Nilpotent operators and weighted projective lines

We show a surprising link between singularity theory and the invariant subspace problem of nilpotent operators as recently studied by C. M. Ringel and M. Schmidmeier, a problem with a longstanding history going back to G. Birkhoff. The link is established via weighted projective lines and (stable) categories of vector bundles on those. The setup yields a new approach to attack the subspace problem. In particular, we deduce the main results of Ringel and Schmidmeier for nilpotency degree p from properties of the category of vector bundles on the weighted projective line of weight type (2,3,p), obtained by Serre construction from the triangle singularity x^2+y^3+z^p. For p=6 the Ringel-Schmidmeier classification is thus covered by the classification of vector bundles for tubular type (2,3,6), and then is closely related to Atiyah's classification of vector bundles on a smooth elliptic curve. Returning to the general case, we establish that the stable categories associated to vector bundles or invariant subspaces of nilpotent operators may be naturally identified as triangulated categories. They satisfy Serre duality and also have tilting objects whose endomorphism rings play a role in singularity theory. In fact, we thus obtain a whole sequence of triangulated (fractional) Calabi-Yau categories, indexed by p, which naturally form an ADE-chain.Comment: More details added. 33 page

Coos County Emergency Management Strategic Plan

22 pagesA Community Planning Workshop (CPW) team of four graduate students worked under the direction of a faculty manager to facilitate this planning process. The student team conducted research, facilitated meetings with local steering committees and prepared the final plans and project reports. This project was completed in collaboration with the Oregon Partnership for Disaster Resilience (OPDR). This strategic planning process evaluated Coos County EM’s core mission areas - Planning, Organization, Equipment, Training, and Exercises (i.e. the POETE framework) - against available and forecasted resources. The resulting strategy outlines how the county will sustain EM capabilities through an action plan informed by the processes, stakeholders and influences that drive local Emergency Management.Coos County Emergency Management Strategic Steering Committe

Postnatal depression is associated with detrimental life-long and multi-generational impacts on relationship quality

Postnatal depression (PND) is known to be associated with a range of detrimental child and adolescent outcomes, resulting from its disruptive impact on mother-child relationship quality. However, until now little has been known about the impact of PND on the longer-term relationships between mothers and their children, and any intergenerational effects this may have. Mother-child relationship quality is of interest from an evolutionary perspective as it plays a role in the accrual of offspring embodied capital, thus affecting offspring quality and offspring’s capacity to subsequently invest in their own children. Relationships with offspring also mediate grandparent-grandchild relations; if PND negatively affects long-term mother–offspring relationship quality, it is also likely to negatively affect grandmaternal investment via reduced grandmother–grandchild relationship quality. Here, we use responses to a retrospective questionnaire study of postmenopausal women, largely from the UK and US, to assess the impact of PND occurring in generation 1 on mother–child relationship quality across the life course of the child (generation 2) with whom it was associated, and also on the relationship quality with grandchildren (generation 3) from that child. Average mother-child relationship quality was lower when the child’s birth was associated with PND. Multi-level regression modelling found that mother-child relationship quality decreased as PND symptom severity increased after controlling for individual effects and a variety of other factors known to influence relationship quality (individual mothers n = 296, mother-child dyads n = 646). Additionally, intergenerational relationships appear to be affected, with PND negatively associated with grandmother-grandchild relations (individual grandmothers n = 125, relations with grandchildren from n = 197 grandmother-parent dyads). That PND has long-term detrimental consequences for mother-child relationships, well beyond adolescence, highlights the need for investment in strategies to prevent PND and its cascade of negative multigenerational effects