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

乔恩·皮茨将继续用变分技术研究极小曲面。这是一个目前正在经历戏剧性发展的领域。皮茨将把注意力主要集中在三维流形中的极小曲面上，而不是欧几里得空间。这项研究的计算方面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，与其他研究人员的当代工作非常一致。三维流形中极小曲面的研究是瑟斯顿几何化猜想的一个重要方面。这里的研究将涉及到构造可容纳无限多个极小球面和环面的度量的可能性。该项目的另一部分将致力于寻找非紧双曲流形上的条件，以保证它支持紧致极小曲面。皮茨还在开发逼近极小曲面的数值方法。他将在软件中实现这些算法，并生成表面的立体图像的图形。可视化这些表面的能力正在迅速成为当前对这些变分问题的研究的重要组成部分。

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.