Computational algebraic geometry

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…ogramming, or if you program robots' spatial reasoning, algebraic geometry might pay off - though, AG doesn't have as easy an entry. (One exception might be via the book Computational Algebraic Geometry by Hal Schenck). As far as number theory "paying off": 1) Many proofs in number theory are algorithmic in nature; 2) Computers understand integers with greater facility than they do the psuedo-reals we call floats - often times efficient integer approximations will be more appropriate than slower floating point solutions, and knowing the integers will help you develop/understand these approximations; 3) If cryptography is your bag, number theory is a must; 4) Martin Davis, Yuri Matiyasevich, Julia Robinson, Hilbert's 10th problem, Turing Machines, computability theory, and (the number theory bit) diophantine equations (sorry to be cryptic,…