Anlaytic Geometry and Representation Theory

解析几何与表示论

• 批准号: 1006353
• 负责人: Joseph Landsberg
• 金额: $ 27.88万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-08-15 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1006353&HistoricalAwards=false
• 关键词: Anlaytic; Geometry; Representation; Theory

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).

兰茨伯格和罗伯斯将继续共同致力于应用和推进外差速系统 (EDS)。特别是，他们将完成对理性齐次簇的射影刚性的确定，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？），并进一步建立 Bernstein-Gelfand-Gelfand 之间的联系(BGG) 决议和 EDS。 Robles 将确定黎曼 4 流形上度量曲率的必要和充分条件，以允许正交坐标，并确定黎曼流形中同源类的几何表示。兰兹伯格将致力于研究源自复杂性理论和信号处理的问题，特别是确定同质簇的割线簇的方程，以及建立与行列式和永久相关的 Mulmuley-Sohoni 簇的性质。微分几何可以被认为是研究“显微镜下”的几何物体（即无限小）。兰茨伯格和罗伯斯对对称的几何物体特别感兴趣。在过去 40 年里，表示论取得了重大进展，该理论系统地研究了对称性（例如 BGG 机制），PI 将应用这些进展来建立微分几何中出现的偏微分方程组的性质。 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 (Landsberg 和 Robles) 和信号处理 (Landsberg) 中的问题。