The geometry Anosov subgroups in Lie groups

李群中的几何阿诺索夫子群

• 批准号: RGPIN-2020-05557
• 负责人: Burelle, JeanPhilippe
• 金额: $ 1.31万
• 依托单位: Université de Sherbrooke
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759314
• 关键词: geometry; Anosov; subgroups; Lie; groups

The field of locally homogeneous geometric structures on manifolds generalizes the field of cartography, in the following sense : drawing an atlas of the world is like trying to describe the shape of the earth using flat geometry, and a locally homogeneous geometric structure is like trying to describe the shape of a more general world (or manifold) using some homogeneous geometry. One of the first theorems in the subject states that it is in fact impossible to construct a complete atlas of the world without distortion, that is, that there exists no Euclidean (flat) structure on the two-dimensional sphere. Already in the study of flat structures, the Bieberbach theorems exemplify the tight relationship between geometric structures and groups of symmetries. These theorems imply that, up to finite covers, the only finite-area flat structures in 2 dimensions are similar to the world of Pac-man, that is, they are obtained by identifying parallel edges of a parallelogram. It is no coincidence that the parallelogram can tile the plane by translating multiple copies of itself, creating a tiling with translational symmetry. The study of symmetries between objects or concepts is foundational to a range of topics in pure and applied mathematics and in other sciences. We say that an object has continuous symmetry if its symmetries can be very small motions, for instance a very small rotation of a sphere is a symmetry, so it has continuous symmetry. We say that it has discrete symmetry if this is not the case, for instance a cube has discrete symmetry because it only has 90 degree rotational symmetry. The symmetries of the cube are a subset of the symmetries of the sphere. Similarly, the plane has continuous symmetry but a parallelogram tiling has discrete symmetry within the symmetries of the plane. My field lies at the intersection between the study of geometric structures and the study of discrete subgroups of continuous groups, which generalize the symmetry discussion above. I will investigate a range of examples which are generalizations of the above and work towards solutions to classification problems in those cases. These examples can then be used to formulate general conjectures and theorems. Some of the examples I am interested in have the local geometry of Einstein's static universe, a toy model of spacetime in 3 dimensions. Some more examples have local projective geometry, the type of geometry used in 3D computer graphics. Having a better understanding of these more general notions of geometry and symmetry provides important insight into the fundamental relationship between the two.

流形上的局部齐次几何结构领域概括了制图学领域，在以下意义上：绘制世界地图集就像尝试使用平面几何来描述地球的形状，而局部齐次几何结构就像尝试使用某种齐次几何来描述更一般的世界（或流形）的形状，该主题中的第一个定理指出，实际上不可能构建一个不扭曲的完整世界图集，也就是说，存在存在。二维球面上没有欧几里得（平面）结构 在平面结构的研究中，比伯巴赫定理举例说明了几何结构和对称群之间的紧密关系，这些定理意味着，直到有限覆盖，唯一的有限-二维区域平面结构与吃豆人的世界类似，即通过识别平行四边形的平行边来获得。平行四边形可以通过平移来平铺平面并非巧合。物体或概念之间的对称性研究是纯数学和应用数学以及其他科学中一系列主题的基础，如果一个物体的对称性可以是连续的，那么它就具有连续对称性。非常小的运动，例如球体的非常小的旋转是对称的，因此它具有连续对称性，如果不是这种情况，我们说它具有离散对称性，例如立方体具有离散对称性，因为它只有 90 度。立方体的对称性是球体对称性的子集。类似地，平面具有连续对称性，但平行四边形平铺在平面对称性中具有离散对称性。结构和连续群的离散子群的研究，概括了上面的对称性讨论。我将研究一系列示例，这些示例是上述内容的概括，然后可以使用这些示例来解决分类问题。我感兴趣的一些例子有爱因斯坦静态宇宙的局部几何，这是一个 3 维时空的玩具模型，还有一些例子有局部射影几何，这是 3D 计算机图形学中使用的几何类型。更好地理解这些更一般的几何和对称概念可以为了解两者之间的基本关系提供重要的见解。