A new method to derive star formation histories of galaxies from their star cluster distributions

Star formation happens in a clustered way which is why the star cluster population of a particular galaxy is closely related to the star formation history of this galaxy. From the probabilistic nature of a mass function follows that the mass of the most-massive cluster of a complete population, M_max, has a distribution with the total mass of the population as a parameter. The total mass of the population is connected to the star formation rate (SFR) by the length of a formation epoch. Since due to evolutionary effects only massive star clusters are observable up to high ages it is convenient to use this M_max(SFR) relation for the reconstruction of a star formation history. The age-distribution of the most-massive clusters can therefore be used to constrain the star formation history of a galaxy. The method, including an assessment of the inherent uncertainties, is introduced with this contribution, while following papers will apply this method to a number of galaxies.Comment: MNRAS: in press, 10 pages, 9 figure

The totally nonnegative Grassmannian is a ball

We prove that three spaces of importance in topological combinatorics are homeomorphic to closed balls: the totally nonnegative Grassmannian, the compactification of the space of electrical networks, and the cyclically symmetric amplituhedron.Comment: 19 pages. v2: Exposition improved in many place

Regularity theorem for totally nonnegative flag varieties

We show that the totally nonnegative part of a partial flag variety $G/P$ (in the sense of Lusztig) is a regular CW complex, confirming a conjecture of Williams. In particular, the closure of each positroid cell inside the totally nonnegative Grassmannian is homeomorphic to a ball, confirming a conjecture of Postnikov.Comment: 63 pages, 2 figures; v2: Minor changes; v3: Final version to appear in J. Amer. Math. So

Root system chip-firing II: Central-firing

Jim Propp recently proposed a labeled version of chip-firing on a line and conjectured that this process is confluent from some initial configurations. This was proved by Hopkins-McConville-Propp. We reinterpret Propp's labeled chip-firing moves in terms of root systems: a "central-firing" move consists of replacing a weight $\lambda$ by $\lambda+\alpha$ for any positive root $\alpha$ that is orthogonal to $\lambda$. We show that central-firing is always confluent from any initial weight after modding out by the Weyl group, giving a generalization of unlabeled chip-firing on a line to other types. For simply-laced root systems we describe this unlabeled chip-firing as a number game on the Dynkin diagram. We also offer a conjectural classification of when central-firing is confluent from the origin or a fundamental weight.Comment: 30 pages, 6 figures, 1 table; v2, v3: minor revision

Root system chip-firing I: Interval-firing

Jim Propp recently introduced a variant of chip-firing on a line where the chips are given distinct integer labels. Hopkins, McConville, and Propp showed that this process is confluent from some (but not all) initial configurations of chips. We recast their set-up in terms of root systems: labeled chip-firing can be seen as a root-firing process which allows the moves $\lambda \to \lambda + \alpha$ for $\alpha\in \Phi^{+}$ whenever $\langle\lambda,\alpha^\vee\rangle = 0$, where $\Phi^{+}$ is the set of positive roots of a root system of Type A and $\lambda$ is a weight of this root system. We are thus motivated to study the exact same root-firing process for an arbitrary root system. Actually, this central root-firing process is the subject of a sequel to this paper. In the present paper, we instead study the interval root-firing processes determined by $\lambda \to \lambda + \alpha$ for $\alpha\in \Phi^{+}$ whenever $\langle\lambda,\alpha^\vee\rangle \in [-k-1,k-1]$ or $\langle\lambda,\alpha^\vee\rangle \in [-k,k-1]$, for any $k \geq 0$. We prove that these interval-firing processes are always confluent, from any initial weight. We also show that there is a natural way to consistently label the stable points of these interval-firing processes across all values of $k$ so that the number of weights with given stabilization is a polynomial in $k$. We conjecture that these Ehrhart-like polynomials have nonnegative integer coefficients.Comment: 54 pages, 12 figures, 2 tables; v2: major revisions to improve exposition; v3: to appear in Mathematische Zeitschrift (Math. Z.

La mesure de la pastorale

Le roman pastoral des xvie et xviie siècles raconte le destin d’êtres humains qui, oscillant entre la constance et la faiblesse, passent du dévouement au caprice et de l’inconstance à la fidélité. Bien que l’amour entraîne ces personnages vers la source du Bien, du Vrai et de l’Un, ils demeurent en proie aux troubles intérieurs et aux perturbations engendrés par le désir. La tension entre l’imperfection humaine et l’élan vers l’idéal se reflète dans l’alternance, typique pour le roman pastoral, de la narration en prose et du transport poétique. Or, en dépit de la coexistence de ces grandeurs incommensurables, des oeuvres comme l’Ameto de Boccace, l’Arcadie de Sannazaro, La Diane de Montemayor, Galatée de Cervantès, l’Arcadie de Sir Philip Sydney et L’Astrée d’Honoré d’Urfé dégagent une remarquable impression de mesure et d’équilibre, qui réunit la force de l’idéal aux impulsions les plus intimes de l’âme humaine. Cet équilibre est perceptible dans tous les ingrédients du roman pastoral : le cadre idyllique, l’intrigue, la psychologie morale et l’élégance du style. Située dans une Arcadie primitive qui ignore les conflits, la pastorale inclut également des épisodes appartenant au monde des rivalités sociales. Elle ne se contente pas d’une seule histoire, mais incorpore une gamme de situations où l’amour est confronté à la Fortune, à la duplicité du coeur humain et, enfin, à l’énigme de l’union entre corps et âme. Concernant la psychologie, la pastorale ne s’attarde pas aux replis du coeur humain mais peint des âmes que l’élan amoureux emporte au-delà des circonstances amères et boueuses de leur vie. Enfin, l’alternance de la prose et des passages en vers participe elle aussi à cette mesure sans doute fragile, à cet équilibre souvent instable, en soulignant à sa manière l’incommensurabilité de la faiblesse humaine avec le Beau et le Bien qui l’attirent.In Renaissance pastoral novels, characters oscillate between weakness and constancy, caprice and resolve, fickleness and perfect loyalty. Although True Love inevitably leads these characters towards the invisible source of the Good, pastoral heroes are nevertheless prey to the turmoil and distraction engendered by earthly passions. This clash between the contradictory nature of human frailty and the neo-platonic desire for an ideal unity is further mirrored by the mix of prose and poetry, a typical stylistic feature of the genre. In spite of the tension between these elements, a strong impression of equilibrium emerges from works like Boccaccio’s Ameto, Sannazaro’s Arcadia, Montemayor’s Diana, Cervantes’ Galatea, Sir Philip Sidney’s Old Arcadia and Honoré d’Urfé’s Astrea. An elusive, yet durable balance unites the movement towards the Ideal and the innermost contradictions of the soul. This balance is perceptible at various levels : the idyllic setting, the plot, the moral psychology and graceful discourse of the characters. Set in Arcadia, a legendary land assumed to be free of greed and discord, all pastoral novels include episodes which take place beyond Arcadia’s borders in the familiar world of wealth, social rank, rivalry and conflict. In addition to the dignified love story (or stories) narrated by the main plot, pastoral novels always stage a variety of situations depicting clashes between love and Fortune, the duplicity of the human heart, and the enigmatic links between body and soul. Yet these novels rarely linger on the inner struggles of the soul ; instead, they highlight True Love’s extraordinary capacity to carry it away, far beyond the often murky and bitter circumstances of its daily existence. Finally, the blend of prose and poetry reinforces the pastoral’s equilibrium and its inner measure, calling attention, in its own way, to the incommensurability between human imperfection and the irresistible attraction of the Ideal