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.
