Workshop on Group Theory and Numerical Analysis

这笔赠款将用于支持美国研究人员、博士后和研究生参加2003年5月26日至31日在蒙特利尔大学数学研究中心举办的群论和数值分析研讨会，并在研讨会上发表演讲和海报。研讨会展示了新兴的“几何积分”理论的一个方面，该理论主张将微分方程组的已知定性特征，如不变量、对称性、守恒定律、相空间动力学和辛结构，纳入其数值求解算法中。特别是，微分方程组的对称性已被证明是理解其解的定性性质以及便于确定精确解析解的一种非常强大的手段。这种特殊解通常作为原型，控制着更复杂情况下的渐近性，也是开发数值积分格式的有用的测试基础。最近，微分方程的对称性分析和可积性方法被用于分析离散方程，并取得了相当大的成功。另一方面，在应用中产生的各种微分方程组的不变几何数值格式的设计中引入对称性是最近日益活跃的研究活动和进展的主题。研讨会的主要目的是将群论应用于离散方程的专家和几何数值方法的实践者聚集在一起，从而促进互惠互利的思想交流和最新发展，并促进这些密切相关学科之间的交叉培养。群论是对称性概念的数学形式化。在物理、工程、化学、计算机视觉、生物学等领域的应用中出现的大多数有趣的系统都具有丰富的各种对称性。虽然利用连续系统的对称性已经取得了漫长而杰出的成就，但直到最近才在离散系统中取得了重大进展。事实上，计算机使用的标准数值解方法通常不能适当地尊重基本的对称性。本研讨会将以离散系统对称性分析的最新发展和保持对称性的数值解技术的发展为基础。参加者是对称方法和几何数值积分方面的专家。讲习班将促进这两个学科的互动和重大进展，以及它们在数学和科学中的应用。研讨会的成果如会谈和讨论中所述，将总结成一本书，重点突出，具有永久价值。此外，将在小组讨论和将在网上分发和张贴的白皮书中重点评估这些主题的最新技术状况和由此产生的新的科学机会。

The funds in this grant will be used to support U.S. researchers, postdocs and graduate students to attend and present lectures and posters in a Workshop on Group Theory and Numerical Analysis at the Centre de Recherches Mathematiques, Universite de Montreal, during the week May 26-31, 2003. The workshop features one aspect of the emerging theory of "geometric integration," that advocates incorporating known qualitative features of systems of differential equations, such as invariants, symmetries, conservation laws, phase-space dynamics, and symplectic structure, in their numerical solution algorithms. In particular, the symmetry properties of a system of differential equations have proven to be a very powerful means of understanding the qualitative properties of its solutions as well as facilitating the determination of exact analytical solutions. Such special solutions often serve as prototypes and govern the asymptotics for more complicated situations, and also as useful testing grounds for developing numerical integration schemes. The methods of symmetry analysis and integrability for differential equations have, more recently, been adapted with considerable success to analyze discrete equations. On the other hand, the incorporation of symmetry in the design of invariant geometric numerical schemes for a variety of differential equations arising in applications has recently been the subject of increasing research activity and advances. The key objective of the workshop is to bring experts on applications of group theory to discrete equations together with practitioners of geometric numerical methods, thereby stimulating a mutually beneficial exchange of ideas and recent developments and fostering a cross-fertilization between such closely related subjects.Group theory is the mathematical formalization of the concept of symmetry. Most interesting systems that arise in applications in physics, engineering, chemistry, computer vision, biology, etc., possess a rich variety of symmetries. While exploiting the symmetries of continuous systems has a long and distinguished record of achievement, only recently have significant developments occurred within the discrete regime. Indeed, standard numerical solution methods used by computers typically fail to properly respect the underlying symmetry properties. This workshop will build on recent developments in symmetry analysis of discrete systems and the development of symmetry-preserving numerical solution techniques. The participants are experts in symmetry methods and in geometric numerical integration. The workshop will promote interactions and significant progress in both subjects, as well as their applications in mathematics and the sciences. The results of the workshop, as presented in the talks and discussions, will be summed up in a book to be so focused as to be of permanent value. In addition, an assessment of the current state of the art in the subjects and new scientific opportunities stemming from this workshop will be the focus of a panel discussion and white paper to be distributed and posted on the web.