Research and Education in Physical Mathematics

建议：DMS-0305873 PI：Michael P. Brenna机构：哈佛大学标题：物理数学的研究和教育摘要该提案涉及应用数学的分析工具与数值计算的耦合，在研究和教育的背景下。该研究通过近似分析和计算的结合解决了几个未解决的科学问题。具体项目包括：（1）研究人员将确定低体积分数沉积物的连续方程，特别注意理解颗粒扩散的正确方法，以及内在噪声在动力学中的作用;（2）研究人员将试图揭示最近观察到的违反疏水表面上无滑移边界条件的起源。特别是研究人员将解决的稳定性和小气泡粘附到固体表面的动态后果，如实验观察到的;（3）研究人员将研究弹性应力的作用，影响生长组织的形态，特别是发展数学模型的形态生长酵母菌落;（4）研究人员将开发任意势中Fokker Plank方程的精确短时解，作为波动环境中动力学数值计算的新算法的手段。教育计划围绕教学“近似艺术”（包括渐近和数值方法）的统一观点，以各级学生为中心。将开发研究生水平和本科生水平的新课程，以教授如何有效地将联合收割机分析与计算结合起来，以理解困难的数学问题。该提案包括两个部分：要调查的研究问题解决当前感兴趣的重要问题，其解决方案可能导致重大的科学和技术进步，对应用数学，材料科学，生物物理学和生物学等领域产生影响。具体的问题范围从开发数学模型来理解流体何时粘在表面上，到开发模拟经历布朗运动的复杂物体的新方法，再到为一组落入流体中的粒子开发适当的数学描述。这些都是基本的问题，其解决方案可以显着推进纳米技术材料的设计和开发。例如，在底层流体中的所有纳米级物体都经历布朗运动;缺乏有效的数值算法来模拟这些物体严重阻碍了计算设计工作。该提案的第二部分是教育性的，重点是开发教授近似艺术的方法（将计算和近似分析结合在一个框架中）。通过找到培训各级学生有效地将联合收割机分析和计算结合起来的方法，他们将受到更好的教育，以解决当今的重要问题。编写的材料将通过网络和传统文本广泛传播。

Proposal: DMS-0305873PI: Michael P. BrennerInstitution: Harvard UniversityTitle: Research and Education in Physical MathematicsABSTRACTThis proposal addresses the coupling of the analytic tools of applied mathematics to numerical computation, in the context of both research and education. The research addresses several unsolved scientific problems through a combination of approximate analysis and computation. The specific projects include: (1) The investigator will determine the continuum equations for a sediment at low volume fraction, with special attention to the proper way to understand particle diffusion, and the role of intrinsic noise in the dynamics; (2) The investigator will attempt to uncover the origin of recently observed violations of the no slip boundary condition on hydrophobic surfaces. In particular the investigator will address both the stability and the dynamical consequences of small bubbles adhering to the solid surface, as observed experimentally; (3) The investigator will study the role of elastic stresses in influencing the morphology of growing tissue, and in particular develop mathematical models for the morphology of growing yeast colonies; (4) The investigator will develop accurate short time solutions to the Fokker Plank equation in arbitrary potentials as a means towards a novel algorithm for numerical computation of dynamics in fluctuating environments. The educational programs center around teaching a unified view of ''the art of approximation'' (including both asymptotic and numerical methods) to students at all levels. New courses at the graduate level and the beginning undergraduate level will be developed to teach how to effectively combine analysis with computation, for understanding difficult mathematics problems. The proposal consists of two parts: the research questions to be investigated address important problems of current interest, whose solution could lead to significant scientific and technological advances, with impact on fields ranging from applied mathematics, to materials science, to geophysics and biology. The specific problems range from developing mathematical models for understanding when fluid sticks to surfaces to developing new ways of simulating complex objects undergoing Brownian motion to developing a proper mathematical description for a set of particles falling in a fluid. These are all fundamental questions whose solution could significantly advance the design and development of materials for nanotechnology. For example, all nano-scale objects in an underlying fluid undergo Brownian motion; the lack of an efficient numerical algorithm for simulating these objects severely hampers computational design efforts. The second part of the proposal is educational, focusing on developing methodologies for teaching the art of approximation (combining computation and approximate analysis in a single framework). By finding ways of training students at all levels to effectively combine analysis and computation, they will be better educated to address the important questions of today. The materials that are developed will be broadly disseminated, both through web-based and traditional texts.