Bifurcations: traveling waves in reaction-diffusion systems, and realizability for delay differential equations

微分方程是我们在日常生活中经历的几乎每一种物理现象的数学模型。我的研究计划包括研究这些类型的方程，揭示其解决方案的基本属性，从而提供洞察正在建模的物理现象。我对控制电波传播的微分方程模型感兴趣，电波使人类心脏收缩和泵血，或通过神经系统传输信号。我在这一领域的研究项目涉及数学研究这些波的传播上的缺陷（例如病变组织）的影响。 例如，在心脏中，这些缺陷的影响可能导致严重的情况，如心动过速和心室纤颤（通过折返波），这可能导致死亡。 我的研究计划还涉及研究一类特殊的微分方程，称为延迟微分方程，这是经常被用来作为模型的生物系统中存在的时间延迟。人类的神经系统就是这种系统的一个很好的例子，其中延迟微分方程模拟了从大脑到感觉器官再返回的电脉冲的传播（与这种来回传播相关的是有限的时滞或延迟）。这些方程有时也被用来模拟给药的病人，机器抖动的工具，瞳孔光反射，疾病爆发和疫苗接种策略等，我在这方面的努力集中在开发分析方法来研究这些类型的方程，最终目标是阐明这些方程所模拟的生物或物理系统的行为。 正如提案中所述，这项研究计划将继续（像过去一样）为若干本科生、硕士生、博士生和博士后的科学培训做出贡献。

Differential equations are mathematical models for almost every physical phenomenon we experience in everyday life. My research program involves studying these types of equations, uncovering basic properties about their solutions, and thus providing insight into the physical phenomenon which is being modelled. I am interested in differential equation models governing the propagation of electrical waves which make the human heart contract and pump blood, or transmit signals through the nervous system. My research projects in this area involve studying mathematically the effects of imperfections (e.g. diseased tissue) on the propagation of these waves. For example, in the heart, the effects of these imperfections can lead to serious conditions such as tachycardia and ventricular fibrillation (via re-entrant waves) which can cause death. My research program also involves studying a special class of differential equations, called delay-differential equations, which are frequently used as models for biological systems in which time-delays are present. The human nervous system is a wonderful example of such a system, where the delay-differential equation models the propagation of pulses of electricity that travel from the brain to the sensory organs and back again (there is a finite time-lag, or delay, associated with this to-and-fro propagation). These equations are also sometimes used to model drug delivery in patients, machine chattering of tools, the pupil-light reflex, disease outbreaks and vaccination strategies, etc. My efforts in this area have concentrated on developing analytical methods to study these types of equations, with the ultimate goal of shedding some light on the behavior of the biological or physical systems which are modelled by these equations. This research program will continue to contribute (as it has in the past) to the scientific training of several undergraduate, masters, doctoral and post-doctoral students, as is described in the proposal.