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

会议：变分方法：开放问题、最新进展和数值算法，2002 年 6 月 11-14 日，北亚利桑那大学

• 批准号: 0124121
• 负责人: John Neuberger
• 金额: $ 1万
• 依托单位: Northern Arizona University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-03-15 至 2003-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0124121&HistoricalAwards=false
• 关键词: Conference; Variational; Methods; Open; Problems

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.

0124121 Neuberger这笔赠款将用于支持将于2001年6月11日至14日在亚利桑那州弗拉格斯塔夫的北方亚利桑那大学举行的变分方法：开放问题，最新进展和数值研究会议。标题中的变分方法一词指的是偏微分方程的丰富领域和历史上用于研究它们的一系列方法。在上个世纪，许多著名的数学家在这一领域作出了重大贡献，但人们普遍认为，进行新一轮发现的时机已经成熟。这些问题的解决方案通常依赖于广泛的数学领域的工具，并可能对重要的物理应用研究产生重大影响。这次会议的目的是聚集尽可能多的专家在这个研究领域，以讨论和目录该地区最重要的开放（未解决）的问题和可能用于攻击他们的方法。数值研究（使用计算机来可视化和近似解及其性质）在数学界越来越受到尊重，但它们的力量还没有被许多既定的研究世界完全理解，也没有被许多欣赏这种力量的人所理解。这种现代方法将通过数字讲座和动手技术演示与传统（非计算机相关）讨论相结合。特别关注的是物理应用，以促进我们对现实世界的科学做出贡献，并为我们的努力提供新的灵感来源。本次会议的驱动力是希望为那些积极研究该领域和刚刚开始职业生涯的年轻数学家提供高度可用的，实际上是必不可少的参考。这两个小组将大大受益于有一个单一的简洁的目录良好的研究问题连同介绍最富有成效的相关数学工具和详尽的参考书目。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份文件的建议。会议将鼓励非正式讨论我们领域的未来，它对科学做出重大贡献的潜力，以及如何最好地把所有这些信息放在一个单独的会议记录卷。讨论的研究重点将是非线性椭圆偏微分方程，变分方法的关键应用。一般来说，非线性方程对于描述物理现象是至关重要的，而椭圆方程（粗略地说）描述了其中的三分之一。生物质量的有效获取、星星出生和死亡的建模以及量子物理学的基本方程都依赖于椭圆方程。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实的影响。其目的是激发人们对解决这类问题的兴趣和努力。会议和相应的诉讼卷将平等的部分研究和教育为导向。开始的研究人员将接触到数学研究领域的物理重要性，未解决的问题将建议为他们的追求，分析和数值工具必不可少的研究将被引入。这些新的研究人员谁参加，或更普遍地说，有机会获得所产生的会议记录卷，将在长期内的关键贡献者。其目的是鼓励进行非正式讨论，进行真实的数学计算，但指导这一进程的重点是编写一份连贯一致的文件这一最终目标。