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I have recently retired after being a maths teacher for 35 years. I am interested in finding out what has happened in my subject since I was a student in the early 70's. I am particularly interested in finite algebra and combinatorics. How can I find other people like myself to correspond with and what are good books to start reading?

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If you have not already done so you should have a look at: arxiv.org –  Bruce Westbury Aug 24 '11 at 11:00
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You can't go wrong with Graham, Knuth, and Patashnik's "Concrete Mathematics", which includes the phrase "Way back before people were fluent with binomial coefficients (the early '70s)". Also, mathunion.org/ICM-search/answer.cgi?first=+combinatorics –  Allen Knutson Aug 24 '11 at 11:00
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This thread has some wonderful suggestions for practicing mathematics as hobby and might be of relevance mathoverflow.net/questions/20386/mathematics-as-a-hobby –  user11000 Aug 24 '11 at 11:17
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Unlike Bruce, I would not recommend the arxiv: it's a great repository, and works wonderfully when you know what you're looking for, but I doubt you would get much joy from just browsing it. These days, it seems like blogs are the way to go for good exposition. In addition to Wilf's blog, I would recommend the ones by Terry Tao and Gil Kalai. If you're into computational complexity aspects at all, you might also want to check out the blogs by Dick Lipton and the joint one of Lance Fortnow and Bill Gasarch. No links really needed; Google ought to lead you straight to them. –  Thierry Zell Aug 24 '11 at 14:47
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As blog recomendations go: Tim Gowers also writes a great blog gowers.wordpress.com, his work is related to combinatorics. You should also look at the mathematical discussions on his webpage dpmms.cam.ac.uk/~wtg10 maybe reading Two culures of mathematics where he writes about Combinatorics –  sisn Aug 24 '11 at 17:41
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6 Answers 6

As far as reading is concerned, there are many areas of combinatorics which either didn't exist in the early 1970s or hardly existed compared to today:

*Additive combinatorics: -Terence Tao, Van Vu. "Additive Combinatorics". Cambridge University Press. revised ed. 2009

*Analytic combinatorics: -Philippe Flajolet, Robert Sedgewick. "Analytic Combinatorics". Cambridge University Press. 2008. Free online edition: http://algo.inria.fr/flajolet/Publications/book.pdf

*Algebraic combinatorics:
-Christopher David Godsil. "Algebraic combinatorics". Chapman & Hall. 1993
-Lowell W. Beineke, Robin J. Wilson. "Topics In Algebraic Graph Theory". Cambridge University Press. 2004

*Geometric combinatorics:
-Ezra Miller, Victor Reiner, Bernd Sturmfels. "Geometric Combinatorics". AMS. 2007

*Topological combinatorics: -Jiří Matoušek. "Using the Borsuk-Ulam Theorem". Springer. 2003

*Combinatorics on words: -Jean Berstel, Juhani Karhumäki. "Combinatorics on words - a tutorial". http://www-igm.univ-mlv.fr/~berstel/Articles/2003TutorialCoWdec03.pdf

*Category-theoretic combinatorics: -François Bergeron, Gilbert Labelle, Pierre Leroux. "Combinatorial Species and Tree-like Structures". Cambridge University Press. 1998

*The C-finite Ansatz: -Doron Zeilberger. http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/cfinite.html

*Model-theoretic combinatorics:
-Martin Grohe, Johann A. Makowsky. "Model Theoretic Methods in Finite Combinatorics". AMS. 2011
-Erich Grädel. "Finite model theory and its applications". Springer. 2007


Modern books on more classical areas of combinatorics:

*Enumerative combinatorics: -Richard P. Stanley. "Enumerative Combinatorics", Volumes 1 and 2. Cambridge University Press. 1997, 1999, online draft of 2nd Ed of vol 1 2012

*Probabilistic combinatorics: -Noga Alon, Joel H. Spencer. "The Probabilistic Method" 3rd ed. Wiley. 2008

*Extremal combinatorics:
-Béla Bollobás. "Extremal graph theory". Academic Press. 1978. (Dover 2004)
-Alexander Soifer. "Ramsey Theory: Yesterday, Today, and Tomorrow". Springer. 2010
-Ian Anderson. "Combinatorics of Finite Sets", Dover reprint. 2002
-Konrad Engel. "Sperner Theory". Cambridge University Press. 1997

*Matroids: -Neil White. "Theory of Matroids". Cambridge University Press. 2008

*Designs: -Thomas Beth, Dieter Jungnickel, Hanfried Lenz. "Design theory", Volumes 1 and 2. Cambridge University Press, 1999.


Finite algebra

For finite algebra and combinatorics together: -Warwick De Launey, Diane Flannery. "Algebraic Design Theory". AMS. 2011

Possible project: investigate how finite algebraic structures interact with other finite structures: search for finite geometries, finite metric spaces, finite topological spaces, finite dynamical systems.

*Finite groups:
-Michael Aschbacher. "Finite group theory". Cambridge University Press. 2000
-Roger William Carter. "Finite groups of Lie type: conjugacy classes and complex characters". Wiley. 1993
-Simon R. Blackburn, P. M. Neumann, Geetha Venkataraman. "Enumeration of finite groups". Cambridge University Press. 2007
-Tullio Ceccherini-Silberstein, Fabio Scarabotti, Filippo Tolli. "Harmonic analysis on finite groups". Cambridge University Press. 2008
-T. Tsuzuku, A. Sevenster, T. Okuyama. "Finite Groups and Finite Geometries". Cambridge University Press. 1982
-Mara D. Neusel, Larry Smith. "Invariant theory of finite groups". AMS. 2002

*Finite fields:
-Rudolf Lidl, Harald Niederreiter, Paul Moritz Cohn. "Finite fields". Cambridge University Press. 1997
-There are regular international conferences on finite fields and applications with proceedings published.


