On the local meromorphic extension of CR meromorphic mappings

Let $M$ be a generic CR submanifold in \C^{m+n}, $m= CRdim M \geq 1$,$n=codim M \geq 1$, $d=dim M = 2m+n$. A CR meromorphic mapping (in the sense of Harvey-Lawson) is a triple $(f,{\cal D}_f, [\Gamma_f])$, where: 1. $f: {\cal D}_f \to Y$ is a ${\cal C}^1$-smooth mapping defined over a dense open subset ${\cal D}_f$ of $M$ with values in a projective manifold $Y$; 2. The closure $\Gamma_f$ of its graph in \C^{m+n} \times Y defines a oriented scarred ${\cal C}^1$-smooth CR manifold of CR dimension $m$ (i.e. CR outside a closed thin set) and 3. Such that $d[\Gamma_f]=0$ in the sense of currents. We prove in this paper that $(f,{\cal D}_f, [\Gamma_f])$ extends meromorphically to a wedge attached to $M$ if $M$ is everywhere minimal and ${\cal C}^{\omega}$ (real analytic) or if $M$ is a ${\cal C}^{2,\alpha}$ globally minimal hypersurface.Comment: 25 pages, LaTeX. To appear in Ann. Pol. Math. 199

On removable singularities for integrable CR functions

We endeavour a systematic approach for the removal of singularities for CR functions on an arbitrary embeddable CR manifold.Comment: 38 pages, LaTeX. To appear in Indiana Univ. Math. J. 199

Metrically thin singularities of integrable CR functions

In this article, we consider metrically thin singularities A of the tangential Cauchy-Riemann operator on smoothly embedded Cauchy-Riemann manifolds M. The main result states removability within the space of locally integrable functions on M under the hypothesis that the (dim M-2)-dimensional Hausdorff volume of A is zero and that the CR-orbits of M and M-A are comparable

Characteristic foliations on maximally real submanifolds of C^n and envelopes of holomorphy

Let S be an arbitrary real surface, with or without boundary, contained in a hypersurface M of the complex euclidean space \C^2, with S and M of class C^{2, a}, where 0 < a < 1. If M is globally minimal, if S is totally real except at finitely many complex tangencies which are hyperbolic in the sense of E. Bishop and if the union of separatrices is a tree of curves without cycles, we show that every compact K of S is CR-, W- and L^p-removable (Theorem~1.3). We treat this seemingly global problem by means of purely local techniques, namely by means of families of small analytic discs partially attached to maximally real submanifolds of C^n and by means of a thorough study of the relative disposition of the characteristic foliation with respect to the track on M of a certain half-wedge attached to M. This localization procedure enables us to answer an open problem raised by B. J\"oricke: under a certain nontransversality condition with respect to the characteristic foliation, we show that every closed subset C of a C^{2,a}-smooth maximally real submanifold M^1 of a (n-1)-codimensional generic C^{2,a}-smooth submanifold of \C^n is CR-, W- and L^p-removable (Theorem~1.2'). The known removability results in CR dimension at least two appear to be logical consequences of Theorem~1.2'. The main proof (65p.) is written directly in arbitrary codimension. Finally, we produce an example of a nonremovable 2-torus contained in a maximally real 3-dimensional maximally real submanifold, showing that the nontransversality condition is optimal for universal removability. Numerous figures are included to help readers who are not insiders of higher codimensional geometry.Comment: 113 pages, 24 figures, LaTe

Online civic intervention: A new form of political participation under conditions of a disruptive online discourse

In the everyday practice of online communication, we observe users deliberately reporting abusive content or opposing hate speech through counterspeech, while at the same time, online platforms are increasingly relying on and supporting this kind of user action to fight disruptive online behavior. We refer to this type of user engagement as online civic intervention (OCI) and regard it as a new form of user-based political participation in the digital sphere that contributes to an accessible and reasoned public discourse. Because OCI has received little scholarly attention thus far, this article conceptualizes low- and high-threshold types of OCI as different kinds of user responses to common disruptive online behavior such as hate speech or hostility toward the media. Against the background of participation research, we propose a theoretically grounded individual-level model that serves to explain OCI

FACE Peace Design Brief #1: Communities of Practice On/Offline

The FACE Peace Initiative at the Joan B. Kroc Institute for Peace and Justice intends to help peacebuilders answer questions about in-person and online collaboration with intention and care. This design brief combines desk research on best practices from other fields with observation of peacebuilding organizations to identify key debates and concerns and provide insight into how to navigate trade-offs between in-person and distanced peacebuilding activities and events. Peacebuilding organizations often attempt to gather members of the field into “communities of practice” (“CoPs”), which intend to increase skills and knowledge among members through long-term information-sharing and reciprocal mentorship. Facilitators of practice communities in peacebuilding and other fields frequently complain that the community falls moribund over time. This FACE Peace design brief considers the question of practice community success from the perspective of hybrid work and the tensions peacebuilders have come to feel between digital and in-person interactions in a truly global field. What does in-person interaction between practice community members accomplish? When are these benefits essential for success? When are they simply “nice to have”? What are the best ways to recreate the benefits of in-person meetings at a distance? Are there benefits only distanced work can provide? Answers depend in part on the goals, constraints and characteristics of the practice community. This design brief offers insights on two related questions. First, how should the facilitators of practice communities decide what happens in person and what happens at a distance? Second, how can facilitators administer the in-person and online aspects of their practice communities to maximum effect?https://digital.sandiego.edu/ipj-research/1059/thumbnail.jp

IREX and the Community Solutions Program

The community of practice (CoP) made up of the alumni Community Solutions Program (CSP) caters to a diverse and globally distributed group of professionals in the peacebuilding, humanitarian and development fields. Facilitating such a community without substantial reliance on communications technology and distanced relationship-building would be impossible. Yet this kind of distanced practice community faces a trust hurdle: members will only contribute quality content if they believe others will do the same. This creates a classic “free-rider” problem that leads to the death of many distanced practice communities. Despite this challenge, the CSP alumni that make up the program’s community of practice contribute enthusiastically. Why? This case study examines five practices IREX employs to increase trust, overcome the free-rider challenge, and therefore inspire participation and the creation of content that provides value to others in the community. The first four of these practices accord with best practices from a variety of other fields known for hosting practice communities. The last practice, focused on building a “community of care” as opposed to simply creating professional value for members, represents an innovation that is well-suited to peacebuilding practice communities.https://digital.sandiego.edu/ipj-research/1078/thumbnail.jp

Holomorphic Extension from Weakly Pseudoconcave CR Manifolds

Let M be a smooth locally embeddable CR manifold, having some CR dimension m and some CR codimension d. We find an improved local geometric condition on M which guarantees, at a point p on M, that germs of CR distributions are smooth functions, and have extensions to germs of holomorphic functions on a full ambient neighborhood of p. Our condition is a form of weak pseudoconcavity, closely related to essential pseudoconcavity as introduced in [HN1]. Applications are made to CR meromorphic functions and mappings. Explicit examples are given which satisfy our new condition,but which are not pseudoconcave in the strong sense. These results demonstrate that for codimension d > 1, there are additional phenomena which are invisible when d = 1