Real-life NUMB3RS
In December 2005, while the IMA was in the midst of a thematic program on imaging science, a homicide investigator from Richmond, Virginia, contacted the institute about an unsolved murder. The best clue available seemed to be a service station security camera video showing the perpetrator fleeing the crime scene in a car. Unfortunately the quality of the video was terrible, and it seemed...
Mind-Bending Math
When viewed from the outside, a human brain appears as a volume with a highly wrinkled surface having numerous long crevices. Sulcal fundi are 3D curves that lie in the depths of the cerebral cortex; informally, the fundus of a sulcus is the curve of maximal average depth that spans the length of the sulcus. The sulcal fundi serve as anatomical landmarks, ‘segmenting’ the cortex into...
Symmetry in 248 Dimensions
A Calculation the Size of Manhattan Mathematicians have mapped the inner workings of one of the most complicated structures ever studied: the object known as the exceptional Lie group E8. This achievement is significant both as an advance in basic knowledge and because of the many connections between E8 and other areas, including string theory and geometry. The magnitude of the calculation is...
Neuronal Network
Modeling the Dynamic Range of a Neuronal Network for Breathing For humans and other mammals, breathing is essential to life. The breathing rhythm relies on an area of the brain stem known as the pre-Bötzinger complex, a network of neurons exhibiting rhythmic bursts of activity that initiate inspiration. The frequency of the rhythm varies in response to such challenges as exercise, sleep, or...
New Tools for Old Problems
When mathematicians notice connections between two distinct areas of their discipline, you can be sure something interesting will develop. That is exactly what happened at a recent workshop at the American Institute of Mathematics (AIM). The scientific advisory board of AIM noticed that new methods in ergodic theory, a subject arising from mathematical physics, may have applications in other...
Honeybee Olfaction
A honeybee may forage on 1,000s of flowers for nectar and pollen in its lifetime. Scent is one of the primary means that it uses for identifying rewarding flowers. How honeybees and other animals learn to associate complex and variable scents with important events is still not well understood. Honeybees are an excellent model system for studying olfaction because their physiology and behavior...
Low X-ray Exposure Imaging
The ability to “see” inside a human body has been of tremendous value to medical diagnostic. For example, it is almost impossible to properly diagnose brain tumor without being able to image the patient’s head, and very difficult to plan the complex surgery of removing the tumor without accurate three-dimensional visualization of the brain itself. A prevalent method to see our way inside a...
Stratification Learning
Data in high dimensions is becoming ubiquitous, from image analysis and finances to computational biology and neuroscience. This data is often given or represented as samples embedded in a high dimensional Euclidean space, point cloud data, though it is assumed to belong to lower dimensional manifolds. Thus, in recent years, there have been significant efforts in the development of methods to...
Homological Mirror Symmetry
During the academic year of 2006-2007 the School of Mathematics conducted a special program in algebraic geometry. This subject, with deep classical roots, is one of the most active areas in contemporary mathematics. Especially notable are its interconnections with number theory, mathematical physics and topology. The scientific activities during the year reflected the depth and breadth of...
Traffic Gridlock
In studying traffic flow in a large city grid, the following questions arise. Is there a largest traffic density that permits free flow? Is there a density above which gridlock is inevitable? In 1992 Biham, Middleton and Levine introduced a simplified model for the study of these questions, called the BML model. This simplified model proved to be surprisingly complex – in a series of papers...