Mathematical Sciences: RUI: Cusp Volumes in Hyperbolic      3-Manifolds
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{'SignBlockName': 'Ralph M. Krause', 'PO_EMAI': None, 'PO_PHON': None}
Adams will continue the study of cusp volumes in                           hyperbolic 3-manifolds.  It has been conjectured by W.                          Thurston that all 3-manifolds can be decomposed into                            components, each of which has one of eight geometries.  He                      has already proven this conjecture for large classes of 3-                      manifolds.  A complete classification of the 3-manifolds with                   seven of those geometries has already been completed.  The                      largest and richest case corresponds to those manifolds with                    the eighth geometry, namely hyperbolic geometry.                                     A hyperbolic 3-manifold has a well-defined volume, which                   is an invariant for the topology of the manifold.  The set of                   volumes of all finite volume hyperbolic 3-manifolds is a                        well-ordered subset of the reals.  Hence it is of interest to                   determine some of the hyperbolic manifolds of low volume, a                     problem suggested by Thurston.                                                       A noncompact hyperbolic 3-manifold has additional                          invariants called cusp volumes.  They offer a means of                          determining manifolds of low volume.  This idea has already                     been utilized by the investigator to determine the smallest                     hyperbolic manifolds of one or two cusps.                                            The project will continue investigations along these                       lines both by extending current computer programs in order to                   create more examples for study and also by utilizing the                        examples to point the way to proofs that certain manifolds                      are the smallest manifolds in their classes.  In particular,                    a goal of the project is to prove that the figure-eight knot                    complement and its sister manifold are the noncompact                           orientable hyperbolic 3-manifolds of minimal volume, also a                     conjecture of Thurston's.
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