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

PI受到一些雄心勃勃的刚性猜想的启发，例如（1）洛伦兹-李希内洛维奇猜想及其推广;（2）某些几何结构的“开-密”猜想;（3）光滑加农猜想。 PI和她的合作者在Cartan几何的设置中开发了新的动力学技术，这些技术对证明支持公式（1）和（2）的结果非常有效。 然而，最近的结果和PI目前的一些工作表明，这些理论并不像最初推测的那样具有广泛的普遍性。 对于许多这些问题，洛伦兹流形似乎是边界情况。 下面的附加问题在洛伦兹的情况下有肯定的答案，但在更高的签名是开放的：（4）是一个紧凑的，平坦的，伪黎曼流形总是完整的？ PI的计划是通过证明或反例来解决问题（1）-（4）。PI在伪黎曼几何和洛伦兹几何中工作，这是一个数学领域，是广义相对论和时空物理的基础。 洛伦兹流形的等距变换或保角变换对应于物理学中的守恒定律，它们在大多数时空模型中都很重要。上面的猜想（1）和（2）来自于Zimmer和Gromov在20世纪80年代发起的一个雄心勃勃的计划，对流形上保持微分几何结构的群作用进行分类。 猜想（3）出现在格罗莫夫的基本工作的δ-双曲群，和（4）是一个重要的情况下，马库斯猜想平坦仿射流形。该项目的教育部分包括：（a）进一步为华盛顿地区的数学研究生举办写作讲习班;（B）让本科生参与与上述专题（2）有关的暑期研究项目;（c）在马里兰州大学指导阅读方案中为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.