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Warwick EPSRC Symposium on Fluctuation-driven Phenomena and Large Deviations

沃里克 EPSRC 波动驱动现象和大偏差研讨会

基本信息

批准号：
EP/M003620/1
负责人：
Colm Connaughton
金额：
$ 20.58万
依托单位：
University of Warwick
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2015
资助国家：
英国
起止时间：
2015 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FM003620%2F1
关键词：
Warwick EPSRC Symposium Fluctuation driven

项目摘要

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.

该沃里克EPSRC数学研讨会每年举办的沃里克大学的支持下，EPSRC的利益数学科学界在英国。它汇集了领先的国家和国际专家与英国研究人员在为期一年的研究活动计划集中在一个新兴的主题在数学科学。2015-16学年拟议的专题讨论会将集中讨论“波动驱动的现象和大偏差”的主题。在非常一般的条款，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，在数学科学，工程和工业的许多领域经常性的挑战。随机过程中发挥了基本作用的数学科学，无论是作为工具，用于构建模型和抽象的数学结构本身的权利。然而，当引入随机过程之间的非线性相互作用时，从随机分析的角度对所得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数学方法，通常在数学之外用来处理这个问题。平均场理论以某种方式通常涉及用随机变量的平均值来代替随机变量，并假设平均值的波动近似为高斯分布。在某些情况下，这些模型提供了对原始系统的良好描述，并且可以被严格证明。在许多情况下，他们并不这样做。理解后一种情况，即平均场模型失效的情况，是本次研讨会的核心挑战。我们用“涨落驱动现象”来描述当平均场理论失效时所观察到的各种效应，挑战来自于这样一个事实，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）在随机背景下反映为各种有趣的，有时是不直观的行为：长程相关，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。这种现象在整个数学中都可以找到，包括纯数学和应用数学，物理，生物和工程科学，以及向工业家和政策制定者提出特殊问题。诸如极端天气事件的预报、海洋基础设施的设计以抵御所谓的“巨浪”、量化波动驱动的气候系统过渡或“临界点”的概率或估计确保基础设施系统抵御冲击所需的冗余度等当代问题，都要求我们在建模和预测这种波动驱动的现象的能力方面发生重大变化。因此，本次研讨会的研究活动方案将从非常理论化的到非常实用的。在理论上，我们有随机矩阵理论，它是最近出现的一个强有力的工具，用于分析统计的随机过程，这是强非高斯，而不需要去通过微扰技术开发的物理科学，如重正化组。在应用端，我们有一些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固有风险的相互作用。在这两者之间，我们有大偏差理论和非平衡统计力学，地球科学中的极端事件，生物科学中的随机性和计算罕见事件的最新数值算法之间的联系的研究，这是一个近年来增长强劲的主题。