The Geometry of PDEs
In the twentieth century, many of the advances in the theory of patial differential equations were the result of studying specific geometric problems; for example, minimal surfaces. The program in Geometric Partial Differential Equations demonstrated that this interaction between geometry and PDE continues to flourish and to illuminate the connections between seemingly unrelated problems....
Algebraic Geometry, Space-time, and Boxes
How can piles of boxes help us understand the structure of space-time? Algebraic geometry—the subject of MSRI’s Spring 2009 scientific program—provides the link, through the notions of moduli spaces and Donaldson-Thomas invariants. String theory predicts that space-time is 10-dimensional: Four of the dimensions are the usual three of space and one of time, and the remaining six are curled up...
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): $$ \min |u|_1 \ \ \text{such that} \ \ Au = f.$$ In...
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...
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 arises in so many applications,...
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...