Warwick EPSRC Symposium on Fluctuation-driven Phenomena and Large Deviations

华威EPSRC数学研讨会由华威大学在EPSRC的支持下每年组织一次，旨在为英国的数学科学界带来利益。它将领先的国内和国际专家与英国研究人员聚集在一起，进行为期一年的研究活动计划，重点关注数学科学中的一个新兴主题。拟在2015-16学年举办的研讨会将以“波动驱动现象和大偏差”为主题。在非常一般的术语，研讨会将构成一个跨学科的重点，了解随机性和非线性之间的相互作用的后果，一个反复出现的挑战，在数学科学，工程和工业的许多领域。随机过程在数学科学中扮演着重要的角色，它既是构建模型的工具，也是抽象的数学结构。然而，当引入随机过程之间的非线性相互作用时，根据随机分析对结果方程的严格理解变得非常具有挑战性。平均场理论是一种有用的启发式方法，通常在数学之外用于处理这一问题。平均场理论以这样或那样的方式通常涉及用随机变量的平均值代替随机变量，并假设平均值的波动近似为高斯分布。在某些情况下，这样的模型提供了原始系统的良好描述，并且可以严格地证明。在许多情况下，他们没有。理解后一种情况，即平均场模型失败的情况，是本次研讨会的核心挑战。我们使用“波动驱动现象”作为一个通用术语来描述平均场理论失效时所观察到的各种效应。挑战源于这样一个事实，即确定性非线性动力学的丰富现象学（奇点，非线性共振，混沌等）在随机环境中通过各种有趣的，有时不直观的行为反映出来：长距离相关性，强非高斯统计，相干结构，吸收态相变，重尾概率分布和大偏差的增强概率。这种现象在纯数学和应用数学、物理科学、生物科学和工程科学中都可以找到，同时也向实业家和决策者提出了特殊的问题。预测极端天气事件、设计海洋基础设施以抵御所谓的“异常巨浪”、量化气候系统中波动驱动的过渡或“引爆点”的概率或估计确保基础设施系统对冲击具有弹性所需的冗余度等当代问题，都需要我们逐步提高建模和预测此类波动驱动现象的能力。因此，构成这次专题讨论会的研究活动方案将包括从理论到应用的各个方面。在理论方面，我们有随机矩阵理论，它最近作为一种强大的工具出现，用于分析强非高斯随机过程的统计数据，而不需要通过物理科学中开发的微扰技术，如重整化组。在应用方面，我们面临着对保险业至关重要的问题，比如如何计算极端自然灾害的风险成本，以及如何量化它们与人为系统中固有风险的相互作用。在此期间，我们研究了大偏差理论与非平衡统计力学之间的联系，地球科学中的极端事件，生物科学中的随机性以及计算罕见事件的最新数值算法，这是近年来增长强劲的主题。

The Warwick EPSRC mathematics symposium is organised annually by the University of Warwick with the support of the EPSRC for the benefit of the mathematical sciences community in the UK. It brings leading national and international experts together with UK researchers in a year-long programme of research activities focused on an emerging theme in the mathematical sciences. The proposed symposium for the 2015-16 academic year will concentrate on the theme of "Fluctuation-driven phenomena and large deviations". In very general terms, the symposium will constitute an interdisciplinary focus on understanding the consequences of the interplay between stochasticity and nonlinearity, a recurrent challenge in many areas of the mathematical sciences, engineering and industry.Stochastic processes play a fundamental role in the mathematical sciences, both as tools for constructing models and as abstract mathematical structures in their own right. When nonlinear interactions between stochastic processes are introduced, however, the rigorous understanding of the resulting equations in terms of stochastic analysis becomes very challenging. Mean field theories are useful heuristics which are commonly employed outside of mathematics for dealing with this problem. Mean field theories in one way or another usually involve replacing random variables by their mean and assuming that fluctuations about the mean are approximately Gaussian distributed. In some cases, such models provide a good description of the original system and can be rigorously justified. In many cases they do not. Understanding the latter case, where mean-field models fail, is the central challenge of this symposium. We use "fluctuation driven phenomena" as a generic term to describe the kinds of effects which are observed when mean field theories fail.The challenges stem from the fact that the rich phenomenology of deterministic nonlinear dynamics (singularities, nonlinear resonance, chaos and so forth) is reflected in the stochastic context by a variety of interesting and sometimes unintuitive behaviours: long range correlations, strongly non-Gaussian statistics, coherent structures, absorbing state phase transitions, heavy-tailed probability distributions and enhanced probabilities of large deviations. Such phenomena are found throughout mathematics, both pure and applied, the physical, biological and engineering sciences as well as presenting particular problems to industrialists and policymakers. Contemporary problems such as the forecasting of extreme weather events, the design of marine infrastructure to withstand so-called "rogue waves", quantifying the probability of fluctuation driven transitions or "tipping points" in the climate system or estimating the redundancy required to ensure that infrastructure systems are resilient to shocks all require a step change in our ability to model and predict such fluctuation-driven phenomena. The programme of research activities constituting this symposium will therefore range from the very theoretical to the very applied.At the theoretical end we have random matrix theory which has recently emerged as a powerful tool for analysing the statistics of stochastic processes which are strongly non-Gaussian without the need to go via perturbative techniques developed in the physical sciences such as the renormalisation group. At the applied end we have questions of existential importance to the insurance industry such as how to cost the risk of extreme natural disasters and quantify their interaction with risks inherent in human-built systems. In between we have research on the connections between large deviation theory and nonequilibrium statistical mechanics, extreme events in the Earth sciences, randomness in the biological sciences and the latest numerical algorithms for computing rare events, a topic which has seen strong growth recent years.