Anlaytic Geometry and Representation Theory

兰茨贝格和罗伯斯将继续他们的联合工作，在应用和推进外部差分系统（EDS）。特别是，他们将完成他们的合理齐次品种的投影刚性的测定，研究有关不变微分算子的问题（例如，黎曼流形上的Killing算子的图像是什么？），并进一步探讨了Bernstein-Gelfand-Gelfand（BGG）分解与EDS的关系。罗伯斯将确定必要和充分条件曲率度量的黎曼4流形承认正交坐标，并确定几何代表的同源类黎曼流形。兰茨贝格将致力于复杂性理论和信号处理中产生的问题，特别是确定方程的割线品种的齐次品种，并建立性质的Mulmuley-Sohoni品种相关联的行列式和永久。微分几何可以被认为是几何对象的研究“在显微镜下”（即，无穷小）。兰茨贝格和罗伯斯特别感兴趣的几何物体的对称性。在过去的40年里，表征理论取得了重大进展，它系统地研究了对称性（例如，BGG机器），这些进展将被PI应用于建立微分几何中产生的偏微分方程系统的属性。 个别地，PI还将使用微分几何和表示理论来解决拓扑学（Robles），理论计算机科学（兰茨贝格和Robles）和信号处理（兰茨贝格）中的问题。

Landsberg and Robles will continue their joint work in applying and advancing exterior differential systems (EDS). In particular they will complete their determination of the projective rigidity of rational homogeneous varieties, study questions regarding invariant differential operators (e.g., what is the image of the Killing operator on a Riemannian manifold?), and further the connections between Bernstein-Gelfand-Gelfand (BGG) resolutions and EDS. Robles will will identify necessary and sufficient conditions on curvature for the metric on a Riemannian 4-manifold to admit orthogonal coordinates, and determine geometric representatives of homology classes in Riemannian manifolds. Landsberg will work on problems originating in complexity theory and signal processing, in particular determining equations for secant varieties of homogeneous varieties, and establishing properties of the Mulmuley-Sohoni varieties associated to the determinant and permanant.Differential geometry may be thought of as the study of geometric objects "under a microscope" (i.e., infinitesimally). Landsberg and Robles are especially interested in geometric objects with symmetry. Over the last 40 years significant advances have been made in representation theory, which systematically studies symmetry (e.g., the BGG machinery), and these advances will be applied by the PIs to establish properties of systems of partial differential equations arising in differential geometry. Individually, the PIs will use also use differential geometry and representation theory to work on problems in topology (Robles), in theoretical computer science (Landsberg and Robles) and signal processing (Landsberg).