Research Highlights
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Multiscale Representation of Macromolecular Motion
May 2011How many variables do we need to describe macromolecular motion? The answer depends on the time and length scale of interest. By analyzing the geometry of the space defined by long molecular dynamics trajectories of macromolecular systems, we find that different regions of the configurational space have different local intrinsic dimension and […]
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Secret Identity of Supercharacters
May 2011The AIM workshop “Supercharacters and combinatorial Hopf algebras,” which took place in May 2010, solved one of the open problems in supercharacter theory and resulted in a paper authored by all 28 of the workshop participants. The character theory of p-groups, such as U(n), the group of n x n uppertriangular matrices […]
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Inverse Problems and Invisibility
April 2011In inverse problems one probes an object with a particular type of field and measures the response. From these measurements one aims to determine the object’s properties or geometrical structure. Typically, the interaction is restricted to a bounded domain with boundary: a part of the human body, the solid earth, the atmosphere, an airplane, etc. […]
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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 […]
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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, L-functions (and with these […]
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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 non-inflammatory nature of AAM, the lung can be considered as an immunosuppressive organ for respiratory pathogens. However, as infection progresses […]
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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 computer-controlled safety system this would most likely lead to […]
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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. […]
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Algebraic Geometry, Space-time, and Boxes
June 2009How 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 […]
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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): […]