III.27 The Fourier Transform, Terence Tao, III.28 Fuchsian Groups, Jeremy Gray. Pages: 206 – 209

Plane waves

Linear Operations on Functions

The Fourier transform diagonalizes the Laplacian.

Functional Calculus.

An important tool for understanding the randomness and uniform distribution properties in probability theory, harmonic analysis and number theory.

Many directions for generalizations are possible.

Fuchsian groups

The torus is defined using various methods.

Fuchsian groups are used to create multiple holed tori and the sphere in a similar way.

A Fuchsian group is a subgroup of the the group of Mobius transformations that map A disk to itself that moves some polygon around “en bloc” and thereby tiles the disk.

“The formal definition of a Fuchsian group is as follows.  A subgroup H of the group of Mobius transformations is said to act discontinuously if, for every compact set K in the disk D the sets h(K) and K are disjoint except for finitely many h in H.  A Fuchsian group is a subgroup H of the group of all Mobius transformations that acts discontinuously on the disk D.”

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