Mathematical Sciences: Geodesic Flows On Manifolds With No  Conjugate Points and Related Topics
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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.
{'FirstName': 'Keith', 'LastName': 'Burns', 'PI_MID_INIT': None, 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Keith Burns', 'EmailAddress': 'burns@math.northwestern.edu', 'NSF_ID': '000297942', 'StartDate': '05/01/1987', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
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{'Code': '1265', 'Text': 'GEOMETRIC ANALYSIS'}
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