Hitchin systems: tame, twisted, and wild

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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.