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**The bispectral problem**

**Abstract**

Under the umbrella of "the bispectral problem" one can place a number of problems with connections to special functions, mathematical physics, algebro-geometric aspects of rings of differential operators, integrable systems, etc.

I will try to adress some of these points with particular emphasis on open problems: I would like to stress the still unclear relation of this problem with the motivating problem of commuting integral-differential operators arising in the the work of Shannon in signal processing (circa 1950) as well as a pair of mostly untouched areas involving matrix versions of the problem as well as some several variables versions of it.