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As the title might indicate , I would like to look for recommendations for mathematical book that present open problems in depth with commentary. The only book of this type that I've come across is Arnold's Problems by V.I Arnold, are there any similar ones?

The subjects that these problems belong to should preferably be the following:

Algebraic Geometry.

Algebraic Topology.

Intersections of pure mathematics with theoretical computer science or ''theoretical'' biology.

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    $\begingroup$ In my opinion this is too broad and too vague. $\endgroup$
    – user9072
    Commented May 4, 2015 at 15:17
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    $\begingroup$ Theristo: MO has many threads on open problems. Perhaps look over them and see if there is much missing from what you are looking for. Presumably you are not interested in open problems in every subject related to mathematics? That would be an enormous amount of reading. $\endgroup$
    – Ryan Budney
    Commented May 4, 2015 at 15:22
  • $\begingroup$ In my view it is better to "learn about X" rather than "find lists of open problems in XYZ" $\endgroup$
    – Yemon Choi
    Commented May 4, 2015 at 16:06

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The following four have non-zero (but not large) overlap with your edit narrowing to algebraic geometry & topology.

(1) Brass, Peter, William OJ Moser, and János Pach. Research problems in discrete geometry. Springer Science & Business Media, 2005. (Springer link.)

(2) Guy, Richard. Unsolved problems in number theory. Vol. 1. Springer Science & Business Media, 2004. (Springer link.)

(3) Croft, Hallard T., Kenneth J. Falconer, and Richard K. Guy. Unsolved problems in geometry. (1991): 107-130. (Springer link.)

(4) Guy, Richard K., and Richard J. Nowakowski. "Unsolved problems in combinatorial games." More Games of No Chance, 42 (2002): 457-473. (Cambridge link.)

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  • $\begingroup$ I am not sure that number (3) has any overlap with alg geom and alg top, but I guess you are more familiar with the contents than I am. Similar comments apply to (2) and (4) $\endgroup$
    – Yemon Choi
    Commented May 4, 2015 at 16:05
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    $\begingroup$ @YemonChoi: Point taken. I started preparing this response before the OP clarified the areas of focus. $\endgroup$
    – Joseph O'Rourke
    Commented May 4, 2015 at 16:25

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