Speaker: David H. Bailey
Date, Time: Fri, 04 Sep 2009 15:00
High-precision arithmetic has
been called the "electron microscope" of experimental
mathematics (the application of high-performance computing
technology to research 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 in
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.