Kristian Strommen

Abstract:

Let (Formula presented.) be a field with (Formula presented.), where (Formula presented.) denotes the maximal pro-2 quotient of the absolute Galois group of a field (Formula presented.). We prove that then (Formula presented.) admits a (non-trivial) valuation (Formula presented.) which is 2-henselian and has residue field (Formula presented.). Furthermore, (Formula presented.) is a minimal positive element in the value group (Formula presented.) and (Formula presented.). This forms the first positive result on a more general conjecture about recovering (Formula presented.) -adic valuations from pro- (Formula presented.) Galois groups which we formulate precisely. As an application, we show how this result can be used to easily obtain number-theoretic information, by giving an independent proof of a strong version of the birational section conjecture for smooth, complete curves (Formula presented.) over (Formula presented.), as well as an analogue for varieties.

Abstract:

It has been demonstrated that decadal variations in the North Atlantic Oscillation (NAO) can be predicted by current forecast models. While Atlantic Multidecadal Variability (AMV) in sea surface temperatures (SSTs) has been hypothesised as the source of this skill, the validity of this hypothesis and the pathways involved remain unclear. We show, using reanalysis and data from two forecast models, that the decadal predictability of the NAO can be entirely accounted for by the predictability of decadal variations in the speed of the North Atlantic eddy-driven jet, with no predictability of decadal variations in the jet latitude. The sub-polar North Atlantic (SPNA) is identified as the only obvious common source of an SST-based signal across the models and reanalysis, and the predictability of the jet speed is shown to be consistent with a forcing from the SPNA visible already within a single season. The pathway is argued to be tropospheric in nature, with the SPNA-associated heating extending up to the mid-troposphere, which alters the meridional temperature gradient around the climatological jet core. The relative roles of anthropogenic aerosol emissions and the Atlantic Meridional Overturning Circulation (AMOC) at generating predictable SPNA variability are also discussed. The analysis is extensively supported by the novel use of a set of seasonal hindcasts spanning the 20th century and forced with prescribed SSTs.

Abstract:

The ‘signal-to-noise paradox’ for seasonal forecasts of the winter NAO is often described as an ‘underconfident’ forecast and measured using the ratio-of-predictable components metric (RPC). However, comparison of RPC with other measures of forecast confidence, such as spread-error ratios, can give conflicting impressions, challenging this informal description. We show, using a linear statistical model, that the ‘paradox’ is equivalent to a situation where the reliability diagram of any percentile forecast has a slope exceeding 1. The relationship with spread-error ratios is shown to be far less direct. We furthermore compute reliability diagrams of winter NAO forecasts using seasonal hindcasts from the European Centre for Medium-range Weather Forecasts and the UK Meteoro logical Office. While these broadly exhibit slopes exceeding 1, there is evidence of asymmetry between upper and lower terciles, indicating a potential violation of linearity/Gaussianity. The limitations and benefits of reliability diagrams as a diagnostic tool are discussed.

Abstract:

It has long been suggested that the mid-latitude atmospheric circulation possesses what has come to be known as ‘weather regimes’, loosely categorised as regions of phase space with above-average density and/or extended persistence. Their existence and behaviour has been extensively studied in meteorology and climate science, due to their potential for drastically simplifying the complex and chaotic mid-latitude dynamics. Several well-known, simple non-linear dynamical systems have been used as toy-models of the atmosphere in order to understand and exemplify such regime behaviour. Nevertheless, no agreed-upon and clear-cut definition of a ‘regime’ exists in the literature, and unambiguously detecting their existence in the atmospheric circulation is stymied by the high dimensionality of the system. We argue here for an approach which equates the existence of regimes in a dynamical system with the existence of non-trivial topological structure of the system’s attractor. We show using persistent homology, an algorithmic tool in topological data analysis, that this approach is computationally tractable, practically informative, and identifies the relevant regime structure across a range of examples.