The skeleton of the Jacobian, the Jacobian of the skeleton, and lifting meromorphic functions from tropical to algebraic curves

Let K be an algebraically closed field which is complete with respect to a nontrivial, non-Archimedean valuation and let \Lambda be its value group. Given a smooth, proper, connected K-curve X and a skeleton \Gamma of the Berkovich analytification X^\an, there are two natural real tori which one can consider: the tropical Jacobian Jac(\Gamma) and the skeleton of the Berkovich analytification Jac(X)^\an. We show that the skeleton of the Jacobian is canonically isomorphic to the Jacobian of the skeleton as principally polarized tropical abelian varieties. In addition, we show that the tropicalization of a classical Abel-Jacobi map is a tropical Abel-Jacobi map. As a consequence of these results, we deduce that \Lambda-rational principal divisors on \Gamma, in the sense of tropical geometry, are exactly the retractions of principal divisors on X. We actually prove a more precise result which says that, although zeros and poles of divisors can cancel under the retraction map, in order to lift a \Lambda-rational principal divisor on \Gamma to a principal divisor on X it is never necessary to add more than g extra zeros and g extra poles. Our results imply that a continuous function F:\Gamma -> R is the restriction to \Gamma of -log|f| for some nonzero meromorphic function f on X if and only if F is a \Lambda-rational tropical meromorphic function, and we use this fact to prove that there is a rational map f : X --> P^3 whose tropicalization, when restricted to \Gamma, is an isometry onto its image.Comment: 21 pages, 1 figur

Perception in Aristotle’s Ethics

Perception in Aristotle's Ethics seeks to demonstrate that living an ethical life requires a mode of perception that is best called ethical perception. Specifically, drawing primarily on Aristotle’s accounts of perception and ethics in De anima and Nicomachean Ethics, Eve Rabinoff argues that the faculty of perception (aisthesis), which is often thought to be an entirely physical phenomenon, is informed by intellect and has an ethical dimension insofar as it involves the perception of particulars in their ethical significance, as things that are good or bad in themselves and as occasions to act. Further, she contends, virtuous action requires this ethical perception, according to Aristotle, and ethical development consists in the achievement of the harmony of the intellectual and perceptual, rational and nonrational, parts of the soul. Rabinoff's project is philosophically motivated both by the details of Aristotle’s thought and more generally by an increasing philosophical awareness that the ethical agent is an embodied, situated individual, rather than primarily a disembodied, abstract rational will