Mathematical Sciences: Coding and Convergence in Ergodic    Theory
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Ergodic theory is an active, central area of Modern                     Analysis.  Its origins lie in the theoretical formulation for                   classical statistical mechanics at the end of the nineteenth                    century.  The modern point of view derives from the profound                    mathematical theory of recurrence and ergodicity, as developed by               Poincare, Birkhoff, von Neumann and other giants earlier in this                century.  As the subject has evolved and sophistication                         continually increased, ergodic theory has acquired close                        relationships with other branches of mathematics, such as                       dynamical systems, probability theory, functional analysis,                     number theory, differential topology, and differential geometry,                and with applications as far ranging as mathematical physics,                   information theory, and computer design.                                                Professor Petersen has worked on several aspects of ergodic             theory, especially those connected with harmonic anlaysis,                      probability, and more recently information theory.  A major part                of his work has concerned maximal functions and almost everywhere               convergence; for example, the refinement of the maximal ergodic                 theorem to an equality, speed of convergence in the ergodic                     theorem, the ergodic Hilbert transform, and the relationship with               spectral theory and harmonic analysis.  He has also conducted                   research in the applications of ergodic theory, specifically                    symbolic dynamics, to efficient coding of information across                    noisy channels.  Here, Professor Petersen proposes to continue                  his research into both symbolic dynamics and almost everywhere                  convergence.  In the former subject, he will consider general                   properties of symbolic dynamical systems on an infinite alphabet                as well as a class of examples that relate to the control of                    signals for magnetic recording.  In the latter area, his research               will emphasize connections between the ergodic Hilbert transform                and the spectral measure of a measure-preserving transformation.
{'FirstName': 'Karl', 'LastName': 'Petersen', 'PI_MID_INIT': 'E', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Karl E Petersen', 'EmailAddress': 'petersen@math.unc.edu', 'NSF_ID': '000201433', 'StartDate': '07/01/1987', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
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{'Code': '1262', 'Text': 'MODERN ANALYSIS'}
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