CAREER: Frontiers of rigidity in pseudo-Riemannian, conformal, and parabolic geometries

PI的灵感来自于一些雄心勃勃的刚性猜想，例如：(1)洛伦兹的Lihnerowicz猜想及其推广；(2)某些几何结构的“开-密”猜想；(3)光滑大炮猜想。PI和她的合作者在Cartan几何的背景下开发了新的动力学技术，这些技术在证明支持猜想(1)和(2)的结果方面非常有效。然而，最近的结果和PI的一些当前工作表明，这些猜想并不符合最初推测的广泛普遍性。对于这些问题中的许多问题，洛伦兹流形似乎是边缘情况。下面的附加问题在洛伦兹情形中有一个肯定的答案，但在更高的签名中是开放的：(4)紧致、平坦的伪黎曼流形总是完备的吗？PI的计划是通过证明或反例来解决猜想(1)-(4)。PI的工作是伪黎曼几何和洛伦兹几何，这是广义相对论和时空物理学的基础数学领域。洛伦兹流形的等距变换或共形变换对应于物理学中的守恒定律，它们是大多数时空模型中的特征。上面的猜想(1)和(2)源于一项雄心勃勃的计划，该计划由Zimmer和Gromov在20世纪80年代发起，目的是对流形上保持微分几何结构的群作用进行分类。猜想(3)出现在Gromov关于Delta-双曲群的基本工作中，(4)是平坦仿射流形上的Markus猜想的一个重要例子。该项目的教育部分包括：(A)为华盛顿特区地区的数学研究生进一步发展写作讲习班；(B)让本科生参与与上文(2)主题有关的暑期研究项目；(C)为马里兰大学定向阅读计划的研究生和本科生提供咨询。

The PI is inspired by some ambitious rigidity conjectures, such as (1) the Lorentzian Lichnerowicz Conjecture and its generalizations; (2) an "Open-Dense" Conjecture for certain geometric structures; and (3) the Smooth Cannon Conjecture. The PI and her collaborators have developed new dynamical techniques in the setting of Cartan geometries that have been quite effective toward proving results supporting conjectures (1) and (2). Recent results and some current work of the PI, however, indicate that these conjectures do not hold in the broad generality originally speculated. For many of these questions, Lorentzian manifolds seem to be the borderline cases. The following additional question has an affirmative answer in the Lorentzian case, but is open in higher signatures: (4) is a compact, flat, pseudo-Riemannian manifold always complete? The PI's plan is to work towards settling conjectures (1)-(4), by proof or counterexample.The PI works in pseudo-Riemannian and Lorentzian geometry, an area of mathematics that that underlies general relativity and the physics of spacetime. Isometries or conformal transformations of Lorentzian manifolds correspond to conservation laws in physics, and they feature in most models of spacetime. Conjectures (1) and (2) above arise from an ambitious program, initiated by Zimmer and Gromov in the 1980s, to classify group actions on manifolds preserving differential-geometric structures. Conjecture (3) appears in Gromov's fundamental work on delta-hyperbolic groups, and (4) is an important case of the Markus Conjecture on flat affine manifolds. The educational component of the project comprises (a) further development of writing workshops for DC area math graduate students; (b) involving undergraduates in summer research projects related to topic (2) above; and (c) advising graduate and undergraduate students in the Directed Reading Program at the University of Maryland.