Experimental Mathematics Meets Mathematical Physics
by David H Bailey
Abstract: High-precision arithmetic has been called the "electron microscope"
of experimental mathematics. The general approach is to compute some
mathematical expression to very high precision (typically several hundred digits)
for some specific choice of parameters, then apply an integer relation algorithm
such as "PSLQ" to find a relation linking this object or expression and other
known mathematical entities. Relations and formulas that are numerically
discovered in this manner must then be proven rigorously.
One particularly fruitful area for this methodology is the evaluation of definite integrals,
such as those that arise in mathematical physics. Literally hundreds of new and
intriguing results, specific and general, have been found in this manner, including
results in Ising theory, quantum field theory and even computational biology.
Progress in this arenas has been hampered by long run times required to evaluate
high-dimensional integrals. However, with the increasing availability of highly parallel
computer systems, many of these integrals can now be evaluated.
Nonetheless, new techniques are required to further advance the state of the art.
For More Information: contact Iwan Jensen. email: I.Jensen@ms.unimelb.edu.au