The Unified Transform, Imaging and Asymptotics

过多的物理、化学、生物甚至社会过程，都可以用数学方程来模拟。这些过程中有许多涉及到连续的变化，然后相关的方程就会以微分方程组的形式出现。在包含一个以上变量的模型中，大多数情况下，相关的方程称为偏微分方程(PDE)。鉴于这些方程对模拟我们周围的世界很有帮助，开发适当的工具来求解偏微分方程组是至关重要的，以便能够适当地分析相关的模型。偏微分方程分为两大类：线性和非线性。19世纪初，伟大的法国数学家傅立叶提出了一种求解线性偏微分方程组的通用方法。非线性偏微分方程解起来要困难得多。1997年，PI引入了一种新的方法来求解一大类非线性偏微分方程组。在意想不到的发展中，这些结果推动了求解二维线性偏微分方程组的一种全新方法的发展。这是值得注意的，因为在此之前，人们一直认为傅立叶等人在18世纪开发的方法无法改进。三位作者在2014年3月出版的《SIAM评论》杂志上发表了题为《解线性偏微分方程组的福卡斯方法》的文章，并在随附的一篇社论中对这种方法进行了评论，并将这种方法在解线性偏微分方程组方面的重要性与跳高中“背越式跳高”的重要性进行了比较。该项目的第一部分包括完成上述方法在二维线性和非线性问题上的实现，然后将该方法从二维扩展到三维。几种医学成像技术，包括计算机层析成像、正电子发射层析成像(PET)和单光子发射计算机层析成像(SPECT)都是基于一类特殊的数学问题的解，称为逆问题。在这个项目的第二部分，将对PET和SPECT实施新的数值和分析技术。黎曼函数出现在许多不同的数学领域。与黎曼函数有关的几个猜想仍然是开放的，包括著名的黎曼假设和林德罗夫假设。该项目的第三部分涉及黎曼函数和相关函数的渐近性的分析，这有望增进我们对相关的、最重要的数学结构的理解。

A plethora of physical, chemical, biological and even social processes, can be modelled by mathematical equations. Many of these processes involve continuous change, and then the relevant equations take the form of differential equations. In models containing more than one variable, which is the great majority of situations, the relevant equations are called partial differential equations (PDEs). Given that these equations are instrumental in modelling the world around us, it is crucial that appropriate tools are developed for solving PDEs so that the associated models can be properly analysed. PDEs come into two broad categories: linear and non-linear. A general technique for solving linear PDEs was developed by the great French mathematician Fourier in the early 1800s. Non-linear PDEs are much more difficult to solve analytically. In 1997 the PI introduced a new method for solving a large class of non-linear PDEs. In an unexpected development, these results have motivated the development of a completely new method for solving linear PDEs in two dimensions. This is remarkable, since until then it was thought that the methods developed by Fourier and others in the 18th century could not be improved. This method is reviewed by three authors in the March 2014 issue of the Journal SIAM Review in the article titled "The Method of Fokas for solving linear PDEs", and in an accompanied editorial the importance of this method for solving linear PDEs is compared with the importance of the "Fosbury flop" in the high jump. The first part of this project involves completing the implementation of the above method to some important linear and non-linear problems in two dimensions, and then extending this method from 2 to 3 dimensions.Several medical imaging techniques, including Computed Tomography, Positron Emission Tomography (PET) and Single Photon Emission Computed Tomography (SPECT) are based on the solution of a particular class of mathematical problems, called inverse problems. In the second part of this project, new numerical and analytical techniques will be implemented for PET and SPECT.The Riemann function occurs in many different areas of mathematics. Several conjectures related to the Riemann function remain open, including the famous Riemann hypothesis and the Lindeloef hypothesis. The third part of the project involves the analysis of the asymptotics of the Riemann and related functions, which is expected to enhance our understanding of the relevant, most important mathematical structures.