Experimental mathematics

The advent of powerful computers enables mathematicians to look for patterns, correspondences, in fact, make up conjectures which have been verified in several computable cases. This activity is often referred to as "experimental mathematics" . By its very nature, this activity cannot be reported on in the rigorous theorem-proof style of exposition which is standard in mathematics. For this reason, experimental mathematics has created some interest groups of its own, e.g., around the Journal "Experimental Mathematics" , and in mathematical programming.

Yet, the possibility of performing more and more advanced computations by means of little human (programming) effort also had its effect on the standard mathematical rigour of exposition: nowadays conjectures are verified to rather a significantly high degree of computational complexity before being brought forward as such. See [a3], [a1] for more applied examples.

Two examples in pure mathematics are the "Moonshine conjectures" , where modular functions are related to representations of sporadic finite simple groups (cf. [a2]), and the "GUE hypothesis" , where joint distributions of zeros of the Riemann zeta-function are equated to those of the eigenvalues of matrices from GUE, the Gaussian unitary ensemble of large-dimensional random Hermitian matrices (cf. [a4]).

#### References[edit]

[a1] |  P.D. Lax, "Mathematics and computing" J. McKenna (ed.) R. Temam (ed.) , ICIAM87 , SIAM (Soc. Industrial Applied Math.) (1988) pp. 137–143

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[a2] |  R.E. Borcherds, "What is Moonshine" , Proc. Internat. Congress Mathem. (Berlin, 1998) (1998) pp. 607–615 (Doc. Math. Extra Vol. 1)
[a3] |  M. Hazewinkel, "Experimental Mathematics" , CWI Monogr. , 1 , North-Holland (1986) pp. 193–234
[a4] |  P.J. Forrester, A.M. Odlyzko, "A nonlinear equation and its aplication to nearest neighbor spacings for zeros of the zeta function and eigenvalues of random matrices" , Organic Math. , CMS Conf. Proc. , 20 , Amer. Math. Soc. (1997) pp. 239–251