Analytic Geometry and Representation Theory

《解析几何与表示理论》，J.M.兰茨伯格和罗伯斯著。这个项目将在几何学和理论计算机科学中产生的问题上取得重大进展。每个问题都将使用位于三个数学领域的界面上的技术来解决：微分几何(子流形，外微分系统)，代数几何(射影变种，交换代数的几何方面)和表示理论(轨道闭包，李代数上同调，不变理论)。我们强调，这种合成产生的异花授粉显著增加了结果的丰富度、价值和影响。Landsberg和Robles将继续有关(1)关于代数几何中Fano簇的Hwang-Mok程序，(2)P=NP？复杂性理论中的问题，以及(3)表示论中轨道的几何结构。这些看似不同的项目被原则调查人员的方法统一了起来。在每一种情况下，射影簇上(并与之密切相连)的线都有一个基本的微分几何。对这一几何的透彻理解将加深我们对这些问题的理解(在某些情况下还可以解决)。这种几何是通过从微分几何、代数几何和表示论中汲取的技术来研究的。虽然所使用的几何和表示理论工具是经典的，但实现和由此产生的应用是创新的，是由主要研究人员发起的。其他项目包括：(1)张量空间中的变元的研究。这项工作借鉴了几何和表示理论，并将在统计学、计算机科学和复杂性理论中得到应用。(2)标定几何图形。这些几何不仅是数学研究的一个活跃领域，而且作为弦理论模型的组成部分，物理学家也很感兴趣。(3)紧致G2-流形的构造。

"Analytic Geometry and Representation Theory" by J.M. Landsberg & C. Robles. This project will make significant progress on problems originating in geometry and theoretical computer science. Each of the questions will be approached using techniques that lie on the interface of 3 areas of mathematics: differential geometry (submanifolds, exterior differential systems), algebraic geometry (projective varieties, geometric aspects of commutative algebra) and representation theory (orbit closures, Lie algebra cohomology, invariant theory). We emphasize that this synthesis yields a cross-pollination that significantly increases the richness, value an influence of the results. Landsberg and Robles will continue work related to (1) the Hwang-Mok program on Fano varieties in algebraic geometry, (2) the P = NP? problem in complexity theory, and (3) the geometric structure of orbits in representation theory. These seemingly disparate projects are unified by the Principle Investigators' approach. In each case there is an underlying differential geometry of lines on (and osculating to) a projective variety. A through understanding of this geometry will deepen our understanding of (and in some cases solve) these problems. This geometry is studied via techniques drawn from differential geometry, algebraic geometry and representation theory. While the geometric and representation theoretic tools used are classical, the implementation and resulting applications are innovative and originated with the principal investigators. Additional projects include: (1) A study of varieties in spaces of tensors. This work draws on geometry and representation theory, and will have applications to statistics, computer science and complexity theory. (2) Calibrated geometries. Not only an active area of research in mathematics, these geometries are of interest to physicists as components of string theory models. (3) The construction of compact G2-manifolds.