Evolution equations displaying complex spatiotemporal behaviour

空间扩展的物理、化学或生物系统的时间演化通常由非线性偏微分方程组(PDE)来模拟，其解往往表现出复杂的时空动力学和图案形成。所观察到的行为往往源于许多不稳定自由度的非线性相互作用、一系列空间尺度上的复杂动力学、能量级联以及空间和时间上的混沌。 对于大多数感兴趣的非线性、高维系统，如描述流体流动的纳维斯托克斯方程，详细的分析理解仍然遥不可及。然而，即使在那时，有时也可以获得对流动的总体性质的数学上的严格估计。我对瑞利-贝纳德问题特别感兴趣，瑞利-伯纳德问题是两个水平板之间的流体层，从下面加热，作为对流模型在许多天体物理、地球物理和工程背景下都有应用。经过几十年密集的理论和实验研究，整体换热与平板温降的完全相关性仍不完全清楚。当数学定理可用时，可以通过聚焦于本质并对非严格的现象学理论的预测建立约束来帮助澄清这种情况。我的工作主要涉及板的热性能的影响：我以前已经展示了如何将这些系统地结合到数学形式中，并建议继续在这些结果的基础上证明严格的界限，从而帮助理解在各种流动情况下热传输的标度。 我的提议的另一个主要主题是研究显示时空复杂和混乱行为的一维PDE模型。通过只包含与正在研究的现象相关的基本项，这些方程可以产生与更复杂和更高维度的问题相关的见解，在这些问题中，比方说，特定对称或不稳定的影响可能很难分离出来。我的方法是将分析工具--同样包括对平均数量的严格估计--与仔细的数字相结合，以寻求对动态的详细理解。目前，我对存在某种对称性(伽利略不变性)的模式形成模型特别感兴趣。我们之前的工作展示了异常的标度和出人意料的丰富的动力学，表面上看起来似乎是一个相对“简单”的方程系统；正在进行的研究旨在更深入地理解这种明显新颖的时空混沌类型。 在另一个方向上，作为隶属于不列颠哥伦比亚省艾滋病毒/艾滋病卓越中心的跨学科研究小组(Impact-HIV)的成员，我致力于与艾滋病毒有关的数学模型。现代抗逆转录病毒治疗可以通过减少病毒载量从而降低艾滋病毒阳性个体的传染性来帮助减缓艾滋病毒的传播；这促使目前加紧努力建立治疗即预防计划：扩大检测和早期治疗，以抗击流行病。我们的流行病学模拟研究将分析和计算方法与人口调查数据相结合，以模拟流行病的发展和治疗的效果，目的是评估不同的方法，从而帮助设计最佳干预策略。

The time evolution of spatially extended physical, chemical or biological systems is commonly modelled by nonlinear partial differential equations (PDEs), whose solutions frequently display complex spatial and temporal dynamics and pattern formation. The observed behaviour often arises from the nonlinear interaction of many unstable degrees of freedom, complicated dynamics over a range of spatial scales, energy cascades, and chaos in space and time. For most nonlinear, high-dimensional systems of interest, such as the Navier-Stokes equations describing fluid flow, a detailed analytical understanding remains well out of reach. Nevertheless, even then mathematically rigorous estimates on bulk properties of the flows may sometimes be obtained. I am particularly interested in the Rayleigh-Bénard problem of a fluid layer between two horizontal plates, heated from below, which as a model for convection has applications in many astrophysical, geophysical and engineering contexts. After decades of intensive theoretical and experimental study, the full dependence of the bulk heat transfer in terms of the temperature drop across the plates remains incompletely understood. Mathematical theorems, when available, can help clarify such a situation by focussing on the essentials and establishing constraints on the predictions of nonrigorous phenomenological theories. My work mainly concerns the influence of the thermal properties of the plates: I have previously shown how to incorporate these systematically into the mathematical formalism, and propose to continue building on those results to prove rigorous bounds and thereby help understand the scaling of heat transport in a variety of flow situations. An additional major theme of my proposal is the study of model one-dimensional PDEs displaying spatiotemporally complex and chaotic behaviour. By containing only the essential terms relevant to the phenomena being investigated, such equations can yield insights relevant to more complicated and higher-dimensional problems in which the effects of, say, a particular symmetry or instability may be difficult to isolate. My approach is to combine analytical tools - including, again, rigorous estimates on averaged quantities - with careful numerics to seek a detailed understanding of the dynamics. I am currently particularly interested in a model for pattern formation in the presence of a certain symmetry (Galilean invariance). Our previous work demonstrated anomalous scaling and unexpectedly rich dynamics in what appeared, on the surface, to be a relatively "simple-looking" system of equations; and ongoing research is aimed at a deeper understanding of this apparently novel type of spatiotemporal chaos. In a separate direction, as part of an interdisciplinary research group (IMPACT-HIV) affiliated with the BC Centre for Excellence in HIV/AIDS, I work on mathematical models related to HIV. Modern antiretroviral treatments can help slow the spread of HIV, by reducing viral loads and hence the infectivity of HIV-positive individuals; this inspires current intensive efforts to build Treatment as Prevention programs: expanded testing and early treatment to combat the epidemic. Our epidemiological modelling research combines analytical and computational approaches with population survey data to model the progression of the epidemic and the effects of treatment, with the goal of assessing different approaches and thereby helping to design optimal intervention strategies.