simple fashion, through a shifting factor. Analogous diagrams exist for the nth monochromatic layer. They are also symmetric and have known shifting factors.
Nevertheless, this offers no more than a partial solution to the basic problem, which again is to list all spaces that are homotopy equivalent. Michael Hopkins of the Massachusetts Institute of Technology notes that even within the Shimomura diagram ''there are many suggestive patterns, but there is no really good theory. One of the most important problems in homotopy theory today is to find a theory that predicts the qualitative features of this diagram, and its analogs for the higher chromatic layers."
Another topic in topology involves knots and unknots. An unknot is a simple closed curve in three-space that can be deformed into a round circle. Alan Hatcher has shown that the deforming or untangling is continuous and involves no choices. A knot lacks this property; it cannot untangle into the simple loop of a circle.
Michael Freedman and Zheng-Xu He define the energy of a closed curve, E, by introducing a 1/r2 potential. Two similar curves have equal energy, and if two strands come close, as if to cross, then E blows up. A curve then can untangle by following the gradient of E. Freedman and He find a theorem: If E < C, where C is a constant, the curve is an unknot. C > 22, but C is not well known; Freedman says it could be about 70. For a round circle, E = 4; hence, only unknots exist for 4 < E < C.
A related issue is the average number of crossings that a curve makes, when one projects its shape onto a randomly oriented plane. If the curve forms a knot, the number is at least 3; you can see this with a loop of string. Both for knots and unknots, Freedman and He find that this number has an upper bound in terms of energy: 11E/12π + 1/π. This theorem draws on the work of Gauss.
The energy criterion for an unknot recalls a similar result due to John Milnor. Milnor defines a total curvature of a loop, T. For a round circle, T = 2π. Milnor has shown that for T < 4π, the loop or curve is an unknot. This then raises a question: In addition to the energy and total curvature, do other integral quantities exist whose values distinguish a knot from an unknot?
In semiconductor physics a significant topic involves preparation of materials in unusually small volumes and thin layers and using them to fabricate new devices. At AT&T Bell Labs, Louis Brus has directed