Mutations of fake weighted projective planes

In previous work by Coates, Galkin, and the authors, the notion of mutation between lattice polytopes was introduced. Such a mutation gives rise to a deformation between the corresponding toric varieties. In this paper we study one-step mutations that correspond to deformations between weighted projective planes, giving a complete characterisation of such mutations in terms of T-singularities. We show also that the weights involved satisfy Diophantine equations, generalising results of Hacking-Prokhorov

Roots of Ehrhart Polynomials of Smooth Fano Polytopes

V. Golyshev conjectured that for any smooth polytope P of dimension at most five, the roots z\in\C of the Ehrhart polynomial for P have real part equal to -1/2. An elementary proof is given, and in each dimension the roots are described explicitly. We also present examples which demonstrate that this result cannot be extended to dimension six.Comment: 10 page

Few smooth d-polytopes with n lattice points

We prove that, for fixed n there exist only finitely many embeddings of Q-factorial toric varieties X into P^n that are induced by a complete linear system. The proof is based on a combinatorial result that for fixed nonnegative integers d and n, there are only finitely many smooth d-polytopes with n lattice points. We also enumerate all smooth 3-polytopes with at most 12 lattice points. In fact, it is sufficient to bound the singularities and the number of lattice points on edges to prove finiteness.Comment: 20+2 pages; major revision: new author, new structure, new result

The Next Frontier: Making Research More Reproducible

Science and engineering rest on the concept of reproducibility. An important question for any study is: are the results reproducible? Can the results be recreated independently by other researchers or professionals? Research results need to be independently reproduced and validated before they are accepted as fact or theory. Across numerous fields like psychology, computer systems, and water resources there are problems to reproduce research results (Aarts et al. 2015; Collberg et al. 2014; Hutton et al. 2016; Stagge et al. 2019; Stodden et al. 2018). This editorial examines the challenges to reproduce research results and suggests community practices to overcome these challenges. Coordination is needed among the authors, journals, funders and institutions that produce, publish, and report research. Making research more reproducible will allow researchers, professionals, and students to more quickly understand and apply research in follow-on efforts and advance the field

Temporal and spatiotemporal autocorrelation of daily concentrations of Alnus, Betula, and Corylus pollen in Poland

The aim of the study was to determine the characteristics of temporal and space–time autocorrelation of pollen counts of Alnus, Betula, and Corylus in the air of eight cities in Poland. Daily average pollen concentrations were monitored over 8 years (2001–2005 and 2009–2011) using Hirst-designed volumetric spore traps. The spatial and temporal coherence of data was investigated using the autocorrelation and cross-correlation functions. The calculation and mathematical modelling of 61 correlograms were performed for up to 25 days back. The study revealed an association between temporal variations in Alnus, Betula, and Corylus pollen counts in Poland and three main groups of factors such as: (1) air mass exchange after the passage of a single weather front (30–40 % of pollen count variation); (2) long-lasting factors (50–60 %); and (3) random factors, including diurnal variations and measurements errors (10 %). These results can help to improve the quality of forecasting models

Hilbert Series, Machine Learning, and Applications to Physics

We describe how simple machine learning methods successfully predict geometric properties from Hilbert series (HS). Regressors predict embedding weights in projective space to ${\sim}1$ mean absolute error, whilst classifiers predict dimension and Gorenstein index to $>90\%$ accuracy with ${\sim}0.5\%$ standard error. Binary random forest classifiers managed to distinguish whether the underlying HS describes a complete intersection with high accuracies exceeding $95\%$. Neural networks (NNs) exhibited success identifying HS from a Gorenstein ring to the same order of accuracy, whilst generation of 'fake' HS proved trivial for NNs to distinguish from those associated to the three-dimensional Fano varieties considered