We prove that a set of N not necessarily distinct points in the plane determine a unique, real analytic solution to the first order Ginzburg-Landau equations with vortex number N. This solution has the property that the Higgs field vanishes only at the points in the set and the order of vanishing at a given point is determined by the multiplicity of that point in the set. We prove further that these are the only C°° solutions to the first order Ginzburg-Landau equations.

This note describes a subcomplex F of the de Rham complex of parametrized knot space, which is combinatorial over a number of universal ‘‘Anomaly Integrals.’’ The self-linking integrals of Guadaguini, Martellini, and Mintchev [‘‘Perturbative aspects of Chern–Simons field theory,’’ Phys. Lett. B 227, 111 (1989)] and Bar-Natan [‘‘Perturbative aspects of the Chern–Simons topological quantum field theory,’’ Ph.D. thesis, Princeton University, 1991; also ‘‘On the Vassiliev Knot Invariants’’ (to appear in Topology)] are seen to represent the first nontrivial element in H0(F)—occurring at level 4, and are anomaly free. However, already at the next level an anomalous term is possible.

Recently, Seiberg and Witten (see [SW1], [SW2], [W]) introduced a remarkable new equation which gives differential-topological invariants for a compact, oriented 4-manifold with a distinguished integral cohomology class. A brief mathematical description of these new invariants is given in the recent preprint [KM].My purpose here is to prove the following theorem:Main Theorem. Let X be a compact, oriented, 4 dimensional manifold with b 2+ ≥ 2. Let ω be a symplectic form on X with ω ∧ ω giving the orientation. Then the first Chern class of the associated almost complex structure on X has Seiberg-Witten invariant equal to ±1.(Note: There are no symplectic forms on X unless b 2+ 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, Corollary.Connect sums of 4-manifolds with non-negative definite intersection forms do not admit symplectic forms which are compatible with the given orientation. For example, when n > 1 and m ≥ 0, then ( #n CP 2 )#( #m CP 2 ) has no symplectic form which defines the given orientation.The Main Theorem also implies that the Seiberg-Witten invariant for the canonical class of a complex surface with b 2+ ≥ 3 is equal to ±1. However, this result is easy to prove directly, as there is just one nondegenerate solution in this case.

Hybrids between species are often characterized by novel gene-expression patterns. A recent study on allele-specific gene expression in hybrids between species of Drosophila revealed cases in which cis-and transregulatory elements within species had coevolved in such a way that changes in cis-regulatory elements are compensated by changes in trans-regulatory elements. We hypothesized that such coevolution should often lead to gene misexpression in the hybrid. To test this hypothesis, we estimated allele-specific expression and overall expression levels for 31 genes in D. melanogaster, D. simulans, and their F 1 hybrid. We found that 13 genes with cis-trans compensatory evolution are in fact misexpressed in the hybrid. These represent candidate genes whose dysregulation might be the consequence of coevolution of cis-and trans-regulatory elements within species. Using a mathematical model for the regulation of gene expression, we explored the conditions under which cis-trans compensatory evolution can lead to misexpression in interspecific hybrids.

The purpose of this article is to explain how pseudo-holomorphic curves in a symplectic 4-manifold can be constructed from solutions to the Seiberg-Witten equations. As such, the main theorem proved here (Theorem 1.3) is an existence theorem for pseudo-holomorphic curves. This article thus provides a proof of roughly half of the main theorem in the announcement [T1]. That theorem, Theorem 4.1, asserts an equivalence between the Seiberg-Witten invariants for a symplectic manifold and a certain Gromov invariant which counts (with signs) the number of pseudoholomorphic curves in a given homology class.The Seiberg-Witten invariants were introduced to mathematicians by Witten at least 2, then these invariants define a diffeomorphism invariant map, SW, from the set of equivalence classes, Spin, of Spin C structures on X to Z. Note that the set Spin has naturally the structure of a principal H 2 (X; Z) bundle over a point. A symplectic 4-manifold is a pair (X, ω), where X is a 4-manifold and where ω is a closed 2-form with ω ∧ ω nowhere zero. Thus, a symplectic 4-manifold has a canonical orientation. A symplectic 4-manifold also has a complex line bundle, K, (called the canonical bundle) which is canonical up to isomorphism. And, as explained in [T1] or [T2], a symplectic 4-manifold has a canonical equivalence class of Spin C structure. The latter endows Spin with a base point and so gives the identification(Both the identification of Spin and the choice of orientation do not change under a continuous deformation of the symplectic form.) In the ensuing discussion, the canonical orientation for a symplectic manifold and the identification in (0.2) will be assumed implicitly. Thus, with (0.2) understood, SW defines a map SW : H 2 (X; Z) → Z. (0.3)