Eigenvalue Comparison and Integral Curvature

数学物理的各种问题可以用拉普拉斯方程或更一般的薛定谔方程来模拟。拉普拉斯量的前两个本征值之间的差异被称为基本能隙，它代表了在量子力学中将一个粒子从基态激发到下一个能级所需的能量。在第一个项目中，首席研究员将估计各种空间的基本差距。第二个项目涉及体积熵，它是紧致光滑流形的一个基本几何不变量。这一概念与动力系统中发现的其他熵概念密切相关，并在微分几何和几何群论等领域发挥着重要作用。关于熵刚性的工作与最优输运、信息几何和离散几何有关。该项目还将通过指导研究生和博士后；招募妇女和其他代表性不足的群体；组织促进年轻学者的研讨会、讲习班和研究方案，支持教育活动和多样性。提案中讨论的材料将成为2021年秋季UCSB高级研究生课程的主题。该项目包括三个部分。第一部分是关于局部对称空间中凸域上具有Dirichlet边界条件的拉普拉斯方程的特征值和基本间隙估计，并与一些合适的一维模型进行了比较。第二个问题是关于曲率下界度量空间的体熵比较和刚性。最后一项是使用RICCI流研究临界功率的积分曲率收缩。这一奖项反映了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。

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.