More open problems

Question

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.

@ Kevin Lin: I think this is a great question actually, provided that the "big-list" tag is added. — Max Muller, Dec 06 '10 at 14:20
Here is collection of open problems in mathworld: http://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: http://mathoverflow.net/questions/96202/open-problems-questions-in-representation-theory-and-around — Alexander Chervov, Jun 29 '12 at 12:15
Maybe some guys thought that there are no more open problems :D — Cristi Stoica, Jun 30 '12 at 00:18
May be this should be added to the list http://mathoverflow.net/questions/38890/status-of-quillens-conjecture-on-elementary-abelian-p-groups Quillen’s conjecture on elementary abelian p-groups — Alexander Chervov, Jul 09 '12 at 11:57
I think this is a very useful question. Don't really understand why it has been closed. — Loreno Heer, Jan 12 '15 at 13:18

Kelly Davis · Answer 1 · 2010-12-04T22:13:01.847

34

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.

edited Dec 04 '10 at 22:13

answered Dec 04 '10 at 22:00

Kelly Davis

1,897

Link does not work? – Rexcirus Sep 26 '22 at 22:16

score 21 · Answer 2 · answered Dec 04 '10 at 20:19

21

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).

answered Dec 04 '10 at 20:19

Mike Spivey

3,253

score 21 · Answer 3 · answered Dec 04 '10 at 21:24

21

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

answered Dec 04 '10 at 21:24

Jonathan Kiehlmann

1,016

2

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 03 '11 at 19:44

Joseph O'Rourke · Answer 4 · 2017-09-22T20:31:36.507

21

... 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.

edited Sep 22 '17 at 20:31

answered Dec 05 '10 at 02:15

Joseph O'Rourke

149,182
34
342
933

1

Thanks and congratulations for your list, it is very nice. – Cristi Stoica Dec 05 '10 at 07:19

score 17 · Answer 5 · answered Dec 04 '10 at 20:50

17

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/

answered Dec 04 '10 at 20:50

Felipe Voloch

30,227

8

See also the book Erdos on Graphs by Chung and Graham for a compilation of Erdos's open problems on graphs. – Timothy Chow Dec 04 '10 at 22:31

score 17 · Answer 6 · answered Dec 05 '10 at 06:11

17

Group theory: Open problems in combinatorial and geometric group theory, Problems in geometric group theory.

Topology: Open problems in topology

answered Dec 05 '10 at 06:11

Denis Osin

2,508

score 14 · Answer 7 · answered Dec 04 '10 at 23:22

14

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.

answered Dec 04 '10 at 23:22

Daniel Moskovich

21,887

Here is a new link: https://msp.org/gtm/2002/04/p024.xhtml - the one in the answer no longer works. DOI: 10.2140/gtm.2002.4.37 redirects to the old location at the moment. (I posted this in a comment rather than editing the answer - to avoid bumping an old question to the frontpage.) – Martin Sleziak Mar 05 '24 at 10:58

Seva · Answer 8 · 2010-12-08T12:19:47.950

14

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

edited Dec 08 '10 at 12:19

answered Dec 08 '10 at 12:12

Seva

22,827

score 11 · Answer 9 · answered Dec 06 '10 at 12:19

11

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

answered Dec 06 '10 at 12:19

Wadim Zudilin

13,404

score 11 · Answer 10 · edited Apr 11 '11 at 19:41

A (very) mixed bag:

A. Auel, E. Brussel, S. Garibaldi and U. Vishne, Open problems on central simple algebras, http://arxiv.org/abs/1006.3304
Many entries in the famous On-line Encyclopedia of Integer Sequences may be considered as problems (to compute further terms, etc.)
Arnold's Problems, 2nd ed., Springer, 2005 (translation from Russian) http://dx.doi.org/10.1007/b138219
(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
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).
B. Sturmfels, Open problems in algebraic statistics, arXiv:0707.4558
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

Re 4: would the Scottish Book be worth giving as a separate answer? (That is, if the question were reopened.) https://en.wikipedia.org/wiki/Scottish_Book — Todd Trimble, Sep 22 '17 at 21:10

score 10 · Answer 11 · answered Dec 05 '10 at 06:25

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.

score 9 · Answer 12 · answered Dec 09 '10 at 00:21

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)

score 8 · Answer 13 · answered Dec 05 '10 at 07:16

8

Mike Boyle's open problems in symbolic dynamics.

answered Dec 05 '10 at 07:16

subshift

1,110

score 7 · Answer 14 · answered Mar 29 '11 at 03:27

7

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/

answered Mar 29 '11 at 03:27

Dan Ramras

8,498

Roy Maclean · Answer 15 · 2010-12-04T21:37:35.813

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

score 6 · Answer 16 · answered Dec 04 '10 at 22:27

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.

score 6 · Answer 17 · answered Dec 04 '10 at 22:43

6

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).

answered Dec 04 '10 at 22:43

Gerry Myerson

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This year's URL is http://www.math.byu.edu/~doud/WNTC.html and includes links to past problem sets. – Paul Jenkins Dec 05 '10 at 06:52

score 6 · Answer 18 · answered Mar 29 '11 at 08:09

6

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

answered Mar 29 '11 at 08:09

Benoît Kloeckner

14,198

score 5 · Answer 19 · answered Dec 05 '10 at 02:30

5

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.

answered Dec 05 '10 at 02:30

partition_of_unity

511

score 5 · Answer 20 · answered Dec 09 '10 at 16:30

5

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.

answered Dec 09 '10 at 16:30

Scott Morrison

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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

score 5 · Answer 21 · answered Mar 29 '11 at 17:39

5

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/

answered Mar 29 '11 at 17:39

Damek Davis

783

Cristi Stoica · Answer 22 · 2010-12-07T21:19:49.100

3

Open Problems In Mathematics And Physics

edited Dec 07 '10 at 21:19

answered Dec 04 '10 at 21:35

Cristi Stoica

4,304

score 3 · Answer 23 · answered Dec 08 '10 at 19:09

3

Problems in analytic number theory around the Riemann Hypothesis

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

answered Dec 08 '10 at 19:09

Stopple

10,820

score 3 · Answer 24 · answered Mar 29 '11 at 03:44

3

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

answered Mar 29 '11 at 03:44

B. Bischof

4,782

score 3 · Answer 25 · answered Mar 29 '11 at 18:00

3

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.

answered Mar 29 '11 at 18:00

Richard Dore

5,205

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another Schindler, another list ... – PatrickT Mar 31 '14 at 18:40

More open problems

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