Computational algebraic geometry

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1. spekcularFeb 2022
   There are some old tutorials that show how to do concrete calculations with these objects with Macaulay. They were an immense help to me when learning algebraic geometry, because they showed me how all this sheaf blah blah blah connected to actual concrete polynomials.

   Try these two: https://faculty.math.illinois.edu/Macaulay2/Book/Computation... https://faculty.math.illinois.edu/Macaulay2/Book/Computation...

   There's more in the book "Computations in algebraic geometry with Macaulay 2," which is free online. See also Schenck's "Computational Algebraic Geometry."

   edit: Also, topos theory is definitely not the way to go here. Getting your hands dirty with algebraic curves is. See e.g. Griffiths' Introduction to Algebraic Curves.

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2. aoristeMar 2010
   Well, unless you're terribly interested in moving on to things like differential geometry, I'd put analysis off a bit longer were I you. If you'd like something stimulating and potentially useful, more advanced combinatorics, number theory, or algebraic geometry might all be good choices. The first two tend to overlap a bit when they are presented in textbooks (with some group theory tossed in as well). If you do crazy graphics programming, 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, this is getting rather long winded).

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