Nonlinear geometric inverse problems

本文考虑了理论物理和微分几何中出现的非线性波动方程反问题的数学研究。我们希望解决的主要问题是：控制波传播的几何结构能否从局部信息中全局确定，或者更物理地说，观察者能否通过局部测量来确定波可以传播和返回的最大区域的几何结构？当几何结构是时空本身，相关的偏微分方程是爱因斯坦方程时，这个问题最近有了进展。本文提出在洛伦兹背景固定的情况下，以重建Yang-Mills场和Higgs场为目标，研究Yang-Mills-Higgs模型。爱因斯坦方程和杨-米尔-希格斯方程的反问题的主要区别在于，要重建的几何结构在前者中出现在主阶项中，而在后一种情况中出现在低阶项中。这种差异带来了新的挑战，因为先导阶的扰动更强，因此更容易从数据中看到。我们的建议将低阶项的重建与先前未研究的问题联系起来，以反演破碎的非阿贝尔x射线变换。

This proposal considers the mathematical study of inverse problems for non-linear wave equations arising in theoretical physics and differential geometry. The main problem we wish to address is the following: can the geometric structures governing the wave propagation be globally determined from local information, or more physically, can an observer do local measurements to determine the geometric structures in the maximal region where the waves can propagate and return back? There has been recent progress on this question when the geometric structure is space-time itself and the relevant partial differential equations are the Einstein equations. Here we propose the study of the Yang-Mills-Higgs model when the Lorentzian background is fixed and the goal is the reconstruction of the Yang-Mills field and the Higgs field. The main difference between the inverse problems for the Einstein and Yang-Mills-Higgs equations is that the geometric structures to be reconstructed appear in the leading order terms in the former case and in the lower order terms in the latter case. This difference poses novel challenges, since a perturbation in theleading order is stronger and therefore easier to see from the data. Our proposal relates the reconstruction of the lower order terms to the previously unstudied problem to invert a broken non-abelian X-ray transform.