Shedding Light on Optics
What you see is what you get… except when the thing you are looking at is either very small or very far away (all a matter of perspective). Take, for instance, the light from a street lamp projected through a hole the size of a fork tine in a screen. If you place another screen just behind the hole what you will see is an image of the hole. But as you move the second screen further and further...
Describing protein motions with nonlinear dimensionality reduction
Twenty years ago, commenting on macromolecular dynamics, Francis Crick wrote that “what seems to physicists a hopelessly complicated process may have been what Nature found simplest.” Indeed, if one uses molecular dynamics simulations (that follow the molecular motion of biological systems in the high dimensional Cartesian space spanned by each and all atomic degrees of freedom) the behavior of...
Therapeutic Use of Oxygen for Chronic Wounds
Chronic wounds, like venous leg and diabetic foot ulcers (see right), are wounds that do not heal in a timely fashion, generally two-to-four weeks. Chronic wounds can cause pain and discomfort and also limit mobility. They are often a consequence of a more serious health problem, such as diabetes, heart disease, or blood circulation disorders. Left untreated, chronic wounds can lead to problems...
Math Dream
New Connections of Representation Theory to Algebraic Geometry and Physics Representation theory is formally a branch of algebra which studies algebraic structure of symmetries. Its role in contemporary mathematics is due to the fact – realized through major discoveries of the 19th and 20th centuries mathematics – that the structure of symmetries holds key to understanding basic objects of...
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...