Variational Methods in Singular Geometry

物理现象往往遵循最小作用力原则，即沿着使作用力局部静止的轨迹运动。如果作用量是由狄利克雷能量给出的，那么驻点在某些情况下是测地线(最短路径)，在其他情况下是调和函数和这些概念的各种推广，这些概念熟悉于初级物理。在这个项目中，PI将研究与通过最小化Dirichlet能量获得的调和映射或通过最小化Lipschitz常数获得的最佳Lipschitz映射(有时称为无穷调和)相关的定常解。这两个问题在几何拓扑学和群论中都有有趣的应用。该项目还包括对初级数学家(学生、博士后研究人员、初级教师)的重要培训和指导。埃尔斯-桑普森的工作在60年代S引发了黎曼流形之间调和映射研究的爆炸性增长。随后在极小曲面理论、Kaehler几何和流形上的群作用的刚性等方面有许多重要的应用。最近，Gromov-Schoen和Korea vaar-Schoen在度量空间目标的调和映射方面的开创性工作在理解与奇异空间相关的现象方面取得了重大进展，如作用于建筑物上的群的刚性和Teichmueller空间的完备性。在项目的第一部分，PI将研究奇异几何调和映射理论中的几个问题，并将其应用于Teichmueller理论。在第二部分中，PI将研究与黎曼流形之间映射的梯度的超范数有关的泛函的变分。这样的泛函给出了具有极具挑战性的正则性的完全非线性退化偏微分方程解，其奇异集给出了与瑟斯顿理论有关的测地线叶片和薄片等拓扑对象的几何实现。PI还将培训研究生，并为他所在系的所有初级研究人员维持一个强有力的指导计划。后者在一系列与专业发展相关的问题上为博士后研究人员和初级教师提供指导和支持。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Physical phenomena tend to obey the least action principle, namely moving in trajectories that make the action locally stationary. If the action is given by Dirichlet energy, then the stationary points are in some cases geodesics (shortest paths), and in other cases harmonic functions and various generalizations of these notions familiar from elementary physics. In this project, the PI will study stationary solutions either related to harmonic maps obtained by minimizing Dirichlet energy or best Lipschitz maps (sometimes called infinity harmonic) obtained by minimizing Lipschitz constants. Both problems have interesting applications to geometric topology and group theory. The project also includes significant training and mentoring of junior mathematicians (students, post-doctoral researchers, junior faculty) The work of Eells-Sampson in the 60's launched an explosion of research for harmonic maps between Riemannian manifolds. Many important applications followed in minimal surface theory, Kaehler geometry and rigidity of group actions on manifolds among others. More recently, the seminal work of Gromov-Schoen and Korevaar-Schoen on harmonic maps to metric space targets initiated major progress in understanding phenomena associated with singular spaces, like rigidity of groups acting on buildings and the completion of Teichmueller space. In the first part of the project PI will study several problems in harmonic map theory for singular geometry with applications to Teichmueller theory. In the second part PI will study the calculus of variations of functionals associated with the sup-norm of the gradient of maps between Riemannian manifolds. Such functionals yield solutions of fully non-linear degenerate PDE's with very challenging regularity properties and whose singular set gives geometric realizations of topological objects like geodesic foliations and laminations related to Thurston theory. The PI will also train graduate students and maintains a robust mentoring program for all junior researchers in his department. The latter provides guidance and support to post-doctoral researchers and junior faculty on a range of issues related to professional development.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.