Research and Education in Physical Mathematics

该奖项是根据2009年美国复苏和投资法（公法111-5）资助的。研究者和他的同事们研究了应用数学中的一系列问题，重点是在研究和教育的背景下，分析方法与数值计算的耦合。提出的研究项目涉及工程和科学领域的一系列问题，采用近似分析、计算和实验相结合的方法。具体项目包括：(1)通过关注由涡旋细丝组成的初始数据，寻找欧拉方程的潜在不稳定奇异解；(2)微流体力学方面的问题，包括开发创造薄聚合物纤维的新方法，以及创造真菌孢子射手的仿生模拟物；(3)生物学问题，主要目的是利用理论和数据分析来了解达尔文雀的形态变异。教育活动围绕着如何教授应用数学概念这一普遍问题展开，考虑到计算的普及。采取了两项具体举措：第一项涉及基于维基的应用数学教学方法和开发应用数学教材；第二项是开发材料，向广大读者解释数学的使用方式和影响我们的世界的方式。这个项目的活动使用数学和数学思想来研究，并希望解决对我们生活的世界有广泛影响的问题。一组主题涉及关于流体流动性质的基本未解决问题。流体流动贯穿于整个工程和科学，了解如何预测其特性对于关键的社会问题和需求至关重要，例如能源，环境材料和工程。该项目开发了利用流体流动创造新材料和材料工艺的方法。该项目的另一部分旨在理解生物学中的一个基本问题，即，导致复杂生物体（如我们自己）发育和生长的生物过程背后的逻辑是什么？这部分项目基于研究者和他的同事们的一项发现，该发现定量地解释了达尔文雀的形态多样性，这是适应性辐射的经典例子——一个单一物种产生利用不同生态位的多个物种的过程。研究者将这项工作扩展到其他鸟类物种，并利用所获得的见解创建一个数学框架，以更广泛地理解发展。最后，该项目的核心是教育本科生、研究生和博士后如何有效地应用数学，使其能够在我们的世界中产生实质性的变化。

BrennerDMS-00907985 This award is funded under the American Recovery andReinvestment Act of 2009 (Public Law 111-5). The investigatorand his colleagues study a range of problems in appliedmathematics, focusing on the coupling of the analytical methodsto numerical computation, in the context of both research andeducation. The proposed research projects address a range ofproblems in engineering and science using a combination ofapproximate analysis, computation, and experiments. The specificprojects include (1) a search for potentially unstable singularsolutions of the Euler equation, by focusing on initial dataconsisting of vortex filaments; (2) questions in microfluidmechanics, including developing novel methodologies for creatingthin polymeric fibers, and creating a biomimetic analogue of afungal spore shooter; (3) biological questions, primarily aimedat using theory and data analysis to understand morphologicalvariation in Darwin's Finches. Educational activities centeraround the general issue of how the concepts of appliedmathematics should be taught, given the prevalence ofcomputation. Two specific initiatives are undertaken: The firstinvolves a wiki-based method for teaching applied mathematics anddeveloping applied mathematics texts; the second involves thedevelopment of materials for explaining to broad audiences theways in which mathematics is used and has been used to impact ourworld. The activities of this project use mathematics andmathematical ideas to study, and hopefully solve, problems withbroad impact on the world in which we live. One set of topicsinvolves basic unsolved questions about the nature of fluidflows. Fluid flows occur throughout engineering and thesciences, and understanding how to predict their properties iscritical for critical societal problems and needs, such asenergy, environment materials, and engineering. The projectdevelops methods for using fluid flows to create novel materialsand material processes. Another part of the project aims tounderstand a fundamental question in biology, namely, what is thelogic behind which biological processes lead complex organisms,such as ourselves, to develop and grow? This part of the projectis based on a discovery of the investigator and his colleaguesthat quantitatively explains the morphological diversity ofDarwin's Finches, the classical example of adaptive radiation --the process in which a single species gives rise to multiplespecies that exploit different ecological niches. Theinvestigator extends this work to other bird species and uses theinsights gained to create a mathematical framework forunderstanding development more broadly. Finally, central to theproject is the educational mission of teaching undergraduates,graduate students and postdoctoral fellows how to effectivelyapply mathematics so that it can make a substantive difference inour world.