Function Theory on Symmetric Spaces
and Lie Groups

Professor Koranyi

(from an NSF grant abstract)

Professor Koranyi investigates harmonic analysis on Riemannian symmetric spaces with an emphasis on harmonic functions and relations with complex analysis. He is particularly interested in solving the long-standing problem of a local Fatou theorem for harmonic functions on symmetric spaces. He is also interested in the use of quasiconformal maps in several complex variables.

The analysis involved in this research rests on the theory of Lie groups, named in honor of the Norwegian mathematician Sophus Lie. This theory has been one of the major themes in twentieth century mathematics. As the mathematical vehicle for exploiting the symmetries inherent in a system, the representation theory of Lie groups has had a profound impact upon mathematics itself, particularly in analysis and number theory. It has also had a profound impact upon theoretical physics especially quantum mechanics and elementary particle physics.

The main examples of Lie groups are groups of matrices You learn about matrices in Linear Algebra. For example, all the 3 by 3 orthogonal matrices of determinant one form a Lie Group. This group is called SO(3) and it represents the rigid motions of three dimensional space. To learn more about group theory you can take Algebra and Number Systems (MAT 314). Lie groups are special groups which have a topological structure. You can learn about topology in MAT 305 and more about complex analysis in MAT 423.

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