Integrable systems in Geometry, Asymptotics and Inverse Problems

可积系统存在于一类特殊的超定偏微分方程组中。它们以略微不同的伪装出现在几个背景下，包括随机矩阵理论、黎曼曲面和联络的模空间、随机过程和反问题。所有这些情况的一个共同线索是，根据矩阵解析函数的特定边值问题，或现在通常所说的Riemann-Hilbert问题(RHP)，重新表述的可能性。这项拟议的研究旨在促进对RHP的一般理解以及它们在几个悬而未决的问题上的应用。 一些具体的目标包括研究RHP变形理论背后的泊松几何，也扩展到Riemann表面上的RHP。 在曲线的模空间上的交数(Gromov-Witten不变量)的背景下，母函数也可以与RHP相关联，这具有导致严格的渐近分析的好处。曲线模空间中基本类之间交数的母函数是由矩阵积分(Kontsevich)得到的，该积分提供了KdV族的形式解。从G.Moore的著作中，可以观察到与等单向性变形(间接地，RHP)有形式上的联系。 然而，目前仍缺乏一种分析方法。目的是根据Kontsevich积分及其推广的固定大小的矩阵的适当的RHP来完成该描述。这尤其有助于揭示母函数的非形式化性质，例如非线性斯托克斯现象(解析恢复)。 RHPS的另一个应用是求解反问题。在这里，该项目集中于非线性薛定谔方程的反向(和正向)散射理论：在半经典极限中，由于普朗克常数被送到零，在研究玻色-爱因斯坦凝聚体的拉长相时也考虑了这一理论，在海洋学中，它被认为描述了所谓的无赖波和“三姐妹”无赖波的形成的根本机制。 第二个突出的逆问题源于医学成像(层析成像)领域；为了减少对患者组织的照射，必须解决部分数据的(病态)逆问题。然后，公开的问题是重构的不稳定程度，它被转化为关于某个积分算子的奇异值和奇异函数的渐近行为的问题。

Integrable systems consist in a special class of overdetermined sets of partial differential (or difference) equations. They appear in several contexts in slightly different guises, including Random Matrix Theory, Moduli Spaces of Riemann surfaces and connections, Stochastic Processes and Inverse problems. A common thread to all these instances is the possibility of reformulation in terms of a particular boundary value problem for matrix-valued analytic functions, or what is now commonly referred to as a Riemann-Hilbert problem (RHP). The proposed research seeks to both advance the general understanding of RHPs as well as their application to several outstanding problems. Some of the specific goals include the study of the Poisson geometry underlying deformation theory of RHPs, extending also to RHPs on Riemann surfaces. In the context of intersection numbers on the moduli space of curves (Gromov-Witten invariants), the generating function can be associated also to a RHP and this has the benefit of leading to a rigorous asymptotic analysis. The generating function of intersection numbers between fundamental classes in the moduli space of curves is obtained from a matrix integral (Kontsevich); the integral is known to provide a formal solution of the KdV hierarchy. From works of G. Moore's, the formal connection with isomonodromic deformations (thus, indirectly, RHP) was observed. However an analytic approach is still missing. It is a goal to complete this description in terms of a suitable RHP for a matrix of fixed size of the Kontsevich integral and generalizations thereof. This in particular will shed light on non-formal properties of the generating function, such as the nonlinear Stokes' phenomenon (analytic resummation). Another application of RHPs is in solving inverse problems. Here the project focuses on the inverse (and forward) scattering theory of the nonlinear Schroedinger equation: in the semiclassical limit as Planck's constant is sent to zero is considered also in the study of elongated phases of Bose--Einstein condensates and in oceanography, where it has been proposed as describing the underlying mechanism of formation of the so-called rogue waves and for the ``three sister'' rogue waves. A second outstanding inverse problem originates in the area of medical imaging (tomography); in order to reduce irradiation of patient's tissue, the (ill-posed) problem of inversion with partial data must be addressed. Then the open question is the degree of instability of the reconstruction, which is translated into a question about asymptotic behavior of singular values and singular functions for a certain integral operator.