Tropical covers of curves and their moduli spaces

We define the tropical moduli space of covers of a tropical line in the plane as weighted abstract polyhedral complex, and the tropical branch map recording the images of the simple ramifications. Our main result is the invariance of the degree of the branch map, which enables us to give a tropical intersection-theoretic definition of tropical triple Hurwitz numbers. We show that our intersection-theoretic definition coincides with the one given by Bertrand, Brugall\'e and Mikhalkin in the article "Tropical Open Hurwitz numbers" where a Correspondence Theorem for Hurwitz numbers is proved. Thus we provide a tropical intersection-theoretic justification for the multiplicities with which a tropical cover has to be counted. Our method of proof is to establish a local duality between our tropical moduli spaces and certain moduli spaces of relative stable maps to the projective line.Comment: 24 pages, 10 figure

Tropical mirror symmetry for elliptic curves

Mirror symmetry relates Gromov-Witten invariants of an elliptic curve with certain integrals over Feynman graphs. We prove a tropical generalization of mirror symmetry for elliptic curves, i.e., a statement relating certain labeled Gromov-Witten invariants of a tropical elliptic curve to more refined Feynman integrals. This result easily implies the tropical analogue of the mirror symmetry statement mentioned above and, using the necessary Correspondence Theorem, also the mirror symmetry statement itself. In this way, our tropical generalization leads to an alternative proof of mirror symmetry for elliptic curves. We believe that our approach via tropical mirror symmetry naturally carries the potential of being generalized to more adventurous situations of mirror symmetry. Moreover, our tropical approach has the advantage that all involved invariants are easy to compute. Furthermore, we can use the techniques for computing Feynman integrals to prove that they are quasimodular forms. Also, as a side product, we can give a combinatorial characterization of Feynman graphs for which the corresponding integrals are zero. More generally, the tropical mirror symmetry theorem gives a natural interpretation of the A-model side (i.e., the generating function of Gromov-Witten invariants) in terms of a sum over Feynman graphs. Hence our quasimodularity result becomes meaningful on the A-model side as well. Our theoretical results are complemented by a Singular package including several procedures that can be used to compute Hurwitz numbers of the elliptic curve as integrals over Feynman graphs.Comment: comment on historical development adde

Tropical covers, moduli spaces & mirror symmetry

Classical Hurwitz theory is the theory of ramified coverings of a Riemann surface, which in various ways has influence on other fields of mathematics. Tropical geometry considers degenerations of such objects from algebraic geometry, which can be naturally treated by combinatorial means. In my thesis I unify definitions for tropical Hurwitz theory and give an intrinsic definition of tropical Hurwitz numbers as degree of a tropical branch map. Furthermore I use tropical Hurwitz theory to construct a new, non-physical proof for mirror symmetry of elliptic curves by prooving tropical mirror symmetry for elliptic curves and using existing correspondency theorems that link tropical geometry objects to algebraic geometry objects. In this context tropical mirror symmetry - as also mentioned by Mark Gross - seems to be so naturally, that it recommends the use of tropical geometry for proving mirror symmetry statements.Klassische Hurwitztheorie ist die Theorie der verzweigten Überlagerungen einer Riemannschen Fläche, die auf vielfältige Art in andere Bereiche der Mathematik einfließt. In der tropischen Geometrie betrachtet man Degenerationen derartiger Objekte der algebraischen Geometrie, die auf natürliche Art und Weise mit kombinatorischen Mitteln behandelt werden können. In meiner Arbeit vereinheitliche ich zunächst die in der Literatur zu findenden Definitionen für tropische Hurwitztheorie und gebe eine intrinsische Definition für tropische Hurwitzzahlen als Grad einer tropischen Verzweigungsabbildung. Des Weiteren benutze ich tropische Hurwitztheorie, um einen neuen, nicht-physikalischen Beweis der Spiegelsymmetrie von elliptischen Kurven zu konstruieren, indem ich die tropische Spiegelsymmetrie von elliptischen Kurve beweise und schon vorhandene Korrespondenzsätze zur algebraischen Geometrie nutze. Die tropische Spiegelsymmtrie erscheint in diesem Zusammenhang - wie auch bei Mark Gross - derart natürlich, dass Sie die Verwendung der tropischen Geometrie zum Beweis von Aussagen der Spiegelsymmetrie nahelegt

