Workshop on Geometry and Symmetry in Numerical Computation

现代计算科学中一些最令人兴奋的发展来自于利用传统上与数值计算无关的数学领域的思想。相反，数值技术已被应用于解决“非传统”领域中出现的计算问题。一个很好的例子是计算数学和代数几何之间富有成效的相互作用。奇点理论建立在代数几何的基础上，已经被计算科学家用来计算微分和偏微分方程的多参数系统中临界点的路径。对称性和群作用用于创建特定类型问题的数值方法，具有显着提高的精度和稳定性。代数几何中的技术对于在流形上求解微分方程也非常有用，目前正被应用于开发计算高阶张量分解的算法。另一方面，连续性、同伦和对称性的数值技术为数值代数几何中用于计算多项式方程组的解分量的方法提供了基础。进行多项式系统的数值分解的能力已经在机械工程中产生了应用，包括传输，控制或约束相对运动的机制的理解和设计，机器人，控制理论（极点配置），整数规划和统计。混合精确/近似方法的发展，找到解决方案的多项式方程引起的问题的错误和稳定性，在许多其他情况下面临的数值分析。关于数值计算中的几何和对称性的研讨会将汇集计算数学和代数几何的专家，以探索和开发这一丰富的跨学科领域的潜力。 该方案已专门规划，以介绍和吸引学生和年轻的调查人员到这一领域。每一次会议将开始与介绍性讲座，然后由领先的专家四个会谈。介绍性演讲者将准备一个简短的“指南”，描述一些基本的语言和结果，听众可以在邀请的演讲中使用。一个小时的讲座本身将针对来自不同数学领域的高级研究生和研究人员的观众。我们期望研讨会打破学科障碍，鼓励代数几何和数值分析研究人员之间的交叉施肥。讲座和讨论部分将鼓励学生和年轻的研究人员参与代数几何和数值分析的交叉，并将为那些已经从事这一活动的人提供刺激和支持。潜在的成果范围从改进的方法来计算由偏微分方程系统控制的大型复杂物理系统，到广义相对论计算方法的进步，到分析大型数据集的新几何方法，以及更有效的机器人和控制数值方法。

Some of the most exciting developments in modern computational science have resulted from exploiting ideas in areas of mathematics not traditionally associated with numerical computation. Conversely, numerical techniques have been applied to solve computational problems arising in "non-traditional" fields. A good example is the fruitful interaction between computational mathematics and algebraic geometry. Singularity theory, which builds on ideas of algebraic geometry, has been embraced by computational scientists to compute paths of critical points in multi-parameter systems of differential and partial differential equations. Symmetries and group actions are used to create numerical methods for specific types of problems with significantly improved accuracy and stability properties. Techniques in algebraic geometry are also very useful for finding solutions of differential equations on manifolds, and are currently being applied to develop algorithms to compute decompositions of higher order tensors. On the other hand, numerical techniques for continuation, homotopy and symmetry provide the basis for methods in numerical algebraic geometry that are used to compute solution components of systems of polynomial equations. The ability to carry out the numerical decomposition of polynomial systems has yielded applications in mechanical engineering including the understanding and design of mechanisms that transmit, control, or constrain relative motion, robotics, control theory (pole placement), integer programming, and statistics. The development of hybrid exact/approximate methods for finding solutions of polynomial equations gives rise to issues of errors and stability that confront numerical analysts in many other contexts. The Workshop on geometry and symmetry in numerical computation will bring together experts from computational mathematics and algebraic geometry in order to explore and develop the potential in this rich interdisciplinary area. The program has been planned specifically to introduce and attract students and young investigators to this area. Each session will begin with an introductory lecture followed by four talks by leading experts. The introductory speakers will prepare a short "guide" describing some basic language and results that the audience can use during the invited talks. The one hour lectures themselves will be aimed towards an audience of advanced graduate students and researchers from different areas of mathematics. We expect the Workshop to break down disciplinary barriers and encourage cross-fertilization between researchers from algebraic geometry and numerical analysis. The lectures and discussion sections will encourage students and young researchers to become involved in the intersection of algebraic geometry and numerical analysis, and will provide stimulation and support for those already engaged in this activity. Potential outcomes range from improved methods to compute large complex physical systems governed by systems of partial differential equations, to advances in computational methods for general relativity, to new geometric methods for the analysis of large data sets, and to more efficient numerical methods for robotics and control.