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

Northern Arizona University, Flagstaff AZ

Investigators

Abstract

0124121 Neuberger This 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.