Odd Mathematical Lists

1. What are the five most famous or important unsolved mathematical problems today?
2. What is the most difficult-to-understand areas of mathematics today?
3. Who are the five most influential mathematicians that ever lived?
4. Who are the five most influential mathematicians alive today?
5. Who was the strangest mathematician that ever lived?
6. What are the five most interesting numbers? (And 5 most interesting integers?)
7. What one question would you like seen added to this list?

Note: these questions and many more are now answered in my book Wonders of Numbers!

Some Sample Answers

                                                                           
1 + sqrt(5)         1 - sqrt(5)                                                 
-----------   and   -----------                                                 
     2                   2                                                      

                                                                                
lim           e^n * n!                                                          
n->inf.    ------------- = sqrt(2 * pi)                                         
           n^n * sqrt(n)                                                        

                                                                     
lim  sqrt(n + sqrt(n +...)).. = 1                                               
n->0                                                                            

From Olivier Gerard:
1. A) The Riemann Hypothesis, B) The Goldbach Conjecture, C) The normality of Pi digits in an integer base, D) Kepler Conjecture, E) Poincar=E9 Conjecture.
2. A) Quantum Groups, B) Motivic Cohomology, C) Infinite Banach Spaces, D) local and micro local analysis of large finite groups, E) Large and inaccessible cardinals
3. Euler, Gauss, Hilbert, Dedekind, Galois
4. Vladimir Arnol'd, John Conway, William Thurston, Alain Connes, Ed Witten 5. Ramanujan 6. 0, i, 2, Euler's Gamma, Chaitin's Omega, Aleph 1
I think all that requires some comments: I have deliberately made very "classic" choices for most questions especially 1,3,5. The order is most of the time meaningless. The answers someone could make to this questionnaire are more revealing of his mathematical culture, background and activity than anything else.
1. My list is very "classic" in the sense that most of these problems were already posed before 1900. There is an ongoing study to write out the most important math problems for the 21st century, in the frame of the forthcoming World Math Year 2000. I think this is under the auspices of the IMU.
2. The question is inevitably biased. A theory can be difficult because it is awfully written and notated. It can also be temporarily difficult because some pieces are lacking (such as in a puzzle) or a good synthesizer has not yet put some order in several layers of recent or less recent ciphering themselves mutually. I have very rarely seen genuine or eternal difficulty in mathematics, but often conjectural or cultural one. My answer A), B) and D) are difficult because you must go through a lengthy initiation, review and training process before hoping to say anything useful or slightly new and at present they seem to mainly feed themselves. C) and E) are difficult because off the well-traveled roads of progressive intuition building of "mainstream" mathematics but if you like what others could call "pathology" or "nonsensical" exotism, they have the advantage of being quite stimulating.
5 items is perhaps not a comfortable bound for most topics. One of the trouble for instance with question 4 is related with the Fields Medal. It cannot be presented to more than 4 people at a time because its renown would be diluted, but in the same time, at each International Congress, a good dozen are really worth it. Influential people are less common, but for 4, very influential people are still alive but quite old and not so active as they were. Example: 4old: Henri Cartan, Ren=E9 Thom, B.L. Van der Waerden, Saunders MacLane, (Eilenberg is dead this year), (Gorenstein, the leader of the attack of the classification of simple groups is dead a few yeas ago) Itoh, Andr=E9 Weil, Walter Feit, Gustave Choquet...
4bis: Eugen Dynkin, Jean-Pierre Serre, Donald Knuth, Stephen Smale, Ingrid Daubechies, Doron Zeilberger, Vladimir Drin'feld, Pierre Cartier, Herbert Wilf, George Andrews, B. Mandelbrot, Gerd Faltings, Martin Kruskal, Armand Borel, Jean-Yves Girard, Lawvere, Harvey Friedmann, Andrew Odlyszko, Philippe =46lajeolet, ...
Other World Class Mathematicians: 3bis: Riemann, John von Neumann, Leibniz, Henri Poincar=E9, Allan Turing, Niels Abel, Sophus Lie, Felix Klein, Lagrange, Fermat, Euler, Grothendieck, Carl Siegel, Jacobi, Camille Jordan, Cantor, Hadamard, Hardy, Littlewood, Newton, Girard Desargues, Pascal, Weierstrass, Gentzen, Tschebycheff, Markov, Kolmogorov, Fredholm, Banach, Cayley, Sylvester, Mobius, Cauchy, Laplace, ... (note Grothendieck is still alive but no more mathematically active)
5. Ramanujan is the strangest but at the same time has been much studied. In many ways, the isolated mathematical genius is a source of wonder but when mathematicians live long like Newton, Gauss or Euler, the later achievements remove the drama of their beginnings. The birth of the mathematical abilities of Abel and Galois is not easily explained. The only common factor seems to be a tremendous amount of time and energy devoted to their passion, and a relative intellectual independance.
6. I have put 0 and 2 because I feel there is still much to understand in the process of counting and its ramifications, the notion of an origin, etc.= =2E i is there for these reasons but it is also the symbol of the algebraic closure of reals, a fantastic "passeur" and tool for many fields (as in number theory), and an historical problem of representation. Euler's Gamma is another kind of link : between the exponential/logs on a si= de and number theory on the other. This is one of the best representatives of these recurrent constants which grows a mystical aura. Chaitin's omega shows among other things the contrast between the wide expression power of human (natural or artificial) languages (that we use in definitions), and the very limited machinery of our mathematics (that we use for demonstrations). Aleph 1 is the symbol of the central independence results of set theory, (AC, CH), one of the biggest mathematical events of the 20th century.
Other questions could be: 7n) What are the five mathematical theories you find over-rated or too publicized to be easy to work seriously in or sort the hype ? 8n) What are the five mathematical theories you find under-rated, unknown, underused, ... ? 9n) What are the five mathematicians who do best when communicating their fields and results to non-specialists ? 10n) What are the five (mathematicians or not) who do best when communicating their results (or other's) to non mathematicians ? 11n) Can you quote 5 living mathematicians whose writing or oral skills are particularly poor ? 12n) What are the 5 five most boring question you get asked on mathematics ?