Research Highlights

Smoothing Surfaces with Corners
August 2010While visiting the IMA in early 2001, Tom Duchamp of University of Washington, a geometer who works in computer graphics, attended a talk by postdoc Selim Esedoglu on image denoising. He wondered if the denoising method that Esedoglu described, which preserves image edges, could be modified to smooth surfaces that have edges and corners. The […]

It’s All in the Symmetries
August 2010Analytic and Geometric Number Theory The tools used in modern number theory range over many fields from algebra and algebraic geometry to harmonic analysis, representation theory, ergodic theory and combinatorics. Central to the study of many problems concerning prime numbers, diophantine equations and number fields, are sieve methods, Lfunctions (and with these […]

Flipping a Switch
February 2010Fighting TB Might be a Matter of ‘Flipping a Switch’ in Immune Response In the lung, alternatively activated macrophages (AAM) form the first line of defense against microbial infection. Due to the noninflammatory nature of AAM, the lung can be considered as an immunosuppressive organ for respiratory pathogens. However, as infection progresses […]

Next Generation Auto Safety Systems
August 2009Imagine you are driving at night on a twisty mountain road. A deer jumps out of nowhere ahead of you. You turn your car to avoid it, and all four wheels of your car hit an icy patch. You instinctively slam on the brakes. Without a computercontrolled safety system this would most likely lead to […]

The Geometry of PDEs
August 2009In 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, Spacetime, and Boxes
June 2009How can piles of boxes help us understand the structure of spacetime? Algebraic geometry—the subject of MSRI’s Spring 2009 scientific program—provides the link, through the notions of moduli spaces and DonaldsonThomas invariants. String theory predicts that spacetime is 10dimensional: Four of the dimensions are the usual three of space and one of […]

Bregman Iteration and Applications to Imaging and Sparse Representation
January 2009A 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): […]

Modeling Placental Growth and Structure
January 2009Early 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. […]

Voronoi Diagram of Ellipses
August 2008Innumerable 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 […]

Shedding Light on Optics
August 2008What 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 […]