Properties of solutions of linear and non-linear hyperbolic equations, singular Fourier integral operators, averages over curves

奇异积分算子是现代分析的一个重要分支，现在吸引了许多强大的分析家。在过去的几年里，与奇异正则关系相关的平均算符和傅立叶积分算符的研究取得了很大进展。Andrew Comech在这一领域得到了几个一般性的结果，特别是在Sobolev空间中奇异正则关系的分类和相关的傅立叶积分算子的正则性。该方法的基础是将某些曲率假设融入到现代调和分析工具中。这些结果的应用包括Melrose-Taylor的Radon变换的正则性和双曲方程解的限制的光滑性。在这两种情况下，安德鲁的方法都会得出最优的结论。未来的研究方向是非线性双曲方程解的性质，如非线性双曲方程解的长时间性态和孤子渐近性态。来自调和分析基础的问题，与拟议的研究相关，包括低维变量取平均值的正则性(最重要的是，曲线上的平均算符)。安德鲁·科梅奇将继续他对两个基本数学领域的研究：调和分析和双曲型方程理论。这项研究深入到了数学的现代发展，与非线性微分方程、吸引子和孤立子密切相关。另一方面，这项研究深深植根于自然科学和技术。安德鲁·科梅奇特别感兴趣的是相对论量子物理中的非线性波动方程的性质，这是目前物理学和数学中都广泛研究的问题。同样有趣的是，由于层析成像的应用以及涉及傅立叶分析的数值计算方面的机会。这些都是当今科学研究的活跃和有前途的领域。

Singular integral operators is a vast modern branch of Analysis, whichnow attracts many strong analysts. Particularly intense developmentfor the last several years has been in the field of averagingoperators and Fourier integral operators associated to singularcanonical relations. Andrew Comech obtained several general results inthis field; in particular, the classification of singular canonicalrelations and regularity properties of the associated Fourier integraloperators in the Sobolev spaces. The approach was based onincorporating certain curvature assumptions into the modern tools ofHarmonic Analysis. Among the applications of these results are theregularity properties of the Radon Transform of Melrose-Taylor andsmoothness of restrictions of solutions to hyperbolic equations. Inboth cases Andrew's methods lead to the optimal conclusions. Thefuture research is aimed at the properties of solutions to non-linearhyperbolic equations, such as long-time behavior and solitonasymptotics of solutions to nonlinear hyperbolic equations. Questionsfrom the very foundations of Harmonic Analysis, which are related tothe proposed research, include the regularity properties of theaverages taken over lower-dimensional varieties (most importantly,averaging operators over curves). Andrew Comech is going to continue his research on the wedge of thetwo fundamental mathematical fields: Harmonic Analysis and the Theoryof Hyperbolic Equations. The research grows far into the moderndevelopment of the mathematics, being intimately related to nonlinear differential equations, attractors, and solitons. On the other hand, the research has its roots deep in the natural sciences and technology. Particular interests of Andrew Comech are related to the properties of nonlinear wave equations of relativistic Quantum Physics, which are now widely investigated both in Physics and in Mathematics. Not less interesting are the opportunities open due to applications to Tomography, as well as aspects of numerical computations involving Fourier Analysis. These are active and promising areas of today's scientific research.