Anlaytic Geometry and Representation Theory

兰茨堡和罗伯斯将继续在应用和推进外部差动系统(EDS)方面共同努力。特别是，他们将完成有理齐次簇的射影刚性的确定，研究关于不变微分算子的问题(例如，黎曼流形上的Killing算子的像是什么？)，并进一步研究Bernstein-Gelfand-Gelfand(BGG)解和EDS之间的联系。罗伯斯将为黎曼4-流形上的度量确定曲率允许正交坐标的充要条件，并确定黎曼流形中同调类的几何表示。兰茨伯格将致力于解决起源于复杂性理论和信号处理的问题，特别是确定齐次变元的割群的方程，并建立与行列式和恒等式有关的MulMuley-Sohoni变元的性质。微分几何可以被认为是在“显微镜下”(即无限小)研究几何对象。兰茨伯格和罗伯斯对具有对称性的几何物体特别感兴趣。在过去的40年里，表示理论取得了重大进展，它系统地研究对称性(例如，BGG机制)，这些进展将被PI应用于建立在微分几何中产生的偏微分方程组的性质。在个人方面，PI还将使用微分几何和表示理论来处理拓扑学(罗伯斯)、理论计算机科学(兰德斯堡和罗伯斯)和信号处理(兰德斯堡)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).