Journal•ISSN: 1073-2780
Mathematical Research Letters
International Press of Boston
About: Mathematical Research Letters is an academic journal published by International Press of Boston. The journal publishes majorly in the area(s): Conjecture & Symplectic geometry. It has an ISSN identifier of 1073-2780. Over the lifetime, 2120 publications have been published receiving 55593 citations. The journal is also known as: MRL.
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TL;DR: In this paper, a new viewpoint about Donaldson theory of four manifolds was proposed, where instead of defining four-manifold invariants by counting $SU(2)$ instantons, one can define equivalent four manifold invariants using solutions of a non-linear equation with an abelian gauge group, in which the gauge group is the dual of the maximal torus of the Donaldson invariant.
Abstract: Recent developments in the understanding of $N=2$ supersymmetric Yang-Mills theory in four dimensions suggest a new point of view about Donaldson theory of four manifolds: instead of defining four-manifold invariants by counting $SU(2)$ instantons, one can define equivalent four-manifold invariants by counting solutions of a non-linear equation with an abelian gauge group. This is a ``dual'' equation in which the gauge group is the dual of the maximal torus of $SU(2)$. The new viewpoint suggests many new results about the Donaldson invariants.
820 citations
684 citations
TL;DR: In this article, a vanishing theorem for the Seiberg-Witten invariants of a manifold X was proved, which implies that no such manifold admits a symplectic form unless b and first Betti number of X have opposite parity.
Abstract: (Note: There are no symplectic forms on X unless b and the first Betti number of X have opposite parity.) In a subsequent article with joint authors, a vanishing theorem will be proved for the Seiberg-Witten invariants of a manifold X, as in the theorem, which can be split by an embedded 3-sphere as X−∪X+ where neither X− nor X+ have negative definite intersection forms. Thus, no such manifold admits a symplectic form. That is,
661 citations
TL;DR: In this article, it was shown that the genus of a smooth 2-manifold can be characterized by the formula g = (d − 1)(d − 2)/2.
Abstract: 1. Statement of the result The genus of a smooth algebraic curve of degree d in CP is given by the formula g = (d − 1)(d − 2)/2. A conjecture sometimes attributed to Thom states that the genus of the algebraic curve is a lower bound for the genus of any smooth 2-manifold representing the same homology class. The conjecture has previously been proved for d ≤ 4 and for d = 6, and less sharp lower bounds for the genus are known for all degrees [5,6,7,10,14]. In this note we confirm the conjecture. Theorem 1. Let Σ be an oriented 2-manifold smoothly embedded in CP so as to represent the same homology class as an algebraic curve of degree d. Then the genus g of Σ satisfies g ≥ (d − 1)(d − 2)/2.
562 citations
TL;DR: In this article, anisotropic conductivities with the same Dirichlet-to-Neumann map as a homogeneous isotropic conductivity were constructed, which are singular close to a surface inside the body.
Abstract: We construct anisotropic conductivities with the same Dirichlet-to-Neumann map as a homogeneous isotropic conductivity. These conductivities are singular close to a surface inside the body.
482 citations