Hitchin systems: tame, twisted, and wild

数学一直被用来解决物理学中的问题。但物理学能解决数学中的问题吗？答案是肯定的。在过去的四十年里，对物理学中的方程进行了严格的几何研究，这让我们看到了奇妙的新想法，以前未解决的问题的解决方案，以及曾经基本上被隐藏的数学学科之间的联系。例如，正是通过对20世纪80年代初杨-米尔斯方程的研究，在四维平面空间上发现了奇异的新几何结构。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，这是一个特定版本的方程的所有解的空间。虽然它是一个数学对象，但希钦系统在许多方面与大约200年前发现的罗塞塔石块相似。这块石头上有三种文字：古埃及文字、大众文字和希腊文文字，这使得钱波利安和杨第一次破译了埃及的象形文字。同样，Hitchin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Hitchin系统及其变体是优先事项，因为几何、表示理论和数学物理中的许多问题现在都已用它重新表述。一个最好的例子是朗兰兹计划基本引理的最近证明。最后通过将索赔重写为关于Hitchin系统的索赔，然后利用Hitchin系统的几何性质来完成证明，最终解决了这个问题。这是处于数学突破核心的物理学对象的又一个例子。与此同时，希钦式系统开始在不同的行业中得到应用。填充在普通Hitchin系统中的“希格斯束”最近被应用于现代医学成像核心的逆问题。填充在另一种希钦系统中的“超级多边形”以一种新的观点展示了DNA的动力学。寻求更好地了解希钦制度的时机已经成熟。我计划使用Morse理论来检测低亏格黎曼曲面的希钦系统的某种“扭曲”变体的拓扑或“形状”，即球面和环面。与此同时，我计划通过将超多边形的概念推向无穷远，来探索超多边形空间的渐近几何。最后，我计划找出不同类型的超多边形之间的关系，这些超多边形呼应了物理学中“镜像对称”的概念，在这种情况下，一个空间可以换成另一个空间，物理上是相同的，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.