Publications (81)135.99 Total impact
 [Show abstract] [Hide abstract]
ABSTRACT: Supposing that X is a Riemannian manifold, a Z/2 spinor on X is defined by a data set consisting of a closed set in X to be denoted by Z, a real line bundle over XZ, and a nowhere zero section on XZ of the tensor product of the real line bundle and a spinor bundle. The set Z and the spinor are jointly constrained by the following requirement: The norm of the spinor must extend across Z as a continuous function vanishing on Z. In particular, the vanishing locus of the norm of the spinor is the complement of the set where the real line bundle is defined, and hence where the spinor is defined. The Z/2 spinor is said to be harmonic when it obeys a first order Dirac equation on XZ. This monograph analyzes the structure of the set Z for a Z/2 harmonic spinor on a manifold of dimension either two, three or four.  [Show abstract] [Hide abstract]
ABSTRACT: Uhlenbeck's compactness theorem can be used to analyze sequences of connections with antiself dual curvature on principal SU(2) bundles over oriented 4dimensional manifolds. The theorems in this paper give an extension of Uhlenbeck's theorem for sequences of solutions of certain SL(2,C) analogs of the antiself dual equations.  [Show abstract] [Hide abstract]
ABSTRACT: This is the second of two papers that describe a compactness theorem for sequences of solutions of certain SL(2;C) analogs of the antiself dual equations on oriented, 4dimensional Riemannian manifolds. This paper proves theorems that characterize the singular locus of limits of sequences of solutions to the equations.  [Show abstract] [Hide abstract]
ABSTRACT: Solutions to the SU(2) monopole equations in the Bogolmony limit are constructed that look very much like Bolognesi's conjectured magnetic bag solutions. Three theorems are also stated and proved that give bounds in terms of the topological charge for the radii of balls where the solution's Higgs field has very small norm  [Show abstract] [Hide abstract]
ABSTRACT: Karen Uhlenbeck's compactness theorem for sequences of connections with L2 bounds on curvature applies only to connections on principal bundles with compact structure group. This article states and proves an extension of Uhlenbecks theorem that describes sequences of connections on principal PSL(2;C) bundles over compact three dimensional manifolds.  [Show abstract] [Hide abstract]
ABSTRACT: The evolutionary trajectory of life on earth is one of increasing size and complexity. Yet the standard equations of evolutionary dynamics describe mutation and selection among similar organisms that compete on the same level of organization. Here we begin to outline a mathematical theory that might help to explore how evolution can be constructive, how natural selection can lead from lower to higher levels of organization. We distinguish two fundamental operations, which we call 'staying together' and 'coming together'. Staying together means that individuals form larger units by not separating after reproduction, while coming together means that independent individuals form aggregates. Staying together can lead to specialization and division of labor, but the developmental program must evolve in the basic unit. Coming together can be creative by combining units with different properties. Both operations have been identified in the context of multicellularity, but they have been treated very similarly. Here we point out that staying together and coming together can be found at every level of biological construction and moreover that they face different evolutionary problems. The distinction is particularly clear in the context of cooperation and defection. For staying together the stability of cooperation takes the form of a developmental error threshold, while coming together leads to evolutionary games and requires a mechanism for the evolution of cooperation. We use our models to discuss simple aspects of the evolution of protocells, eukarya, multicellularity and animal societies.  [Show abstract] [Hide abstract]
ABSTRACT: At a time when genomes are being sequenced by the hundreds, much attention has shifted from identifying genes and phenotypes to understanding the networks of interactions among genes. We developed a gene network developmental model expanding on previous models of transcription regulatory networks. In our model, each network is described by a matrix representing the interactions between transcription factors, and a vector of continuous values representing the transcription factor expression in an individual. In this work we used the gene network model to look at the impact of mating as well as insertions and deletions of genes in the evolution of complexity of these networks. We found that the natural process of diploid mating increases the likelihood of maintaining complexity, especially in higher order networks (more than 10 genes). We also show that gene insertion is a very efficient way to add more genes to a network as it provides a much higher chance of developmental stability. The continuous model affords a more complete view of the evolution of interacting genes. The notion of a continuous output vector also incorporates the reality of gene networks and graded concentrations of gene products.  [Show abstract] [Hide abstract]
ABSTRACT: This is the fourth of five papers that construct an isomorphism between the SeibergWitten Floer homology and the Heegaard Floer homology of a given compact, oriented 3manifold. The isomorphism is given as a composition of three isomorphisms; the first of these relates a version of embedded contact homology on an an auxillary manifold to the Heegaard Floer homology on the original. The second isomorphism relates the relevant version of the embedded contact homology on the auxilliary manifold with a version of the SeibergWitten Floer homology on this same manifold. The third isomorphism relates the SeibergWitten Floer homology on the auxilliary manifold with the appropriate version of SeibergWitten Floer homology on the original manifold. The paper describes the second of these isomorphisms. 
