Mathematical Sciences: Computations for Knotted Surfaces

4维空间中的曲面表现出与3维空间中的圆相似的打结行为，并且在数学上是基本理解4维空间的核心。主要的观点是拓扑学。该程序的基本步骤如下。(1)将结点曲面和这些曲面的等值线表示为数据集。传统的拓扑结构不容易产生公式，因此顶点坐标的计算不是一件微不足道的事情。基于纽结能量概念的计算产生了有趣的、有用的，在某种意义上，在四维空间中纽结曲面的自然放置和运动。第二种方法本质上是局部的，并且基于研究者的高维节点移动的方法。第三种方法涉及交互式计算机图形学。(2)以符号方式表示结点曲面。在4维空间中的一个曲面上，可以关联一个称为纽结图的组合对象。对于这样的图有一个类似于经典结点图的演算，目标是实现这一点。从纽结图可以确定纽结曲面的一些代数不变量。结组的演示就是这样一个例子。使用示例(在上面第(1)部分中产生)，将生成并分析更高维图，以期找到其他组合定义的不变量。(3)开发可视化技术，用于搜索和验证实例之间的数学关系。仅仅“看到”这些物体是不够的，人们想要得到一个数学上的理解。一种这样的新可视化技术使用参数化纹理来为更高维度的信息提供自然的视觉提示。另一种是交互操作4维空间中的物体。目前，研究人员正在开发将CAVE与并行计算结合使用，作为这种四维交互环境的最佳当前工具。基本问题是：什么是四维空间？一种是寻找数学答案，重点是计算和可视化。对四个(或更多)维度的研究一点也不深奥。从数学上讲，维度只不过是一个数字。任何需要有四个独立变量来描述它的东西都可以被认为是四维空间的一个子集。最广为人知的例子就是“事件”。一件事发生在特定的地点和时间。需要三个数字来定位事件的位置，第四个维度是时间。四维与三维有很大不同的原因是它很难可视化。这是本研究的重点。我们最能可视化的东西是表面，所以在四维空间中使用表面作为可视化工具是很自然的。此外，曲面还为4维空间提供了强大的数学探测器。这项调查在很大程度上依赖于最先进的设备。为了解决复杂的计算问题，使用了“超级计算机”。为了解决可视化和操控的复杂性，使用了“虚拟现实”系统。总而言之，这个项目的广泛意义在于扩展了使用现代计算工具对复杂信息进行视觉理解的前沿。

Surfaces in 4-dimensional space exhibit knotting behavior similar to circles in 3-space and are mathematically central to basic understanding of 4-space. The primary point of view is topological. Basic steps in the program are as follows. (1) Represent knotted surfaces and isotopies of these surfaces as data sets. Traditional topological constructions do not readily give rise to formulas, thus calculations for coordinates of vertices is a non-trivial matter. Computations based on the concept of an energy of a knot give rise to interesting, useful, and in some sense, natural placements and motions of knotted surfaces in four dimensional space. A second method is local in nature and is based on the investigator's methods of higher dimensional knot moves. A third method involves interactive computer graphics. (2) Represent knotted surfaces in a symbolic way. To a surface in 4-space one can associate a combinatorial object called a knot diagram. There is a calculus for such diagrams similar to that which exists for classical knot diagrams, a goal is to implement this. From a knot diagram one can determine some algebraic invariants of a knotted surface. A presentation of the knot group is one such example. Using examples (produced in part (1), above), higher dimensional diagrams will be generated and analyzed with the hope of finding other combinatorially defined invariants. (3) Develop visualization techniques for the search and verification of mathematical relationships between examples. It is not enough to "see" these objects, one wants to get a mathematical understanding. One such new visualization technique uses a parameterized texture to provide natural visual cues for higher dimensional information. Another involves interactive manipulations objects in 4-space. Currently, the investigator is developing use of the CAVE in conjunction with parallel computation as the best current tool for such a four-dimensional inte ractive environment. The basic question is: what is four dimensional space ? One is looking for mathematical answers with emphasis on computation and visualization. The study of four (and more) dimensions is not at all esoteric. Mathematically, a dimension is nothing more than a number. Anything that needs to have four independent variables to describe it can be considered to be a subset of four-dimensional space. The most well known example is an "event". An event happens at a certain place and a certain time. Three numbers are needed to locate the position of the event and the "fourth dimension" is time. The reason that four-dimensions is very different than three is that it is difficult to visualize. This is a focus of this research. The things that we visualize best are surfaces, so it is natural to use surfaces in four-dimensional space as a visualization tool. In addition surfaces provide powerful mathematical probes into 4-space. This investigation relies heavily on "state of the art" equipment. To grapple with the complexities of the computation, use is made of "super-computers". To grapple with the complexities of visualization and manipulation, use is made of a "virtual reality" system. In summary, the broad significance of this program is the extension of frontiers of visual understanding for complex information using modern computational tools.