Written: March 10, 2000
Physical Science, at least for the last 200 years, had two kinds of research modes: theoretical and experimental. Now one can also add: computational. By contrast, mathematics, whether it was "pure" or "applied" only had the theoretical mode. No one got a Ph.D. for writing a computer program. Even a "conjectures-only" thesis was not acceptable. One had to prove theorems. If the theorems were too hard, one found easier versions, but proving theorems was the sine-qua-non for a math Ph.D.
who expects to earn his Ph.D. this coming May,
is a harbinger of a new kind of math thesis. It is not
"experimental" in the
Akalu did a masterful job in
After Akalu "built" (i.e. wrote) the "equipment" (i.e. software) he "took data" (i.e. used it to DISCOVER) the beautiful MULTI-VARIATE TEFERA INTEGRAL. This is a GENUINE theorem, even today, since, like the Selberg and Mehta integrals, it states something for an arbitrary dimension. Even Tefera's package, Mint, can't prove it in that generality. But for k=1,2,..., 6, Mint can find immediately, BEAUTIFUL WZ-Certificates. In the "analyzing the data" stage, Akalu detected a general pattern and proved, humanly, the general case. But without the equipment (the Maple package Mint), this would not have been possible. However, what's nice about Tefera's "equipment" is that it can be used on many other problems.
Two other 3rd-millennium Ph.D. thesis , that were completed ahead of schedule, way in the last millennium (c. 1998), are Frederic Chyzak's and Axel Riese's futuristic theses. I am sure that there are others, but we need many more, and in fifty years, this mode will be the norm.
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