Quercetin Protects Primary Human Osteoblasts Exposed to Cigarette Smoke through Activation of the Antioxidative Enzymes HO-1 and SOD-1

Smokers frequently suffer from impaired fracture healing often due to poor bone quality and stability. Cigarette smoking harms bone cells and their homeostasis by increased formation of reactive oxygen species (ROS). The aim of this study was to investigate whether Quercetin, a naturally occurring antioxidant, can protect osteoblasts from the toxic effects of smoking. Human osteoblasts exposed to cigarette smoke medium (CSM) rapidly produced ROS and their viability decreased concentration- and time-dependently. Co-, pre- and postincubation with Quercetin dose-dependently improved their viability. Quercetin increased the expression of the anti-oxidative enzymes heme-oxygenase- (HO-) 1 and superoxide-dismutase- (SOD-) 1. Inhibiting HO-1 activity abolished the protective effect of Quercetin. Our results demonstrate that CSM damages human osteoblasts by accumulation of ROS. Quercetin can diminish this damage by scavenging the radicals and by upregulating the expression of HO-1 and SOD-1. Thus, a dietary supplementation with Quercetin could improve bone matter, stability and even fracture healing in smokers

Efficacy of Budesonide Orodispersible Tablets as Induction Therapy for Eosinophilic Esophagitis in a Randomized Placebo-Controlled Trial.

BACKGROUND & AIMS: Swallowed topical-acting corticosteroids are recommended as first-line therapy for eosinophilic esophagitis (EoE). Asthma medications not optimized for esophageal delivery are sometimes effective, although given off-label. We performed a randomized, placebo-controlled trial to assess the effectiveness and tolerability of a budesonide orodispersible tablet (BOT), which allows the drug to be delivered to the esophagus in adults with active EoE. METHODS: We performed a double-blind, parallel study of 88 adults with active EoE in Europe. Patients were randomly assigned to groups that received BOT (1 mg twice daily; n = 59) or placebo (n = 29) for 6 weeks. The primary end point was complete remission, based on clinical and histologic factors, including dysphagia and odynophagia severity ≤2 on a scale of 0-10 on each of the 7 days before the end of the double-blind phase and a peak eosinophil count <5 eosinophils/high power field. Patients who did not achieve complete remission at the end of the 6-week double-blind phase were offered 6 weeks of open-label treatment with BOT (1 mg twice daily). RESULTS: At 6 weeks, 58% of patients given BOT were in complete remission compared with no patients given placebo (P < .0001). The secondary end point of histologic remission was achieved by 93% of patients given BOT vs no patients given placebo (P < .0001). After 12 weeks, 85% of patients had achieved remission. Six-week and 12-week BOT administration were safe and well tolerated; 5% of patients who received BOT developed symptomatic, mild candida, which was easily treated with an oral antifungal agent. CONCLUSIONS: In a randomized trial of adults with active EoE, we found that budesonide oral tablets were significantly more effective than placebo in inducing clinical and histologic remission. Eudra-CT number 2014-001485-99; ClinicalTrials.gov ID NCT02434029

Search for dark matter produced in association with bottom or top quarks in √s = 13 TeV pp collisions with the ATLAS detector

A search for weakly interacting massive particle dark matter produced in association with bottom or top quarks is presented. Final states containing third-generation quarks and miss- ing transverse momentum are considered. The analysis uses 36.1 fb−1 of proton–proton collision data recorded by the ATLAS experiment at √s = 13 TeV in 2015 and 2016. No significant excess of events above the estimated backgrounds is observed. The results are in- terpreted in the framework of simplified models of spin-0 dark-matter mediators. For colour- neutral spin-0 mediators produced in association with top quarks and decaying into a pair of dark-matter particles, mediator masses below 50 GeV are excluded assuming a dark-matter candidate mass of 1 GeV and unitary couplings. For scalar and pseudoscalar mediators produced in association with bottom quarks, the search sets limits on the production cross- section of 300 times the predicted rate for mediators with masses between 10 and 50 GeV and assuming a dark-matter mass of 1 GeV and unitary coupling. Constraints on colour- charged scalar simplified models are also presented. Assuming a dark-matter particle mass of 35 GeV, mediator particles with mass below 1.1 TeV are excluded for couplings yielding a dark-matter relic density consistent with measurements