Biological macromolecules function at highly disparate scales in both space and time. These differences in scale imply that mathematical and numerical descriptions must be developed to focus on the particular scale of the phenomenon of interest. For example, one experimental motif for studying the sequence dependent mechanical properties of DNA is mini-circles which are formed from a closed loop of 100--200 base-pairs of the double helix. Numerical simulation of such systems at full atomistic resolution is still beyond current day capabilities, but as this image portrays, an approximate, effective model based on a highly twisted elastic ribbon can be developed to capture sequence-dependent effects. Image by J.H. Maddocks, R.S. Manning, and R.C. Paffenroth.
Introduction During the academic year 2004-2005, the IMA will host a program aimed at a synthesis of the problems at the interface between mathematics, materials science, condensed matter physics, and biology. We believe that a program on Mathematics and Matter is highly timely as the traditional barriers between the fields mentioned above are slowly disintegrating, providing rich interdisciplinary opportunities for the interplay between mathematics and the more traditional disciplines which have involved the study of matter. We will focus on phenomena that require modeling that integrates the atomic to the continuum scales. Though we acknowledge that much can change in the years between now and the beginning of this program, we have targeted several representative topics which appear to provide particularly exciting opportunities.
A remarkably broad spectrum of modern mathematics is being utilized to understand matter. Several of the most active research efforts in nonlinear partial differential equations have been motivated by the need to model the structure and dynamics of defects and microstructure. Current research in stochastic differential equations is being driven by the need to model microscale and nanoscale devices and phenomena. Topological and geometric concepts are being developed to understand defects in crystals, the structure of DNA, protein folding, and the topology of cellular organisms. Symmetry and representation theory are being utilized to identify order parameters in complex materials and to study phase transitions.
Scientific computation is playing a major role in the development of materials theory and its validation by experiment. Mathematicians and materials scientists are beginning to confront the computational challenges of multiscale modeling, singularities, and disorder. However, contemporary computational algorithms for the study of matter, such as "hyperdynamics," are often developed in the context and language of physical theories that are not part of the traditional education of computational mathematicians. This IMA program will strive to enable interactions and research between mathematicians and materials scientists on such computational problems that are important in the study of matter, but have been given little attention by the mathematics community.
Organization Our goal in the IMA Program is to facilitate the development of multi-disciplinary research efforts for which mathematics can play a role at the cutting edge of research in matter. We believe that the major themes of multiscale modeling, singularities, and disorder must be studied in the context of specific physical and biological systems. We think that our goals for this program can be best achieved by the organization of several focused research groups composed of both senior and junior members that will investigate a physical or biological system starting from elementary aspects and progressing to research projects. During the first weeks of the Fall semester, each IMA post-doc and long-term visitor will join one of more focus groups. We expect that each focus group will meet regularly and will invite short-term visitors to discuss current developments.
Each focus group will organize a workshop in which leading experts will be invited to the IMA for several days to participate in a forum to discuss current theories, identify research opportunities, and make connections to related physical and mathematical theory. We will also welcome the participation of mathematicians and scientists who are not expert in the focus area. The format for the workshops will be structured to faciliate active discussion and exploration. There will be a few general lectures, and then participants will take turns leading the discussion by describing their own work or other ideas that intrigue them. Several members of this organizing committee have participated in very successful workshops with this format.
During the Spring semester, the focus groups will continue to meet, but in addition periods of concentration will be organized on the general themes of multiscale modeling, singularities, and disorder. We will organize new focus groups to investigate these general themes and each focus group will organize a workshop on its theme with the participatory format described above for the Fall semester.
In addition to the focus areas descibed below, we will invite one or more scientific leaders to organize other focus groups during the next year as research opportunities are identified. To maintain the vitality of the research groups and the direction of the post-docs, we expect the focus group organizers to be resident at the IMA for long-term visits.
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