Article: HF=HM III: Holomorphic curves and the differential for the ech/Heegaard Floer correspondence
[Show abstract] [Hide abstract]
ABSTRACT: This is the third of five papers that construct an isomorphism between the SeibergWitten Floer homology and the Heegaard Floer homology of a given compact, oriented 3manifold. The isomorphism is given as a composition of three isomorphisms; the first of these relates a version of embedded contact homology on an an auxillary manifold to the Heegaard Floer homology on the original. This paper describes the relationship between the differential on the embedded contact homology chain complex and the differential on the Heegaard Floer chain complex. The paper also describes the relationship between the various canonical endomorphisms that act on the homology groups of these two complexes.  [Show abstract] [Hide abstract]
ABSTRACT: This is the second of five papers that construct an isomorphism between the SeibergWitten Floer homology and the Heegaard Floer homology of a given compact, oriented 3manifold. The isomorphism is given as a composition of three isomorphisms; the first of these relates a version of embedded contact homology on an an auxillary manifold to the Heegaard Floer homology on the original. This paper describes this auxilliary manifold, its geometry, and the relationship between the generators of the embedded contact homology chain complex and those of the Heegaard Floer chain complex. The pseudoholomorphic curves that define the differential on the embedded contact homology chain complex are also described here as a first step to relate the differential on the latter complex with that on the Heegaard Floer complex. Comment: Various typos corrected  [Show abstract] [Hide abstract]
ABSTRACT: Let M be a closed, connected and oriented 3manifold. This article is the first of a five part series that constructs an isomorphism between the Heegaard Floer homology groups of M and the corresponding SeibergWitten Floer homology groups of M.  [Show abstract] [Hide abstract]
ABSTRACT: In "Proof of the Arnold chord conjecture in three dimensions, I" [12], we deduced the Arnold chord conjecture in three dimensions from another result, which asserts that an exact symplectic cobordism between contact threemanifolds induces a map on (filtered) embedded contact homology satisfying certain axioms. The present paper proves the latter result, thus completing the proof of the threedimensional chord conjecture. We also prove that filtered embedded contact homology does not depend on the choice of almost complex structure used to define it.  [Show abstract] [Hide abstract]
ABSTRACT: Fix a compact 4dimensional manifold with selfdual 2nd Betti number one and with a given symplectic form. This article proves the following: The Frechet space of tamed almost complex structures as defined by the given symplectic form has an open and dense subset whose complex structures are compatible with respect to a symplectic form that is cohomologous to the given one. The theorem is proved by constructing the new symplectic form by integrating over a space of currents that are defined by pseudoholomorphic curves.  [Show abstract] [Hide abstract]
ABSTRACT: Various SeibergWitten Floer cohomologies are defined for a closed, oriented 3manifold; and if it is the mapping torus of an areapreserving surface automorphism, it has an associated periodic Floer homology as defined by Michael Hutchings. We construct an isomorphism between a certain version of SeibergWitten Floer cohomology and the corresponding periodic Floer homology, and describe some immediate consequences.  [Show abstract] [Hide abstract]
ABSTRACT: This is the second of two articles that describe the moduli spaces of pseudoholomorphic, multiply punctured spheres in R x (S^1 x S^2) as defined by a certain natural pair of almost complex structure and symplectic form. The first article in this series described the local structure of the moduli spaces and gave existence theorems. This article describes a stratification of the moduli spaces and gives explicit parametrizations for the various strata.  [Show abstract] [Hide abstract]
ABSTRACT: This is the first of at least two articles that describe the moduli spaces of pseudoholomorphic, multiply punctured spheres in R x (S^1 x S^2) as defined by a certain natural pair of almost complex structure and symplectic form. This article proves that all moduli space components are smooth manifolds. Necessary and sufficient conditions are also given for a collection of closed curves in S^1 x S^2 to appear as the set of s > infinity limits of the constant s in R slices of a pseudoholomorphic, multiply punctured sphere.  [Show abstract] [Hide abstract]
ABSTRACT: Let M be a closed, connected, orientable 3manifold. The purpose of this paper is to study the SeibergWitten Floer homology of M given that S^1 X M admits a symplectic form. In particular, we prove that M fibers over the circle if M has first Betti number 1 and S^1 X M admits a symplectic form with nontorsion canonical class. 

 [Show abstract] [Hide abstract]
ABSTRACT: This paper proves the following: A volume preserving vector field on a compact 3manifold whose dual 2form is exact can not generate uniquely ergodic dynamics unless its asymptotic linking number is zero.
Publication Stats
3k  Citations  
135.99  Total Impact Points  
Top Journals
Institutions

19782013

Harvard University
 • Department of Mathematics
 • Department of Organismic and Evolutionary Biology
 • Department of Physics
Cambridge, Massachusetts, United States


19842006

University of California, Berkeley
 Department of Mathematics
Berkeley, CA, United States
