Compact Riemann surfaces : prime Galois coverings of P¹

The uniqueness of the hyperelliptic involution is well known in the theory of Riemann surfaces. More precisely, we know that if X is a hyperelliptic compact Riemann surface, there is a unique automorphism τ of order 2 such that X/〈τ〉 ≅ ℙ¹ . We wish to generalize the situation slightly. We say X is a prime Galois covering of ℙ¹ if there exists an automorphism τ of (odd) prime order p such that X/〈T〉 ≅ ℙ¹. This leads us to ask the question: When is this automorphism τ unique? We begin by building the necessary background to understand prime Galois coverings of ℙ¹. We then prove a theorem due to Gonzlez-Diez that answers our question about uniqueness. The proof given here follows his proof (given in [G-D]) quite closely, though we elaborate and modify certain details to make it more self contained.Science, Faculty ofMathematics, Department ofGraduat

Quantifying the influence of global warming on unprecedented extreme climate events

Efforts to understand the influence of historical global warming on individual extreme climate events have increased over the past decade. However, despite substantial progress, events that are unprecedented in the local observational record remain a persistent challenge. Leveraging observations and a large climate model ensemble, we quantify uncertainty in the influence of global warming on the severity and probability of the historically hottest month, hottest day, driest year, and wettest 5-d period for different areas of the globe. We find that historical warming has increased the severity and probability of the hottest month and hottest day of the year at >80% of the available observational area. Our framework also suggests that the historical climate forcing has increased the probability of the driest year and wettest 5-d period at 57% and 41% of the observed area, respectively, although we note important caveats. For the most protracted hot and dry events, the strongest and most widespread contributions of anthropogenic climate forcing occur in the tropics, including increases in probability of at least a factor of 4 for the hottest month and at least a factor of 2 for the driest year. We also demonstrate the ability of our framework to systematically evaluate the role of dynamic and thermodynamic factors such as atmospheric circulation patterns and atmospheric water vapor, and find extremely high statistical confidence that anthropogenic forcing increased the probability of record-low Arctic sea ice extent

Quantifying the Influence of Global Warming on Unprecedented Extreme Climate Events

Efforts to understand the influence of historical global warming on individual extreme climate events have increased over the past decade. However, despite substantial progress, events that are unprecedented in the local observational record remain a persistent challenge. Leveraging observations and a large climate model ensemble, we quantify uncertainty in the influence of global warming on the severity and probability of the historically hottest month, hottest day, driest year, and wettest 5-d period for different areas of the globe. We find that historical warming has increased the severity and probability of the hottest month and hottest day of the year at >80% of the available observational area. Our framework also suggests that the historical climate forcing has increased the probability of the driest year and wettest 5-d period at 57% and 41% of the observed area, respectively, although we note important caveats. For the most protracted hot and dry events, the strongest and most widespread contributions of anthropogenic climate forcing occur in the tropics, including increases in probability of at least a factor of 4 for the hottest month and at least a factor of 2 for the driest year. We also demonstrate the ability of our framework to systematically evaluate the role of dynamic and thermodynamic factors such as atmospheric circulation patterns and atmospheric water vapor, and find extremely high statistical confidence that anthropogenic forcing increased the probability of record-low Arctic sea ice extent