Department of Mathematics
Mathematics in Science & Society
by
Prof. Richard E. Blahut

A time-honored observation is that the engineer eventually puts to practical use everything that he encounters. Recently, theoretical algebraic geometry and computational algebraic geometry have found several roles to play in electrical engineering. The topics of error-control codes, cryptography, and signal processing have been enriched by the insights that come from algebraic geometry. In return, I hope that algebraic geometry will be enriched by the insights coming from practical engineering problems.

This talk starts with an introduction to the subject of error control codes, including examples of applications, and some easily stated open problems. Then the role of algebraic geometry will be discussed, though without much algebraic geometry. The concrete case of an algebraic code defined on a Hermitian curve will be treated in more detail. The Sakata algorithm as a method for computing a Groebner basis for the error locator ideal will be described and related to computational commutative algebra. This will lead to a discussion of the Koetter algorithm, and the Mayaguez VLSI chip now being designed.

The talk will conclude with a few remarks about the role of algebraic geometry in public key cryptography and in the phase- retrieval problem for image reconstruction in the presence of incomplete data.