Fractional Dehn twists in knot theory and contact topology

Fractional Dehn twists give a measure of the difference between the relative isotopy class of a homeomorphism of a bordered surface and the Thurston representative of its free isotopy class. We show how to estimate and compute these invariants. We discuss the the relationship of our work to stabilization problems in classical knot theory, general open book decompositions, and contact topology. We include an elementary characterization of overtwistedness for contact structures described by open book decompositions.Comment: We have removed an incorrect assumption about properties of meridional disks of Heegaard decompositions of S^3 and have added a conjecture about stabilizations of knots in S^

Public geographies II: being organic

This second report on ‘public geographies' considers the diverse, emergent and shifting spaces of engaging with and in public/s. Taking as its focus the more ‘organic’ rather than ‘traditional’ approach to doing public geography, as discussed in the first report, it explores the multiple and unorthodox ways in which engagements across academic-public spheres play out, and what such engagements may mean for geography/ers. The report first explores the role of the internet in ‘enabling conversations', generating a range of opportunities for public geography through websites, wikis, blogs, file-sharing sites, discussion forums and more, thinking critically about how technologies may enable/disable certain kinds of publically engaged activities. It then considers issues of process and praxis: how collaborations with groups/communities/organizations beyond academia are often unplanned, serendipitous encounters that evolve organically into research/learning/teaching endeavours; but also that personal politics/positionality bring an agency to bear upon whether we, as academics, follow the leads we may stumble upon. The report concludes with a provocative question – given that many non-academics appear to be doing some amazing and inspiring projects and activities, thoughtful, critical and (arguably) examples of organic public geographies, what then is academia’s role

Weak and strong fillability of higher dimensional contact manifolds

For contact manifolds in dimension three, the notions of weak and strong symplectic fillability and tightness are all known to be inequivalent. We extend these facts to higher dimensions: in particular, we define a natural generalization of weak fillings and prove that it is indeed weaker (at least in dimension five),while also being obstructed by all known manifestations of "overtwistedness". We also find the first examples of contact manifolds in all dimensions that are not symplectically fillable but also cannot be called overtwisted in any reasonable sense. These depend on a higher-dimensional analogue of Giroux torsion, which we define via the existence in all dimensions of exact symplectic manifolds with disconnected contact boundary.Comment: 68 pages, 5 figures. v2: Some attributions clarified, and other minor edits. v3: exposition improved using referee's comments. Published by Invent. Mat

Sex and the Cinema: What American Pie Teaches the Young

This paper focuses upon the wildly successful blockbuster American Pie teenpics, especially American Pie 3 – the Wedding. I argue that these films, which are sited so securely within the visual and pedagogical machinery of Hollywood culture, are specifically designed to appeal to teenage male audiences, and to provide lessons in sex and romance. Movies like this are especially important as they are experienced by far more teenagers than, for example, instructional films or other classroom materials; indeed, as Henry Giroux has observed, "teens and youth learn how to define themselves outside of the traditional sites of instruction, such as the home and the school… Learning in the postmodern age is located elsewhere – in popular spheres that shape their identities, through forms of knowledge and desires that appear absent from what is taught in schools" (Giroux, 1997, p.49). In this paper I discuss whether the American Pie series is actually a "new age" effort which, via insubordinate performances of gender, contests the hegemonic field of signification which regulates the production of sex, gender and desire, or whether it is more accurately described as a retrogressive hetero-conservative opus with a veneer of sexual radicalism. In short, I intend to probe whether this filmic vector for sex education is all about the shaping of responsible, caring, vulnerable men, or is it guiding them to become just like their heterosexual, middle-class fathers? And whether, despite its riotous and raunchy advertising, American Pie really dishes up something spicy or something terribly wholesome instead

Tight contact structures and taut foliations

We show the equivalence of several notions in the theory of taut foliations and the theory of tight contact structures. We prove equivalence, in certain cases, of existence of tight contact structures and taut foliations.Comment: Published in Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol4/paper7.abs.htm

Towards a framework for critical citizenship education

Increasingly countries around the world are promoting forms of "critical" citizenship in the planned curricula of schools. However, the intended meaning behind this term varies markedly and can range from a set of creative and technical skills under the label "critical thinking" to a desire to encourage engagement, action and political emancipation, often labelled "critical pedagogy". This paper distinguishes these manifestations of the "critical" and, based on an analysis of the prevailing models of critical pedagogy and citizenship education, develops a conceptual framework for analysing and comparing the nature of critical citizenship

Brieskorn manifolds as contact branched covers of spheres

We show that Brieskorn manifolds with their standard contact structures are contact branched coverings of spheres. This covering maps a contact open book decomposition of the Brieskorn manifold onto a Milnor open book of the sphere.Comment: 8 pages, 1 figur

Reinventing College Physics for Biologists: Explicating an epistemological curriculum

The University of Maryland Physics Education Research Group (UMd-PERG) carried out a five-year research project to rethink, observe, and reform introductory algebra-based (college) physics. This class is one of the Maryland Physics Department's large service courses, serving primarily life-science majors. After consultation with biologists, we re-focused the class on helping the students learn to think scientifically -- to build coherence, think in terms of mechanism, and to follow the implications of assumptions. We designed the course to tap into students' productive conceptual and epistemological resources, based on a theoretical framework from research on learning. The reformed class retains its traditional structure in terms of time and instructional personnel, but we modified existing best-practices curricular materials, including Peer Instruction, Interactive Lecture Demonstrations, and Tutorials. We provided class-controlled spaces for student collaboration, which allowed us to observe and record students learning directly. We also scanned all written homework and examinations, and we administered pre-post conceptual and epistemological surveys. The reformed class enhanced the strong gains on pre-post conceptual tests produced by the best-practices materials while obtaining unprecedented pre-post gains on epistemological surveys instead of the traditional losses.Comment: 35 pages including a 15 page appendix of supplementary material

Tight Beltrami fields with symmetry

Let $M$ be a compact orientable Seifered fibered 3-manifold without a boundary, and $\alpha$ an $S^1$-invariant contact form on $M$. In a suitable adapted Riemannian metric to $\alpha$, we provide a bound for the volume $\text{Vol}(M)$ and the curvature, which implies the universal tightness of the contact structure $\xi=\ker\alpha$.Comment: 26 page