The logarithmic Picard group and its tropicalization

We construct the logarithmic and tropical Picard groups of a family of logarithmic curves and realize the latter as the quotient of the former by the algebraic Jacobian. We show that the logarithmic Jacobian is a proper family of logarithmic abelian varieties over the moduli space of Deligne-Mumford stable curves, but does not possess an underlying algebraic stack. However, the logarithmic Picard group does have logarithmic modifications that are representable by logarithmic schemes, all of which are obtained by pullback from subdivisions of the tropical Picard group.Comment: 3 figures (2 added), 66 pages; comments welcome

An introduction to moduli stacks, with a view towards Higgs bundles on algebraic curves

This article is based in part on lecture notes prepared for the summer school "The Geometry, Topology and Physics of Moduli Spaces of Higgs Bundles" at the Institute for Mathematical Sciences at the National University of Singapore in July of 2014. The aim is to provide a brief introduction to algebraic stacks, and then to give several constructions of the moduli stack of Higgs bundles on algebraic curves. The first construction is via a "bootstrap" method from the algebraic stack of vector bundles on an algebraic curve. This construction is motivated in part by Nitsure's GIT construction of a projective moduli space of semi-stable Higgs bundles, and we describe the relationship between Nitsure's moduli space and the algebraic stacks constructed here. The third approach is via deformation theory, where we directly construct the stack of Higgs bundles using Artin's criterion.Comment: 145 pages, AMS LaTeX, to appear in the NUS IMS Lecture Note Series on The Geometry, Topology, and Physics of Moduli Spaces of Higgs Bundle

Social Security Programs and Retirement around the World: Fiscal Implications, Introduction and Summary

This is the introduction to and summary of Phase III of an international research project to study the relationship between social security provisions and retirement. The project relies on the work of a large group of economists in 12 countries who conduct the analysis for each of their countries. The first phase described the retirement incentives inherent in plan provisions and documented the strong relationship across countries between social security incentives to retire and the proportion of older persons out of the labor force. The second phase illustrated the large effects that changing plan provisions would have on the labor force participation of older workers. This third phase shows the consequent fiscal implications that extending labor force participation would have on net program costs -- reduced government social security benefit payments less increased government tax revenues. The findings are conveyed by simulating the implications of illustrative reforms. One reform increases benefit eligibility ages by three years. Another illustrative reform reduces actuarially benefits received before the normal retirement age. A common reform prescribes the same provisions in each country. The financial implications of the illustrative reforms are very large in many instances, often as much as 20 to 40 percent of current program costs. The savings amount to as much a 1 percent or more of country GDP. The results make clear that reforms like those considered in this volume can have very large fiscal implications for the cost of social security benefits as well as for government revenues engendered by changes in the labor force participation of older workers.