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.
