The homotopy type of the space of symplectic balls in rational ruled 4-manifolds

Let M:=(M^{4},\om) be a 4-dimensional rational ruled symplectic manifold and denote by w_{M} its Gromov width. Let Emb_{\omega}(B^{4}(c),M) be the space of symplectic embeddings of the standard ball B^4(c) \subset \R^4 of radius r and of capacity c:= \pi r^2 into (M,\om). By the work of Lalonde and Pinsonnault, we know that there exists a critical capacity \ccrit \in (0,w_{M}] such that, for all c\in(0,\ccrit), the embedding space Emb_{\omega}(B^{4}(c),M) is homotopy equivalent to the space of symplectic frames \SFr(M). We also know that the homotopy type of Emb_{\omega}(B^{4}(c),M) changes when c reaches \ccrit and that it remains constant for all c \in [\ccrit,w_{M}). In this paper, we compute the rational homotopy type, the minimal model, and the cohomology with rational coefficients of \Emb_{\omega}(B^{4}(c),M) in the remaining case c \in [\ccrit,w_{M}). In particular, we show that it does not have the homotopy type of a finite CW-complex.Comment: 38 pages; revised versio

Four-dimensional symplectic cobordisms containing three-handles

We construct four-dimensional symplectic cobordisms between contact three-manifolds generalizing an example of Eliashberg. One key feature is that any handlebody decomposition of one of these cobordisms must involve three-handles. The other key feature is that these cobordisms contain chains of symplectically embedded two-spheres of square zero. This, together with standard gauge theory, is used to show that any contact three-manifold of non-zero torsion (in the sense of Giroux) cannot be strongly symplectically fillable. John Etnyre pointed out to the author that the same argument together with compactness results for pseudo-holomorphic curves implies that any contact three-manifold of non-zero torsion satisfies the Weinstein conjecture. We also get examples of weakly symplectically fillable contact three-manifolds which are (strongly) symplectically cobordant to overtwisted contact three-manifolds, shedding new light on the structure of the set of contact three-manifolds equipped with the strong symplectic cobordism partial order.Comment: This is the version published by Geometry & Topology on 28 October 200

On the Equivalence Problem for Toric Contact Structures on S^3-bundles over S^2$

We study the contact equivalence problem for toric contact structures on $S^3$-bundles over $S^2$. That is, given two toric contact structures, one can ask the question: when are they equivalent as contact structures while inequivalent as toric contact structures? In general this appears to be a difficult problem. To find inequivalent toric contact structures that are contact equivalent, we show that the corresponding 3-tori belong to distinct conjugacy classes in the contactomorphism group. To show that two toric contact structures with the same first Chern class are contact inequivalent, we use Morse-Bott contact homology. We treat a subclass of contact structures which include the Sasaki-Einstein contact structures $Y^{p,q}$ studied by physicists. In this subcase we give a complete solution to the contact equivalence problem by showing that $Y^{p,q}$ and $Y^{p'q'}$ are inequivalent as contact structures if and only if $p\neq p'$.Comment: 61 page

Adapting a Curriculum Unit to Facilitate Interaction Between Technology, Mathematics and Science in the Elementary Classroom: Identifying Relevant Criteria

Calls for the integration of subjects continue to emanate from a wide range of professional bodies, including governments and subject associations. Yet as some authors suggest, blurring the boundaries between subjects may be one of the most daunting tasks educators face. The authors have recently begun a research study that will investigate the extent to which (a) relevant mathematics and science can be made explicit in a technology curriculum unit, (b) pupils utilise this mathematics and science learning, and (c) pupils' ability to design is enhanced by making the mathematics and science explicit and useful. This paper reports the results of Phase 1 of the study: an examination of research literature in order to identify criteria to inform the re-writing of an existing technology curriculum (to be used as a research instrument) that previously did not make explicit embedded mathematics and science concepts. Our reading of the literature has identified two essential criteria that must be met during the re-writing: (a) protecting the integrity of the subjects and (b) identifying the nature and purpose of the intended learning

Predicting Online Media Effectiveness Based on Smile Responses Gathered Over the Internet

We present an automated method for classifying “liking” and “desire to view again” based on over 1,500 facial responses to media collected over the Internet. This is a very challenging pattern recognition problem that involves robust detection of smile intensities in uncontrolled settings and classification of naturalistic and spontaneous temporal data with large individual differences. We examine the manifold of responses and analyze the false positives and false negatives that result from classification. The results demonstrate the possibility for an ecologically valid, unobtrusive, evaluation of commercial “liking” and “desire to view again”, strong predictors of marketing success, based only on facial responses. The area under the curve for the best “liking” and “desire to view again” classifiers was 0.8 and 0.78 respectively when using a challenging leave-one-commercial-out testing regime. The technique could be employed in personalizing video ads that are presented to people whilst they view programming over the Internet or in copy testing of ads to unobtrusively quantify effectiveness.MIT Media Lab Consortiu

Mirror Symmetry for Two Parameter Models -- II

We describe in detail the space of the two K\"ahler parameters of the Calabi--Yau manifold $\P_4^{(1,1,1,6,9)}[18]$ by exploiting mirror symmetry. The large complex structure limit of the mirror, which corresponds to the classical large radius limit, is found by studying the monodromy of the periods about the discriminant locus, the boundary of the moduli space corresponding to singular Calabi--Yau manifolds. A symplectic basis of periods is found and the action of the $Sp(6,\Z)$ generators of the modular group is determined. From the mirror map we compute the instanton expansion of the Yukawa couplings and the generalized $N=2$ index, arriving at the numbers of instantons of genus zero and genus one of each degree. We also investigate an $SL(2,\Z)$ symmetry that acts on a boundary of the moduli space.Comment: 57 pages + 9 figures using eps

Semisimplicity of the quantum cohomology for smooth Fano toric varieties associated with facet symmetric polytopes

The degree zero part of the quantum cohomology algebra of a smooth Fano toric symplectic manifold is determined by the superpotential function, W, of its moment polytope. In particular, this algebra is semisimple, i.e. splits as a product of fields, if and only if all the critical points of W are non-degenerate. In this paper we prove that this non-degeneracy holds for all smooth Fano toric varieties with facet-symmetric duals to moment polytopes.Comment: 16 pages; corrected version, published in Electron. Res. Announc. Math. Sc

Topological Field Theory and Rational Curves

We analyze the superstring propagating on a Calabi-Yau threefold. This theory naturally leads to the consideration of Witten's topological non-linear sigma-model and the structure of rational curves on the Calabi-Yau manifold. We study in detail the case of the world-sheet of the string being mapped to a multiple cover of an isolated rational curve and we show that a natural compactification of the moduli space of such a multiple cover leads to a formula in agreement with a conjecture by Candelas, de la Ossa, Green and Parkes.Comment: 20 page

Measuring Small Distances in N=2 Sigma Models

We analyze global aspects of the moduli space of K\"ahler forms for $N$=(2,2) conformal $\sigma$-models. Using algebraic methods and mirror symmetry we study extensions of the mathematical notion of length (as specified by a K\"ahler structure) to conformal field theory and calculate the way in which lengths change as the moduli fields are varied along distinguished paths in the moduli space. We find strong evidence supporting the notion that, in the robust setting of quantum Calabi-Yau moduli space, string theory restricts the set of possible K\"ahler forms by enforcing ``minimal length'' scales, provided that topology change is properly taken into account. Some lengths, however, may shrink to zero. We also compare stringy geometry to classical general relativity in this context.Comment: 62 pp. with 6 figs., LaTeX and epsf.te