Nonlinear parabolic equations and related geometric problems

本项目主要研究非线性抛物方程解的正则性和奇异性分析，主要与微分几何中更复杂的问题有关。这个项目的一个重要部分是致力于理解几何流的全局实时解，如Ricci流、Yamabe流和平均曲率流，以及相关的奇点和孤子分类问题。其他项目包括由金融数学引起的自由边界问题，完全非线性椭圆和抛物方程解的定性行为，以及洛伦兹流形上的里奇流的研究。该项目涉及广泛的数学活跃领域，特别是非线性偏微分方程、几何和经典分析。主要研究数学问题在量子场论、相对论、数学金融学等其他学科的应用。研究结果将通过各种会议和研究论文的出版向研究界传播。为研究生设计并实施将偏微分方程与几何分析联系起来的新课程。

This project addresses questions concerning the regularity of and the analysis of singularities of solutions to nonlinear parabolic equations, primarily in connection with more complex problems in differential geometry. A significant part of this project is devoted to the understanding of global-in-time solutions to geometric flows such as the Ricci flow, the Yamabe flow, and the mean curvature flow, and the related problem of the classification of associated singularities and solitons. Other projects include free boundary problems arising from the mathematics of finance, the qualitative behavior to solutions of fully nonlinear elliptic and parabolic equations, and the study of the Ricci flow on Lorentzian manifolds.The project links a wide range of active fields of mathematics, in particular, nonlinear partial differential equations, geometry, and classical analysis. The principal investigator intends to study the applications of the mathematical problems to other disciplines such as quantum field theory, relativity theory, and mathematical finance. Results will be disseminated to the research community at various meetings and by publication of research articles. New courses linking partial differential equations and geometric analysis for graduate students will be designed and implemented.