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，如：(1)洛伦兹利希纳罗维兹猜想及其推广；(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，这些技术对于证明支持猜想(1)和(2)的结果非常有效。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，洛伦兹流形似乎是一个边缘情况。下面的附加问题在洛伦兹情况下有一个肯定的答案，但在更高的签名中是开放的：(4)紧致的、平坦的、伪黎曼流形是否总是完备的？PI的计划是通过证明或反例来解决猜想(1)-(4)。PI在伪黎曼和洛伦兹几何中工作，这是广义相对论和时空物理学的基础数学领域。洛伦兹流形的等距或共形变换对应于物理学中的守恒定律，并且它们在大多数时空模型中都具有特征。上面的猜想(1)和(2)来自于一个雄心勃勃的计划，由Zimmer和Gromov在20世纪80年代提出，对保持微分几何结构的流形上的群作用进行分类。猜想(3)出现在Gromov关于三角双曲群的基础著作中，猜想(4)是关于平面仿射流形的Markus猜想的一个重要例子。该项目的教育部分包括：(a)进一步发展华盛顿地区数学研究生的写作讲习班；(b)让本科生参与与上述第(2)项有关的暑期研究项目；(c)为马里兰大学指导阅读项目的研究生和本科生提供建议。

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.