information into a larger picture. NSF supports research in a variety of these areas, including:

• Inverse Problems. In the aftermath of an earthquake, scientists are confronted with a plethora of data from numerous seismic recorders scattered around the affected region. The challenge is to combine those measurements to answer key questions: Where was the quake centered and how strong was it?

• Mathematicians can help find the answers by using insights from problems that work in the opposite direction. For example, geologists often use explosive charges or mechanical thumpers to apply a shock at specific locations and record the results with arrays of remote sensors. The combined instrument readings are then used to make deductions about rock structures far below ground. The earthquake problem runs in reverse. That is, the researchers take advantage of many years of accumulated data on geological structure and then derive the location and strength of a shock that could pass through those known rock structures to produce the reported measurements. Because the mathematics required for those conclusions results from reasoning in a somewhat backward fashion--moving from effects toward likely causes--problems of this sort are often called "inverse."
• Image Creation. Medical imaging issues often take the form of inverse problems. A celebrated example is the CAT (computer-assisted tomography) scan, which collates a large number of low-dose X-ray images into a cross-sectional image that doesn’t exist outside the computer. The relevant mathematical technique was developed years before it was needed in radiography. But the computations needed, although theoretically straightforward, were not feasible numerically until digital computers were widely available. Modern mathematical methods allow radiologists to obtain fast, accurate results with less patient exposure to X-rays than ever before.

• A dot or pixel in an X-ray image is largely an estimate of the total density of matter along the straight line through which the X-rays travel from their source to that point. Other technologies for medical imaging, such as PET and MRI, measure different aspects of the body, leading to other kinds of assembly problems that are an active subject of mathematical research.
• Statistics. This field has been dramatically affected by the power of modern computers, which permit ideas that were at best theoretical 50 or 100 years ago to become part of the routine toolkit. One such innovation combines a classic idea from probability theory with a mid-20 th century mathematical technique to produce systems that can determine, for example, which of a number of models is the best fit for a new data set, or how various models fit with each other.

Managing and Modeling Complexity [Next]