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Promoting essential laminations
"... ABSTRACT. We show that a co–orientable taut foliation of a closed, orientable, algebraically atoroidal 3–manifold is either the weak stable foliation of an Anosov flow, or else there are a pair of very full genuine laminations transverse to the foliation. 1. ..."
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Cited by 10 (6 self)
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ABSTRACT. We show that a co–orientable taut foliation of a closed, orientable, algebraically atoroidal 3–manifold is either the weak stable foliation of an Anosov flow, or else there are a pair of very full genuine laminations transverse to the foliation. 1.
Smooth quasiregular mappings with branching
 Publ. Math. Inst. Hautes Études Sci
"... We give an example of a C 3−ǫsmooth quasiregular mapping in 3space with nonempty branch set. Moreover, we show that the branch set of an arbitrary quasiregular mapping in nspace has Hausdorff dimension quantitatively bounded away from n. By using the second result, we establish a new, qualitative ..."
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Cited by 5 (0 self)
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We give an example of a C 3−ǫsmooth quasiregular mapping in 3space with nonempty branch set. Moreover, we show that the branch set of an arbitrary quasiregular mapping in nspace has Hausdorff dimension quantitatively bounded away from n. By using the second result, we establish a new, qualitatively sharp relation between smoothness and branching. 1.
BETWEEN LOWER AND HIGHER DIMENSIONS (in the work of Terry Lawson)
"... There are several approaches to summarizing a mathematician’s research accomplishments, and each has its advantages and disadvantages. This article is based upon a talk given at Tulane that was aimed at a fairly general audience, including faculty members in other areas and graduate students who had ..."
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There are several approaches to summarizing a mathematician’s research accomplishments, and each has its advantages and disadvantages. This article is based upon a talk given at Tulane that was aimed at a fairly general audience, including faculty members in other areas and graduate students who had taken the usual entry level courses. As such, it is meant to be relatively nontechnical and to emphasize qualitative rather than quantitative issues; in keeping with this aim, references will be given for some standard topological notions that are not normally treated in entry level graduate courses. Since this was an hour talk, it was also not feasible to describe every single piece of published mathematical work that Terry Lawson has ever written; in particular, some papers like [42] and [50] would require lengthy digressions that are not easily related to the central themes in his main lines of research. Instead, we shall focus on some ways in which Terry’s work relates to an important thread in geometric topology; namely, the passage from studying problems in a given dimension to studying problems in the next dimensions. Qualitatively speaking, there are fairly welldeveloped theories for very low dimensions and for all sufficiently large dimensions, but between these ranges there are some dimensions in which the answers to many fundamental
On the finiteness and shape of the Universe
, 2004
"... Abstract — Previous plausible theoretical assumptions about the cosmic 3manifold, such as isotropy, orientability, and compactness, have been unable to reduce the number of candidate topologies to a finite set. We now consider several new possible assumptions inspired by relationships between micro ..."
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Abstract — Previous plausible theoretical assumptions about the cosmic 3manifold, such as isotropy, orientability, and compactness, have been unable to reduce the number of candidate topologies to a finite set. We now consider several new possible assumptions inspired by relationships between microscopic physics and cosmic topology. The most important are 1. “Notwist ” assumption that there does not exist a twisted closed geodesic (to allow photons to exist in momentumpolarization eigenstates), 2. At most one isotopy class of nonseparating surface exists (related to charge quantization and seems necessary to allow charge to exist), 3. Orthogonal and/or commuting smooth vector fields exist, either locally or globally (may be needed to
Waldhausen’s Theorem
, 2007
"... This note is an exposition of Waldhausen’s proof of Waldhausen’s Theorem: the threesphere has a single Heegaard splitting, up to isotopy, in every genus. As a necessary step we also give a sketch of the Reidemeister–Singer Theorem. ..."
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This note is an exposition of Waldhausen’s proof of Waldhausen’s Theorem: the threesphere has a single Heegaard splitting, up to isotopy, in every genus. As a necessary step we also give a sketch of the Reidemeister–Singer Theorem.