Eigenvalue Comparison and Integral Curvature

数学物理的各种问题都可以用拉普拉斯方程或更一般的薛定谔方程来建模。拉普拉斯函数的前两个特征值之间的差被称为基本间隙，它代表了量子力学中将粒子从地面激发到下一层所需的能量。在第一个项目中，首席研究员将估算各个空间的基本差距。第二个项目涉及体积熵，它是紧光滑流形的基本几何不变量。这个概念与在动力系统中发现的熵的其他概念密切相关，在微分几何和几何群论等中起着重要作用。熵刚性的研究涉及到最优传输、信息几何和离散几何。该项目还将通过指导研究生和博士后来支持教育活动和多样性；招募妇女和其他代表性不足的群体；组织研讨会、工作坊和研究项目，促进青年学者的发展。提案中讨论的材料将成为UCSB 2021年秋季高级研究生课程的主题。这个项目有三个部分。首先通过与一些合适的1-dim模型的比较，讨论了局部对称空间凸域上具有Dirichlet边界条件的拉普拉斯算子的特征值和基本间隙估计。第二个问题是曲率下界度量空间的体积熵比较和刚性。最后，利用Ricci流研究了临界功率的积分曲率缩紧。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Various problems of mathematical physics can be modeled by the Laplacian or more generally Schrodinger equations. The difference between the first two eigenvalues of the Laplacian is referred to as the fundamental gap, which represents the energy needed to excite a particle from ground level to the next level in quantum mechanics. In the first project, the principal investigator will estimate the fundamental gap for various spaces. The second project relates to volume entropy which is a fundamental geometric invariant for compact smooth manifolds. This concept is closely related to other notions of entropy found in dynamical systems and plays an essential role in differential geometry and geometric group theory among others. The work on entropy rigidity is related to optimal transport, information geometry and discrete geometry. The project will also support educational activities and diversity through mentoring graduate students and postdocs; recruiting women and other underrepresented groups; organizing seminars, workshops and research programs promoting young scholars. The material discussed in the proposal will be the subject of an advanced graduate course at UCSB in Fall 2021.The project has three parts. The first is about eigenvalue and fundamental gap estimates of the Laplacian with Dirichlet boundary conditions on a convex domain in locally symmetric spaces by comparing with some suitable 1-dim model. The second concerns volume entropy comparison and rigidity for metric measure spaces with curvature lower bounds. The last is to study integral curvature pinching for the critical power using Ricci flow.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.