Mathematical Sciences: Local Harmonic Analysis With Applications to Automorphic Representations

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Professor Shelstad will continue her study of the transfer of orbital integrals in endoscopy and twisted endoscopy, along with applications in local harmonic analysis and to distributions associated with comparisons of Arthur - Selberg trace formulas. This has to do with the representation theory of Lie groups. These fundamental mathematical objects arise naturally in several ways. One basic example of a Lie group is the group of rotations of a sphere, where the group operation consists of following one motion by another. Detailed information concerning this group is very helpful in solving mathematical or physical problems in which spherical symmetry is present. Other groups of motions capture other kinds of symmetry. A more algebraic (as opposed to geometric) source of examples of Lie groups comes from the multiplication of matrices. The group of all invertible real (or complex) matrices of a given size is a Lie group, as is just about any subgroup thereof that can be described in a natural manner. It is desirable to be able to go back and forth between the geometric and algebraic points of view, for instance to consider the numerous ways in which the rotation group of the sphere can be realized as a group of invertible matrices. This, roughly, is representation theory. Facts about the representations of a given group tend to store a lot of information very economically. Transfer of this information (between two different ways of keeping track of the representations of a single group, or between two different but mathematically related groups) is the underlying theme of Professor Shelstad's research project.