Cohomology of abelian arrangements

An abelian arrangement is a finite set of codimension one abelian subvarieties (possibly translated) in a complex abelian variety. In this paper, we study the cohomology of the complement of an abelian arrangement. For unimodular abelian arrangements, we provide a combinatorial presentation for a differential graded algebra whose cohomology is isomorphic to the rational cohomology of the complement. Moreover, this DGA has a bi-grading that allows us to compute the mixed Hodge numbers.Comment: 11 pages; to appear in Proceedings of the American Mathematical Societ

Keeping Watch

Undergraduate 2-

Representation stability for the cohomology of arrangements associated to root systems

From a root system, one may consider the arrangement of reflecting hyperplanes, as well as its toric and elliptic analogues. The corresponding Weyl group acts on the complement of the arrangement and hence on its cohomology. We consider a sequence of linear, toric, or elliptic arrangements which arise from a family of root systems of type A, B, C, or D, and we show that the rational cohomology stabilizes as a sequence of Weyl group representations. Our techniques combine a Leray spectral sequence argument similar to that of Church in the type A case along with FI$_W$-module theory which Wilson developed and used in the linear case. A key to the proof relies on a combinatorial description, using labelled partitions, of the poset of connected components of intersections of subvarieties in the arrangement.Comment: 20 pages; improved exposition and minor correction

Jesus Lives, but Should He Live in My Front Yard?

As I drove home from church, I eyed the bright foam sign my 6-year-old daughter held. “Jesus is Alive” it read in kid scrawl. “We’re supposed to put them in our yards!” Noelle beamed, eyeing her creation proudly through pink-rimmed glasses. I imagined our wide, open yard in Pennsylvania, the green grass stretching without fences from one neighbor to the next. Our best friends in the neighborhood, secular humanists, would easily see it. I cringed. What would they think? [excerpt

Unwrapping the Comfort of Sameness With Spanish Immersion Elementary School

I watched my 6-year-old hover around the periphery of the table, unable to find somewhere to sit. The cafeteria was a cacophony of little voices, Spanish and English, tumbling over each other, her classmates sitting close and waiting to be dismissed to homeroom. I couldn’t help but notice how different Noelle looked from most of the children, with her liquid blond hair and saucerlike blue eyes. [excerpt

Crew: Finding Community When Your Dreams Crash

Most young adults at some point experience a personal shipwreck —missing out on the job you wanted, the unexpected end of a relationship, a crisis of faith—that threatens to rip apart the fabric of your identity. What helps navigate a personal shipwreck is to have a crew of reliable people who walk with you through it. In Crew: Finding Community When Your Dreams Crash, Christin Taylor explores how young adults can both find good company during a time of personal shipwreck and be good company for others who might be experiencing their own shipwreck. In the process, you will learn the hope and security that comes from being part of a community. Based on sound scriptural principles and the latest research on young adult spiritual formation, Taylor gives young adults the knowledge and perspective you need to build a community that will help you make your way toward a sense of hope and new meaning. [From the publisher]https://cupola.gettysburg.edu/books/1059/thumbnail.jp

Economic insecurity and cohabitation strategies in Italy

A particular aspect of demographic behavior among young people in Italy is postponement of entering first union. High youth unemployment, a tense housing situation, and a passive welfare state are currently creating a precarious economic situation, in which most young adults are unable to choose cohabitation. Thus, not surprisingly, previous studies found evidence that in Italy cohabitation was only a choice for people who were economically independent. Also of interest is that the percentage of informal unions varies to a considerable extent across Italy, showing higher proportions of cohabitation in the more prosperous regions of the North, unlike the South, where informal unions are much less prevalent and the economic system is affected by mismanagement, unemployment, and the informal economy. This suggests an interrelationship between the diffusion of cohabitation and the regional economic situation. In this qualitative study we are particularly interested in the question of how job insecurity affects cohabitation – or more precisely: How are job insecurity and resulting economic shortages related to the hesitant spread of cohabitation in Italy? For our analysis we investigated two different regional settings: Bologna in the North and Cagliari (Sardinia) in the South. Our findings show that, when compared to their counterparts in Cagliari, couples in Bologna benefited from higher opportunities to access at least temporary job contracts. Benefiting also from the availability of parental support during cohabitation, the Bologna couples faced fewer obstacles when deciding on an informal union. In Cagliari, couples were strongly affected by unstable employment conditions; further, the lack of parental approval of cohabitation often led to decreasing economic support, thereby making cohabitation an expensive choice.Italy, cohabitation, economic resources, qualitative methods

Entry deterrence through cooperative R&D over-investment

In this paper, we highlight new conditions under which R&D agreements may have anti-competitive effects. We focus on cases where two firms compete with each other and with a competitive fringe. R&D activities need a specific input available to all firms on a common market, the price of which increases with demand for the input. In such a context, if a firm increases its R&D expenses, it increases the cost of R&D for its rivals. This induces exit from the fringe and may increase the final price. Therefore, by contrast to the case where the cost of R&D for one firm is independent of its rivals' R&D decisions, cooperation between strategic firms on the upstream market may induce more R&D by strategic firms, in order to exclude firms from the fringe and increase the final price. --Competition policy,Research and Development Agreements,Collusion,Entry deterrence