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Open Problem Garden and Wikipedia are good resources for more or less famous open problems. But many mathematicians will be happy with more specialized problems. They may want to find a research theme, e.g. for their PhD thesis, or they may have one, and want to connect their work to other problems, to find applications. Or they may simply want to check if something is already done about a particular question.

So, I would like to ask you (I apologize in advance if the question is not appropriate for MO):

can you suggest some links to other compilations of open problems, even if they are very specialized and not as famous?

Please note that I would like it to be large and specialized, not to contain only famous conjectures. I think each subject in mathematics has many such open problems. It would be useful to be maintained such a list, containing problems classified on subject. It would be useful to be able to check there if new progress was made, who is working to those particular problems, and how important are these problems (e.g. rated as in Open Problem Garden).

Thank you.

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closed as no longer relevant by Felipe Voloch, Bill Johnson, Ryan Budney, David Roberts, Andres Caicedo Dec 28 '11 at 22:43

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This question feels a bit too broad... –  Kevin H. Lin Dec 4 '10 at 23:04
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@ Kevin Lin: I think this is a great question actually, provided that the "big-list" tag is added. –  Max Muller Dec 6 '10 at 14:20
    
Here is collection of open problems in mathworld: mathworld.wolfram.com/topics/UnsolvedProblems.html –  Alexander Chervov Jun 29 '12 at 12:14
    
This MO question contains references to other MO questions with open problems lists in specific areas: mathoverflow.net/questions/96202/… –  Alexander Chervov Jun 29 '12 at 12:15
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I do not understand why this has been closed. –  Alexander Chervov Jun 29 '12 at 12:17

25 Answers 25

A (very) mixed bag:

  1. A. Auel, E. Brussel, S. Garibaldi and U. Vishne, Open problems on central simple algebras, http://arxiv.org/abs/1006.3304

  2. Many entries in the famous On-line Encyclopedia of Integer Sequences may be considered as problems (to compute further terms, etc.)

  3. Arnold's Problems, 2nd ed., Springer, 2005 (translation from Russian) http://dx.doi.org/10.1007/b138219

  4. (famous?) Ulam's "Problems in Modern Mathematics", and its successor: R.D. Mauldin and S.M. Ulam, Mathematical problems and games, Adv. Appl. Math. 8 (1987), 281-344 http://dx.doi.org/10.1016/0196-8858(87)90026-1

  5. M. Sapir, Some group theory problems, Intern. J. Algebra Computation 17 (2007), 1189-1214, arXiv:0704.2899 (probably related to and/or overlaps with a list provided by Denis Osin).

  6. B. Sturmfels, Open problems in algebraic statistics, arXiv:0707.4558

  7. E. Zelmanov, Some open problems in the theory of infinite dimensional algebras, J. Korean Math. Soc. 44 (2007), 1185-1195 http://www.kms.or.kr/home/journal/RPArticles/View.asp?IDXNo=557&Page=1

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Miller has a list of set theory problems here (upper right).

Schindler also has a list of open problems specifically in inner model theory here.

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another Schindler, another list ... –  PatrickT Mar 31 at 18:40

Igor Shparlinski maintains a large list of open problems in exponential and character sums. You can find it on his website: http://web.science.mq.edu.au/~igor/

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Douglas B. West has a sorted list of open problems in combinatorics and graph theory: http://www.math.uiuc.edu/~west/openp/

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Ismar Volic from Wellesly has a list of problems in Calculus of Functors related to knot theory on his webpage here.

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Mark Hovey maintains a list of open problems in algebraic topology (which, as he points out, hasn't been updated in a while): http://math.wesleyan.edu/~mhovey/problems/

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There's a recent list of open problems in von Neumann algebras, from the recent 2010 Noncommutative Geometry and Operator Algebras Spring Institute at Vanderbilt University.

