Computational Methods in Mathematics and the Physical Sciences

诺曼9626804 研究人员和他的同事在支持现代科学计算和先进计算机图形学的环境中进行跨学科研究和培训。 研究领域涉及偏微分方程，可积系统，几何和拓扑学中的变分问题。 具体专题包括：自旋结构表面上的应用几何浸入，特别是刚性和周期问题;表面，最小化弯曲能受到自然约束;模空间的恒定平均曲率嵌入和浸入，使用力学，辛几何和可积系统;建设嵌入极小曲面的desingularizing浸入极小曲面;弹性曲线及其高阶孤立子类似物;与结和链接相关的静电能，以及它们的梯度流行为;不平衡中不同热力学相之间的曲线和表面界面;以及不可压缩流体流动的欧拉方程的统计平衡解。 几何，分析，数值图形中心（GANG）在支持现代科学计算和先进计算机图形的环境中进行跨学科研究和培训。 GANG的数学工作是由许多有趣的自然系统所激发的，这些系统出现在“应用”和“基础”研究中。 例如，长分子（如蛋白质、DNA或合成聚合物）的折叠和缠结可以使用最初在GANG开发的弹性和静电曲线能量进行动态建模，以研究结和链接的几何形状和拓扑结构;这些模型可能有助于预测聚合物材料的强度或药物的生化效应。 法国物理学家在1995年8月出版的《科学》杂志上报道了一个相当出乎意料的基本新现象--微小磷脂囊泡的“共形扩散”--这是由GANG对表面弯曲能的研究所激发的。 GANG实验室的计算和可视化设施承担着重要且相互关联的三重职责，首先允许进行开创性的数学实验，为学生的教育和培训提供肥沃的环境，并帮助与公众沟通（解释已故物理学家尤金维格纳）数学在自然世界中的显着效果。

Norman 9626804 The investigator and his colleagues pursue interdisciplinary research and training in an environment supporting modern scientific computation and advanced computer graphics. The areas of investigation involve partial differential equations, integrable systems, and variational problems in geometry and topology. Particular topics include: spin structures on surfaces with applications to the geometry of immersions, particularly to rigidity and period problems; surfaces that minimize bending energy subject to natural constraints; the moduli spaces of constant mean curvature embeddings and immersions, using mechanics, symplectic geometry and integrable systems; construction of embedded minimal surfaces by desingularizing immersed minimal surfaces; elastic curves and their higher order soliton analogues; electrostatic energies associated with knots and links, and the behavior of their gradient flows; curve and surface interfaces between distinct thermodynamic phases in disequilibrium; and statistical equilibrium solutions to the Euler equations for incompressible fluid flow. The Center for Geometry, Analysis, Numerics & Graphics (GANG) pursues interdisciplinary research and training in an environment supporting modern scientific computation and advanced computer graphics. The mathematical work at GANG is motivated by - and can shed light upon - many interesting natural systems that arise in both "applied" and "basic" research. For example, the folding and entanglement of long molecules, like proteins, DNA or synthetic polymers, can be dynamically modeled using elastic and electrostatic curve energies that were originally developed at GANG to study the geometry and topology of knots and links; such models may be useful in predicting the strength of polymer materials or the biochemical effect of pharmaceuticals. And a quite unexpected and fundamental new phenomenon - the "conformal diffusion" of tiny phospholipid vesicles, reported by French physicists in the Augu st 1995 issue of Science - was stimulated by work at GANG on the bending energy of surfaces. The computation and visualization facilities at the GANG laboratory serve an important and interrelated triple duty by permitting the pioneering mathematical experiments to be carried out in the first place, providing a fertile environment for the education and training of students, and aiding communication with the general public (to paraphrase the late physicist, Eugene Wigner) on the remarkable effectiveness of mathematics in the natural world.