Mathematical Sciences: Numerical and Theoretical VariationalMethods in the Large and Applications to Geometry and Topology

乔恩·皮茨将继续他对最小表面的研究， 变分技术这是一个目前 经历了戏剧性的发展。皮茨将集中精力 主要关注三维空间中的最小曲面 流形而不是欧几里得空间。计算方面 这一研究为皮茨的研究提供了一个新的方向。 这是一个令人兴奋的发展， 其他研究人员的当代工作。 三维流形中极小曲面的研究 是Thurston几何化猜想的一个重要方面。 这里的调查将涉及的可能性， 构造容许无穷多个极小球面的度量 还有托丽该项目的另一部分将旨在寻找 非紧双曲流形上保证 它支持一个紧凑的极小曲面。皮茨也是 开发用于近似极小曲面的数值方法。 他将在软件中实现这些算法， 表面的立体图像的图形。的能力 可视化这些表面正迅速成为一个重要的 目前对这些变分问题研究的主要内容。

Jon Pitts will continue his study of minimal surfaces by variational techniques. This is an area which is currently experiencing dramatic developments. Pitts will focus his attention primarily on minimal surfaces in three dimensional manifolds rather than Euclidean space. The computational aspects of this research provide a new direction for Pitts' research. This is an exciting development which is very much in line with contemporary work of other researchers. The study of minimal surfaces in three dimensional manifolds is an important aspect of Thurston's geometrization conjecture. Here the investigations will concern the possibility of constructing metrics which admit infinitely many minimal spheres and tori. Another part of the project will be aimed at finding conditions on a noncompact hyperbolic manifold which guarantee that it supports a compact minimal surface. Pitts is also developing numerical methods for approximating minimal surfaces. He will implement these algorithms in software and produce graphics of the stereographic images of the surfaces. The ability to visualize these surfaces is rapidly becoming an essential ingredient of current research into these variational problems.