Hitchin systems: tame, twisted, and wild

数学一直被用来解决物理问题。 但是物理能解决数学问题吗？ 答案是肯定的。 在过去的40年里，对物理学方程的严格的几何研究使我们看到了奇妙的新思想，以前未解决的问题的解决方案，以及曾经在很大程度上被隐藏的数学学科之间的联系。 例如，在20世纪80年代早期，通过对杨-米尔斯方程的研究，人们发现了四维平坦空间中奇异的新几何结构。 几年后，对杨-米尔斯方程的进一步研究导致了希钦系统的发现，希钦系统是一个特定版本的方程的所有解的空间。虽然希钦系统是一个数学对象，但它在许多方面类似于大约200年前发现的罗塞塔石碑。 石头上有三种文字：古埃及文字、世俗文字和希腊文字，这使得商博良和杨首次破译了埃及象形文字。 同样，希钦系统允许几何学家几乎毫不费力地在拓扑、光滑几何和复杂几何之间进行转换。理解希钦系统及其变体是一个优先事项，因为几何学、表示论和数学物理学中的许多问题现在都已经用它来重新表述，一个主要的例子是最近对朗兰兹纲领基本引理的证明。 最后通过将这个命题重写为关于希钦系统的命题，然后利用希钦系统的几何性质来完成证明，从而解决了这个问题。 这是另一个例子，说明物理学中的一个对象是数学突破的核心。与此同时，Hitchin-type系统开始在不同行业中应用。 填充普通希钦系统的“希格斯束”最近被应用于现代医学成像核心的逆问题。 在另一种希钦系统中出现的“超多边形”从一个新的角度展示了DNA的动力学。* 现在是更好地了解希钦系统的时候了。 我计划使用莫尔斯理论来检测低亏格黎曼曲面（即球面和环面）的希钦系统的某种“扭曲”变体的拓扑或“形状”。 与此同时，我计划探索超多边形空间的渐近几何，从字面上推动超多边形的概念到无穷大。 最后，我计划找到不同种类的超多边形之间的关系，这些超多边形与物理学中的“镜像对称”概念相呼应，其中一个空间可以交换为另一个空间，物理学是相同的，但数学更容易。

Mathematics has always been used to solve problems in physics. But can physics solve problems in mathematics? The answer is yes. The rigorous geometric study of equations from physics over the last forty years has opened our eyes to fantastic new ideas, solutions to previously unsolved problems, and connections between mathematical disciplines that were once largely hidden. For example, it was through the study of the Yang-Mills equations in the early 1980s that exotic new geometric structures on four-dimensional flat space were discovered. Several years later, a further investigation of the Yang-Mills equations led to the discovery of the Hitchin system, which is the space of all solutions of a particular version of the equations. While it is a mathematical object, the Hitchin system is in many ways akin to the Rosetta stone discovered some two hundred years earlier. On the stone were three scripts: ancient Egyptian, Demotic, and Greek, allowing Champollion and Young to decipher Egyptian hieroglyphics for the first time. Likewise, the Hitchin system allows geometers to translate almost effortlessly between topology, smooth geometry, and complex geometry.******Understanding the Hitchin system and its variants is a priority because of the number of problems in geometry, representation theory, and mathematical physics that have now been rephrased in terms of it. A prime example is the recent proof of the Fundamental Lemma of the Langlands Program. It was finally settled by rewriting the claim as one about the Hitchin system and then using geometric properties of the Hitchin system to complete the proof. This is yet another example of an object from physics lying at the heart of a breakthrough in mathematics.******At the same time, Hitchin-type systems are beginning to see applications in diverse industries. The "Higgs bundles" that populate the ordinary Hitchin system have recently been applied to inverse problems at the core of modern medical imaging. The "hyperpolygons" that populate another kind of Hitchin system show up in a new point of view on the dynamics of DNA. ******The time is ripe to seek a better understanding of the Hitchin system. I plan to use Morse theory to detect the topology or "shape" of a certain "twisted" variant of the Hitchin system for low-genus Riemann surfaces, namely spheres and tori. At the same time, I plan to probe the asymptotic geometry of spaces of hyperpolygons, by literally pushing the notion of hyperpolygon to infinity. Finally, I plan to find relationships between different kinds of hyperpolygons that echo the notion of "mirror symmetry" in physics, where one space can be swapped for another for which the physics is the same but the mathematics is easier.