Mathematical Sciences: Geodesic Flows On Manifolds With No Conjugate Points and Related Topics

Burn的作品展示了各种令人印象深刻的技术和问题，这些技术和问题源于有意义的物理应用。他有一些优秀的出版物，并与一些非常有能力的同事合作。由于他们的努力，负曲率类黎曼流形的结构性质得到了很好的理解，他们正在探索有意义的物理现象的本质，如熵。当前项目的两个主要目标是将非正曲率流形的最新结果推广到没有共轭点的流形，并进一步研究非正曲率流形。非正曲率紧流形根据秩进行分类。测地线流是遍历的当且仅当流形为秩1，且较高秩的例子是秩1的例子与对称空间乘积的商。非正曲率流形的主要问题是进一步理解秩为1的例子。对于没有共轭点的流形，尚不清楚1阶的模拟是否意味着测地线流的遍历性，即使在2维中也是如此。

Burn's work displays an impressive variety of techniques and questions which stem from meaningful physical applications. He has some fine publications alone and in collaboration with some very able colleagues. Due to their efforts the structural properties of Riemannian manifolds of the negative curvature sort are remakably well understood and they are probing into the nature of physically meaningful phenomena such as entropy. The two main objectives of the current project are to extend recent results for manifolds with non-positive curvature to manifolds with no conjugate points, and to further investigate manifolds with non-positive curvature. Compact manifolds with non-positive curvature are classified by their rank. The geodesic flow is ergodic if and only if the manifold has rank 1, and higher rank examples are quotients of products of rank 1 examples and symmetric spaces. The main problem for manifolds with non- positive curvature is to further understand the examples with rank 1. For manifolds with no conjugate points, it is still unknown whether the analogue of rank 1 will imply ergodicity of the geodesic flow, even in dimension 2.