Conference: Variational Methods: Open Problems, Recent Progress, and Numerical Algorithms, June 11-14, 2002, Northern Arizona University

这笔赠款将支持2001年6月11日至14日在亚利桑那州弗拉格斯塔夫的北亚利桑那大学举行的关于变分方法：开放问题、最新进展和数值研究的会议。标题中的术语变分方法指的是偏微分方程的丰富领域和历史上用来研究它们的一系列方法。许多著名的数学家在上个世纪对这一领域做出了重大贡献，但人们普遍认为，新一轮发现的时机已经成熟。这些问题的解决通常依赖于来自广泛数学领域的工具，并可能对重要物理应用的研究产生重大影响。这次会议的目的是聚集尽可能多的这一研究领域的专家，以便讨论和编目该领域最重要的(未解决的)问题以及可能用来解决这些问题的方法。数值研究(使用计算机来可视化和近似解决方案及其性质)在数学界获得了尊重，但在现有的研究世界中，许多人还没有充分认识到它们的力量，许多真正欣赏这种力量的人也还无法接触到它们。这种现代方法将通过数字讲座和动手技术演示与传统的(非计算机相关的)讨论相结合。会议将特别关注物理应用，以促进我们对现实世界科学的贡献，并为我们的努力提供新的灵感来源。这次会议的驱动力是希望为那些在该领域积极研究的人和刚刚开始他们职业生涯的年轻数学家提供一个高度可用的、事实上必不可少的参考。这两个群体都将受益于拥有一个优秀的研究问题的单一简洁目录，以及对最富有成效的相关数学工具的介绍和详尽的参考书目。一家出版商(美国数学学会当代数学杂志)已经暂时接受了出版这篇论文的提议。这次会议将鼓励非正式讨论我们这个领域的未来，它对科学做出重大贡献的潜力，以及如何最好地将所有这些信息放在一个论文集中。讨论的研究重点将是非线性椭圆型偏微分方程组，变分方法的关键应用。一般说来，非线性方程对于描述物理现象是至关重要的，而椭圆型方程(粗略地说)描述了其中的三分之一。生物质量的有效获取、恒星诞生和死亡的模型以及支配量子物理的基本方程都依赖于椭圆型方程。这一研究领域的重大进展有可能在科学和经济上对整个世界产生真正的影响。这样做的目的是激发人们对解决这类问题的兴趣和努力再次活跃起来。会议和相应的论文集将具有同等的研究和教育导向。初级研究人员将接触到该领域的物理重要性的数学研究，将为他们的追求提出未解决的问题，并将介绍对该研究至关重要的分析和数值工具。这些参加的新研究人员，或者更广泛地说，能够接触到由此产生的论文集，从长远来看将是关键的贡献者。这样做的目的是鼓励进行真正的数学计算的非正式讨论，但指导这一过程的重点是制作一份连贯的文件的最终目标。

0124121NeubergerThis grant will support a conference on Variational Methods: Open Problems, Recent Progress, and Numerical Investigations to be held June 11-14, 2001 at Northern Arizona University in Flagstaff, Arizona. The term Variational Methods in the title refers to a rich area of partial differential equations and a family of methods historically used to investigate them. Many famous mathematicians have made significant contributions in the last century to this area, but it is widely held that the time is ripe for a new round of discoveries to be made. Solutions of these problems typically rely on tools from a wide range of mathematical areas and potentially have a significant impact on the study of important physical applications. The purpose of this conference is to assemble as many experts as is possible in this research area, in order to discuss and catalog the area's most important open (unsolved) problems and the methods that might be used to attack them. Numerical investigations (using a computer to visualize and approximate solutions and their properties) are gaining respect in the mathematical community, but their power is not yet fully appreciated by many in the established research world, nor are they yet accessible to many of those who do appreciate that power. This modern approach will be integrated with traditional (non computer-related) discussions via numerical lectures and hands-on technological demonstrations. Special attention will be given to physical applications, in order to facilitate our contributing to real-world science and to provide a source of renewed inspiration for our endeavors.The driving force for this conference is a desire to produce a highly usable, in fact essential, reference for those actively researching in the area and for young mathematicians just starting their careers. Both groups will benefit greatly from having a single succinct catalog of good research problems together with an introduction to the most fruitful relevant mathematical tools and an exhaustive bibliography. A publisher (American Mathematical Society Journal of Contemporary Mathematics) has already provisionally accepted a proposal for publishing this document. The conference will encourage informal discussion about the future of our field, the potential for it to make significant contributions to the sciences, and how best to put all this information in a single proceedings volume.The focus of the discussed research will be on nonlinear elliptic partial differential equations, a key application of variational methods. Nonlinear equations in general are vital to the description of physical phenomena, and elliptic equations describe (roughly speaking) one third of these. The efficient harvesting of biological mass, modeling of star birth and death, and the fundamental equations governing quantum physics all rely on elliptic equations. Significant progress in this research area has the potential to make a real difference, scientifically and economically, to the world at large. The aim is to trigger a vibrant renewed surge of interest and effort towards solving these types of problems. The conference and corresponding proceedings volume will be of equal parts research and educationally oriented. Beginning researchers will be exposed to research in mathematics to the physical importance of the field, unsolved problems will be suggested for their pursuit, and analytical and numerical tools essential to that research will be introduced. These new researchers who attend or, more generally, have access to the resulting proceedings volume, will in the long run be the key contributors. The aim is to encourage informal discussion, where real mathematics is done, but guide the process with a focus on the ultimate goal of producing a cohesive document.