Mathematical Sciences: Ergodic Theory and Topological       Dynamics
{'Code': '03040108', 'Directorate': {'LongName': 'Direct For Mathematical & Physical Scien'}, 'Division': {'LongName': 'Division Of Mathematical Sciences'}}
{'SignBlockName': 'James Glazebrook'}
The situation of a group acting on a space, for instance the               group of real numbers acting by time evolution on the state space               of a physical system, is encountered quite often in mathematics                 and its applications.  One is interested in what happens overall,               on average, in the long run, which may be quite difficult to                    infer solely from what happens over short intervals of time.  A                 seemingly straightforward example figures prominently in the                    project of Professor Veech, namely the flow described by billiard               balls careening about inside a triangular region. Trajectories                  will either essentially fill out the whole triangle over time or                be periodic, coming back on themselves after a finite number of                 bounces. Classifying and describing the periodic trajectories                   requires some surprisingly sophisticated mathematics. In                        particular, it turns out to be advantageous to relate billiard                  flows to flows and other group actions on more exotic geometric                 objects.                                                                             At a technical level, this has to do with the ergodic theory               and dynamics of the action of SL(2,R) on Teichmueller space.                    Professor Veech is studying the prevalence of points in this                    space whose isotropy subgroups for the action are lattices.                     Certain such points arise from triangular billiards. The                        difficult problem of length spectrum for a flow, i.e. of what                   lengths of periodic orbits can occur, will be studied in various                geometric contexts.
{'FirstName': 'William', 'LastName': 'Veech', 'EmailAddress': 'veech@rice.edu', 'StartDate': '05/15/1989', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
{'Name': 'William Marsh Rice University', 'CityName': 'Houston', 'ZipCode': '770051827', 'PhoneNumber': '7133484820', 'StreetAddress': '6100 MAIN ST', 'CountryName': 'United States', 'StateName': 'Texas', 'StateCode': 'TX'}
[{'Code': '1262', 'Text': 'MODERN ANALYSIS'}, {'Code': '1265', 'Text': 'GEOMETRIC ANALYSIS'}]
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