Research Highlights
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Bregman Iteration and Applications to Imaging and Sparse Representation
A breakthrough in the sparse representation of data was made in the Fall of 2004 by David Donoho and by Emmanuel Candes and Terence Tao, in part at IPAM’s Multiscale Geometry and Analysis program. Their important results reduce a computationally difficult problem to an easily stated, convex minimization problem (the basis pursuit problem): […]
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Modeling Placental Growth and Structure
Early life influences adult diseases. Such serious public health priorities as diabetes, heart disease, breast and prostate cancer, osteoporosis, and depression have all been related by recent medical studies to the development of the fetus. Environmental influences, such as maternal diseases, can cause irreversible metabolic consequences for the fetus, altering susceptibility to later adverse outcomes. […]
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Voronoi Diagram of Ellipses
Innumerable scientific and engineering applications from geology to telecommunications lead to the same problem: given a set of objects, find which object is nearest to any particular point. The Voronoi diagram answers this question for every point on the plane by dividing the plane into cells containing the points nearest each object. Because this question […]
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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 […]
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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 […]
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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. […]
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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 […]
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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.” […]
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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 […]
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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. […]