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Pseudoholomorphic curves in fourorbifolds and some applications, in Geometry and Topology of Manifolds
 Fields Institute Communications 47
, 2005
"... The main purpose of this paper is to summarize the basic ingredients, illustrated with examples, of a pseudoholomorphic curve theory for symplectic 4orbifolds. These are extensions of relevant work of Gromov, McDuff and Taubes on symplectic 4manifolds concerning pseudoholomorphic curves and Seiber ..."
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Cited by 6 (5 self)
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The main purpose of this paper is to summarize the basic ingredients, illustrated with examples, of a pseudoholomorphic curve theory for symplectic 4orbifolds. These are extensions of relevant work of Gromov, McDuff and Taubes on symplectic 4manifolds concerning pseudoholomorphic curves and SeibergWitten theory (cf. [17, 32, 33, 42, 44]). They form the technical backbone of [8, 9] where it was shown that a symplectic scobordism of elliptic 3manifolds (with a canonical contact structure on the boundary) is smoothly a product. We believe that this theory has a broader interest and may find applications in other problems. One interesting feature is that existence of pseudoholomorphic curves gives certain restrictions on the singular points of the 4orbifold contained by the pseudoholomorphic curves. There are four sections. The first one is concerned with the Fredholm theory for pseudoholomorphic curves in symplectic orbifolds, which is based on the theory of maps of orbifolds developed in [7], particularly the topological structure of the corresponding mapping spaces. In the second section, we discuss the orbifold version of adjunction formula and a formula expressing the intersection number of two distinct pseudoholomorphic curves in terms of
SMOOTH sCOBORDISMS OF ELLIPTIC 3MANIFOLDS
, 2004
"... The main result in this paper states that a symplectic scobordism of an elliptic 3manifold to itself, which is equipped with a naturally defined contact structure, is diffeomorphic to a product. (The case of lens spaces was proved in an earlier paper [12].) Based on this theorem, we conjecture tha ..."
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Cited by 4 (4 self)
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The main result in this paper states that a symplectic scobordism of an elliptic 3manifold to itself, which is equipped with a naturally defined contact structure, is diffeomorphic to a product. (The case of lens spaces was proved in an earlier paper [12].) Based on this theorem, we conjecture that a smooth scobordism of elliptic 3manifolds is smoothly a product if its universal cover is smoothly a product. We explain how this conjecture fits naturally into the program of Taubes of constructing symplectic structures on an oriented smooth 4manifold with b + 2 ≥ 1 from generic selfdual closed 2forms.
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
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"... My primary research area is low dimensional topology. Most of my research has been focused on constructing “exotic ” pairs of surfaces in fourmanifolds: surfaces whose embeddings are homeomorphic but not diffeomorphic. The ideas and methods come from SeibergWitten theory, classical surgery theory, ..."
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My primary research area is low dimensional topology. Most of my research has been focused on constructing “exotic ” pairs of surfaces in fourmanifolds: surfaces whose embeddings are homeomorphic but not diffeomorphic. The ideas and methods come from SeibergWitten theory, classical surgery theory, algebraic topology, and use surgical constructions to provide many interesting smooth and symplectic 4manifolds.
ON THE FARRELLJONES AND RELATED CONJECTURES
, 710
"... Abstract. These extended notes are based on a series of six lectures presented at the summer school “Cohomology of groups and algebraic Ktheory ” which took place in Hangzhou, China from July 1 until July 12 in 2007. They give an introduction to the FarrellJones and the BaumConnes Conjecture. Key ..."
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Abstract. These extended notes are based on a series of six lectures presented at the summer school “Cohomology of groups and algebraic Ktheory ” which took place in Hangzhou, China from July 1 until July 12 in 2007. They give an introduction to the FarrellJones and the BaumConnes Conjecture. Key words: K and Lgroups of group rings and group C ∗algebras, Farrell