Mathematical Sciences: Investigations in Analysis
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The study of functions defined on real n-space (i.e.                        the situation in which a quantity depends jointly on some                       number n of variable quantities) has been one of the basic                      endeavors of mathematics for centuries, with frequent feed-                     back from applications. For example, suppose we have an                         inhomogeneous body situated in 3-space; density within the                      body may be considered as a function of position. In                            practice, e.g. in computed tomography, the density function                     is unknown and must be explored indirectly. The mathematical                    procedure for doing this is to "transform" the unknown                          function into another function that one can more easily see.                    (A CAT scanner is an expensive and elaborate implementation                     of a particular sort of transform.) Mathematical investigation                  of a given transform typically addresses questions such as                      the following. How much information is lost in forming the new                  function from the old? What kind (from very smooth to extremely                 chaotic) of new function does one obtain from what kind of                      old function? If you change the old function just a little,                     does the new, transformed function also change just a little,                   or drastically? (In mathematical language, is the transform                     continuous, or not?)                                                                Professor Oberlin's project has mostly to do with the                       question of continuity for a family of transforms indexed by                    real numbers between 1 and infinity. At the two extremes,                       one has, respectively, the Riesz potential and the spherical                    maximal operator, both of whose behavior is well-understood.                    The problem is, what happens in between? This is a hard                         question. One can get a feel for the situation by means of                      computer experiments, but these are necessarily inconclusive.                   Intricate and clever arguments by the mathematician himself                     are required to settle matters finally.
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{'Code': '1262', 'Text': 'MODERN ANALYSIS'}
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