Travel Time Tomography
Inverse Problems are problems where causes for a desired or observed effect are to be determined. They arise in all fields of science and technology. An important example is to determine the density distribution inside a body from measuring the attenuation of X-rays sent through this body, the problem of “X-ray tomography.” Two types of inverse problems have intimate connections with a...
When The Volcano Blows
“I don’t know where I’m gonna go when the volcano blows”. So wrote Jimmy Buffet. The more important challenge for volcanologists, however, is to determine “When should we go before the volcano blows!” Tackling this challenge was a principal focus of a group¹ of statisticians, mathematicians and volcanologists at SAMSI last year. The Soufriere Hills Volcano (right, April 2005) on the island of...
A New (Math) World
A new mathematical object was revealed during a lecture at the American Institute of Mathematics (AIM). Two researchers from the University of Bristol exhibited the first example of a third degree transcendental L-function. These L-functions encode deep underlying connections between many different areas of mathematics. The news caused excitement at the AIM workshop attended by 25 of the...
Better Seismic Imaging
The economic value to the oil industry of 3D seismic imaging is approximately $11 billion annually. How accurately seismic imaging can be done depends on both the quality of the sensing equipment, but also very much on the effectiveness of the mathematical algorithms that are used. Hence it is an important event when seismic imaging algorithms are improved. Data: 80% missing traces...
Sparse Representations
What do researchers studying infrared spectroscopy, seismic imaging, error correcting codes, and MRI’s have in common? They all can get better results if they have the right math. Fourier analysis, discovered in analyzing the flow of heat, revolutionized how mathematics is used in a variety of sciences by breaking complicated functions into a sum of simple parts. Of course, it is even better if...
EvoGeo
Can polyhedral geometry and commutative algebra—usually regarded as pure mathematics—help biologists? We at the IMA certainly think so, and the emerging applications of these mathematical areas to evolutionary biology was a major theme during a workshop bringing 135 mathematicians, statisticians, biologists, and computer scientists to the IMA in March 2007 as part of our year-long thematic...
Spatial Model for Rabies
A human bitten by an animal with rabies will almost certainly die within days unless immediately treated with a multi-stage vaccine regimen initially developed by Louis Pasteur in 1885. To combat the spread of rabies today, which nationwide affects over 40,000 people annually and countless wild and domestic animals, the United States spends over $300 million each year on its prevention and...
Alimentary Math
Meat and soybeans are two important food sources in many parts of the world, and in the form of biodiesel fuel, soybeans are a promising source of renewable energy. Two groups of long-term visitors to the IMA during our 2005-2006 program on imaging are contributing to the improved production of these important foodstuffs. For beef, the rib eye area is an important indicator of the meat quality...
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...