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For combinatorics on words, there are also 3 books by Lothaire, 2 of them being freely accessible from www-igm.univ-mlv.fr/~berstel/Lothaire . –  darij grinberg Aug 24 '11 at 15:37
    
Very nice list! –  quid Aug 24 '11 at 16:04
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A good reference for combinatorics on words is the book Automatic Sequences: Theory, Applications, Generalizations by J.-P. Allouche and J. Shallit (Cambridge University Press, 2003). –  Joel Reyes Noche Aug 25 '11 at 0:28
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Some simple, humble suggestions to address the part about "how can I find other people like myself..."

  1. Talk to more people about this (you already took the first steps!)
  2. Blog
  3. Read survey papers, ask questions on both MO and math.stackexchange on your path to (re)-discovering your interests.
  4. Perhaps try to found an informal mailing list for mathematicians similar to you, with maybe a local meetup group. See if creating such a group via social networking sites helps.
  5. Maybe send emails to math departments, telling them about your "math club", and request them to circulate it to the target group of mathematicians you want to connect with. This requires heavy emailing, and the results will be sparse, but not zero.

Wish you the best.

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Hurray for #3. At times, MO can play the part of tea hour in a good math department, when you can get just the shove in the right direction after you've stumbled onto something that's been bothering you in your readings. Don't be shy! –  Thierry Zell Aug 24 '11 at 19:20
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Gerhard Paseman suggested that I post the comments I made to his post as an answer. I will edit it a bit.)

I suggest that a blog with theme exactly the question of maths in retirement might be quite fun to do. My experience is that some of the web activity which I am involved with is not being done only by full time professional mathematicians. The other contributors have good mathematical credentials and may be retired, out of work(!), or have another job which does not fully fulfill their mathematical interests and they can thus be freer to contribute when the political pressures of having to publish work might otherwise dominate. People who have retired whether from (school) teaching, from lecturing or from industrial mathematics have experience and knowledge that if pooled could be useful for everyone.

As some of you may know, I was `retired' by my university closing the mathematics section in the University of Bangor, but I still do a lot of research and contribute to the n-Lab etc. Retirement does not mean that you stop doing maths if you want to (we all know it is an addiction!) or if your background is in secondary school teaching or in business or industrial mathematics,, that you can not start building up mathematical activity of various sorts. I would not know how to start such a blog, so will not volunteer to do so, but would encourage others to try it out. The exchange of ideas problems etc. could produce some very interesting results.

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As I recall, the AMS and/or MAA sometimes have mini-courses at their national meetings intended for mathematicians who want to enter another branch of mathematics as a field of research.

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At the January 2012 Joint Math Meetings in Boston, here are links for the AMS and the MAA short courses: jointmathematicsmeetings.org/meetings/2138_amssc jointmathematicsmeetings.org/meetings/2138_maasc –  Joseph O'Rourke Aug 25 '11 at 1:22
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In addition to reading weblogs, as some comments have suggested, you should consider starting your own blog.. If you are discreet about it, you could place the occasional link to your writings in other places. At some point your audience will select your writings and can become the correspondents you desire.

Addition: Tim Porter suggested what I think is an excellent theme for your new blog, 'Mathematics In Retirement'. He didn't tell me where he got the idea though. End Addition.

Gerhard "Ask Me About System Design" Paseman, 2011.08.24

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I was going to suggest that a blog with theme exactly the question of maths in retirement might be quite fun to do. –  Tim Porter Aug 24 '11 at 15:44
    
I think you just did. I will edit my answer to make your suggestion more prominent. Gerhard "Ask Me About Answer Improvement" Paseman, 2011.08.24 –  Gerhard Paseman Aug 24 '11 at 15:56
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My point was that some of the web activity which I am involved with is not being done only by full time professional mathematicians. The others have good mathematical credentials and may be retired, out of work(!), or have another job which does not fully fulfill their mathematical interests and they can be freer to contribute when the political pressures of having published work might dominate otherwise. People who have retired whether from (school) teaching or from lecturing have experience and knowledge that pooled could be great for everyone. – Tim Porter 0 secs ago –  Tim Porter Aug 24 '11 at 17:31
    
Sounds good. I think it would be good if you posted those as a separate answer. I don't mind the extra traffic if you decide to keep them as comments here. Gerhard "Ask Me About System Design" Paseman, 2011.08.24 –  Gerhard Paseman Aug 24 '11 at 18:07
    
@Tim I agree :D –  David Roberts Aug 24 '11 at 22:05
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Some more books to read:

*A 2nd edition with a series of appendices outlining recent developments: "General lattice theory", George A. Grätzer, Birkhäuser, 2003

*Kolmogorov Complexity has applications to combinatorics:

-"An introduction to Kolmogorov complexity and its applications", Ming Li, P. M. B. Vitányi, Springer, 2008

*Association schemes have connections to finite groups:

-"Association schemes: designed experiments, algebra, and combinatorics", Rosemary Bailey, Cambridge University Press, 2004

And because there's more to algebra than groups, rings and fields:

-An older book but still: "The Algebraic Theory of Semigroups Vols 1 & 2", A. H. Clifford, G. B. Preston, American Mathematical Soc., 1961

-"An introduction to quasigroups and their representations", Jonathan D. H. Smith, CRC Press, 2007

-"Applications of hyperstructure theory", Authors Piergiulio Corsini, Violeta Leoreanu, Springer, 2003

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