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28/11/2005, 16:30 — 17:30 — Amphitheatre Pa2, Mathematics Building

Jonathan Borwein, *Dalhouisi University, Canada*

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What is High Performance Mathematics?

Seventy-five years ago Kurt Gödel overturned the mathematical apple cart entirely deductively; but he held quite different ideas about legitimate forms of mathematical reasoning:

*If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics.*

*Kurt Gödel, 1951*

This lecture will be a general introduction to Experimental Mathematics, its theory and its “Experimental methodology” that David Bailey and I — among many others — have come to practice over the past two decades. I will focus on the differences between *Discovering Truths* and *Proving Theorems*. We shall explore various of the computational tools available for deciding what to believe in mathematics, and — using accessible examples — illustrate the rich experimental tool-box mathematicians now have access to. In an attempt to explain how mathematicians use High Performance Computing (HPC) and what they have to offer other computational scientists, I will touch upon various * Computational Mathematics Challenge Problems* including $${\int}_{0}^{\mathrm{\infty}}\mathrm{cos}(2x)\prod _{n=1}^{\mathrm{\infty}}\mathrm{cos}\left(\frac{x}{n}\right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{dx}\stackrel{?}{=}\frac{\pi}{8}.$$

This problem set was stimulated by Nick Trefethen”s recent more numerical *SIAM 100 Digit, 100 Dollar Challenge* (*), which I shall also mention.

### Bibliography

- D.H. Bailey and J. M. Borwein,
*Experimental Mathematics: Examples, Methods and Implications*, *Notices Amer. Math. Soc.*, **52** No. 5 (2005), 502–514. [CoLab Preprint 269]. - Jonathan M. Borwein and David H. Bailey,
*Mathematics by Experiment: Plausible Reasoning in the 21st Century*, A.K. Peters, Natick, MA, 2004. - Jonathan M. Borwein, David H. Bailey and Roland Girgensohn,
*Experimentation in Mathematics: Computational Paths to Discovery*, A.K. Peters, Natick, MA, 2004.

All resources are available at http://www.experimentalmath.info.