Topics in low-dimensional topology

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AbstractAward: DMS-0104039Principal Investigator: Darren LongThe proposers plan to continue their work on a variety ofproblems bearing on the understanding of the geometric andtopological aspects of low dimensional manifolds. On thetopological side, the areas of study include Heegaard splittings,unknotting tunnels and related topics; the geometric side dealswith issues related to orbifolds, the construction and melding ofsurface subgroups, developing an understanding of higherdimensional hyperbolic manifolds, as well as some interactionswith other areas, for example algebraic number theory.Manifolds play a central role in physics, mathematics and to someextent in other sciences, since they are objects which on smallscales look like Euclidean space of dimension n. For example,the space that we live in is a three manifold and space-time is afour manifold. For this and other reasons, these dimensions haveattracted a good deal of attention in mathematics and physics.One basic unsolved problem is exactly which manifold is thecorrect model for the universe - there has been some speculationthat it falls into the class of so-called hyperbolic manifolds, acertain class of three dimensional spaces which in some senseappear to be generic. Cosmological theories about the origins ofthe universe put constraints on the shape of space-time and henceconstraints on which manifolds could occur. One of the goals ofthis project is refine current methods (which have already ruledout many possibilities) to narrow the search down further. Thereare also other applications of geometric ideas which are lessobvious but still, in fact, directly important. It is noexaggeration to say that almost any problem which can beformulated qualitatively can be studied with geometric methodsand as a result the powerful tools developed over the last thirtyyears in low-dimensional topology can be brought to bear upon it.