Nonlinear geometric inverse problems

非线性几何反问题

• 批准号: EP/R001898/1
• 负责人: Gabriel Paternain
• 金额: $ 41.49万
• 依托单位: University of Cambridge
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2017
• 资助国家: 英国
• 起止时间: 2017 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FR001898%2F1
• 关键词: Nonlinear; geometric; inverse; problems

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.

这个提议考虑了理论物理和微分几何中产生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：是否可以从局部信息全局地确定支配波传播的几何结构，或者更物理地，观察者是否可以进行局部测量以确定波可以传播和返回的最大区域中的几何结构？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，这个问题最近有了一些进展。在这里，我们提出了研究的杨-米尔斯-希格斯模型时，洛伦兹背景是固定的，目标是重建的杨-米尔斯场和希格斯场。爱因斯坦方程和杨-米尔斯-希格斯方程的反问题之间的主要区别在于，要重建的几何结构在前一种情况下出现在首阶项中，而在后一种情况下出现在低阶项中。这种差异带来了新的挑战，因为在领导秩序的扰动更强，因此更容易从数据中看到。我们的建议涉及到以前未研究过的问题，反演破碎的非阿贝尔X射线变换的低阶项的重建。

项目成果

THE TRANSPORT OKA-GRAUERT PRINCIPLE FOR SIMPLE SURFACES

• DOI: 10.5802/jep.231
• 发表时间: 2023-01-01
• 期刊: JOURNAL DE L ECOLE POLYTECHNIQUE-MATHEMATIQUES
• 影响因子: 1.1
• 作者: Bohr，Jan;Paternain，Gabriel P.
• 通讯作者: Paternain，Gabriel P.

The Sobolev inequalities on real hyperbolic spaces and eigenvalue bounds for Schrödinger operators with complex potentials

实双曲空间上的索博列夫不等式和复势薛定谔算子的特征值界

• DOI: 10.2140/apde.2022.15.1861
• 发表时间: 2022
• 期刊: Analysis & PDE
• 影响因子: 2.2
• 作者: Chen X

Inverse Problem for the Yang-Mills Equations

Yang-Mills 方程的反问题

• DOI: 10.1007/s00220-021-04006-0
• 发表时间: 2021
• 期刊: Communications in Mathematical Physics
• 影响因子: 2.4
• 作者: Chen X

Detection of Hermitian connections in wave equations with cubic non-linearity

• DOI: 10.4171/jems/1136
• 发表时间: 2019-02
• 期刊: Journal of the European Mathematical Society
• 影响因子: 2.6
• 作者: Xi Chen;M. Lassas;L. Oksanen;G. Paternain

The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds

近双曲 3 流形的 Ruelle zeta 函数为零

• DOI: 10.1007/s00222-022-01108-x
• 发表时间: 2022
• 期刊: Inventiones mathematicae
• 影响因子: 3.1
• 作者: Cekić, Mihajlo;Delarue, Benjamin;Dyatlov, Semyon;Paternain, Gabriel P.
• 通讯作者: Paternain, Gabriel P.