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I like the way Problem 3.8 is just sneaked in there, as if "oh, by the way, what about this one?" –  Yemon Choi Mar 29 '11 at 21:03

Many PhD supervisors have lists of open problems in their subject, often hidden somewhere on their web site. Here are some from my university:

Here are 27 research problems in group theory from Rob Wilson: http://www.maths.qmul.ac.uk/~raw/resprob.html

Here are many problems in combinatorics and group theory from Peter Cameron: http://www.maths.qmul.ac.uk/~pjc/oldprobidx.html

(And also his links to other lists of problems: http://www.maths.qmul.ac.uk/~pjc/bcc/links.html#prob)

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Problems in analytic number theory around the Riemann Hypothesis

http://www.aimath.org/WWN/rh/

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A collection of (mostly, still) open problems in additive combinatorics due to Ernie Croot and myself, including a brief historical account.

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These should be really hard: Michel Waldschmidt, Open Diophantine Problems, Moscow Math. Journal 4:1 (2004), 245-305, 312.

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For problems on topics related to the mapping class group, the moduli space of curves, etc, I recommend the collection of papers "Problems on Mapping Class Groups and Related Topics" edited by Benson Farb. It is available on his webpage here.

Another interesting list of problems on related topics is the problem list for the Center for the Topology and Quantization of Moduli Spaces, which is located here.

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For the probabilists out there, the recent books by Yuval Peres seem to contain some open problems. For example, see:

  • Mörters and Peres. Brownian motion. 2010.
  • Levin, Peres, and Wilmer. Markov Chains and Mixing Times. 2008.
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... even if they are very specialized and not as famous

Taking you at your word, may I humbly point to the list I maintain with Joe Mitchell (SUNY Stonybrook) and Eric Demaine (MIT), focusing on discrete and computational geometry: what we call The Open Problems Project.

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Thanks and congratulations for your list, it is very nice. –  Cristi Stoica Dec 5 '10 at 7:19

Tomotada Ohtsuki's Problems on invariants of knots and 3–manifolds is a good compilation of open problems in low-dimensional topology, mainly in quantum topology. I think that it complements Kirby's problem list nicely.

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The Western Number Theory meeting holds a problem session every year. The problems get written up, and for the last 10 years or so they've been posted to the meeting website (for which, alas, I don't have the URL handy).

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This year's URL is math.byu.edu/~doud/WNTC.html and includes links to past problem sets. –  Paul Jenkins Dec 5 '10 at 6:52

The Egres Open is a collection of open problems in combinatorics (theoretical combinatorial optimization) by the Egerváry Research Group.

It currently has about a hundred problems, but is actively maintained, so occasionally new ones will appear and some old ones get solved.

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Kirby's list is large (380 pages), specialized (only contains problems in low-dimensional topology), classified on subject (knot theory, 2-manifolds, 3-manifolds, 4-manifolds...), and contains references indicating who has worked on each problem. Sounds like what you want.

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Thank you, that's nice. –  Cristi Stoica Dec 4 '10 at 22:09

The book Research problems in discrete geometry by Peter Brass, W. O. J. Moser, János Pach, is a very large collection which describes what is known about each problem with a large reference list of papers for each problem.

Also Google Books search: "open problems in" OR "unsolved problems in" OR "research problems in" mathematics

and Google Scholar search: "open problems in" OR "unsolved problems in" OR "research problems in" mathematics

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The Kourovka notebook is published every four years containing unsolved problems in Group Theory.

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Once upon a time, there were also similar "Dniester notebook" (rings and modules) and "Sverdlovsk notebook" (semigroups). Of these, only "Kourovka notebook" seems to survive (the last edition of Dniester notebook appeared in 1993 and of Sverdlovsk notebook - in 1989). –  Pasha Zusmanovich Jan 3 '11 at 19:44

There is a website with pdfs of many (most, all?) papers of P. Erdos. Several of his later papers are compilations of problems. If you are into that sort of thing, it's the place to go:

http://www.renyi.hu/~p_erdos/

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See also the book Erdos on Graphs by Chung and Graham for a compilation of Erdos's open problems on graphs. –  Timothy Chow Dec 4 '10 at 22:31

I have found Richard Guy's Unsolved Problems in Number Theory to be very interesting and useful.

Springer also publishes Unsolved Problems in Geometry (by Croft, Falconer, and Guy).

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Thanks, this is great. –  Cristi Stoica Dec 4 '10 at 20:25

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