Abstract:
Long before current graphic, visualisation and geometric tools were available, John E. Littlewood, 1885-1977, wrote in his delightful Miscellany:
A heavy warning used to be given [by lecturers] that pictures are not rigorous; this has never had its bluff called and has permanently frightened its victims into playing for safety. Some pictures, of course, are not rigorous, but I should say most are (and I use them whenever possible myself). [[L], p. 53]
Over the past decade, the role of visual computing in my own research has expanded dramatically. In part this was made possible by the increasing speed and storage capabilities and the growing ease of programming of modern multi-core computing environments [BSC]. But, at least as much, it has been driven by my groups paying more active attention to the possibilities for graphing, animating or simulating most mathematical research activities.
I shall describe diverse work from my group in transcendental number theory (normality of real numbers [AB3]), in dynamic geometry (iterative reflection methods [AB]), probability (behaviour of short random walks [BS, BSWZ]), and matrix completion problems (especially, applied to protein conformation [ABT]). While all of this involved significant numerical-symbolic computation, I shall focus on the visual and experimental components.
AB F. Aragon and J.M. Borwein, ``Global convergence of a non-convex Douglas-Rachford iteration.’’ J. Global Optimization. 57(3) (2013), 753{769. DOI 10.1007/s10898-012-9958-4.
AB3 F. Aragon, D. H. Bailey, J.M. Borwein and P.B. Borwein, Walking on real numbers." Mathematical Intelligencer. 35(1) (2013), 42{60. See also http://walks.carma.newcastle.edu.au/.
ABT F. Aragon, J. M.Borwein, and M. Tam, ``Douglas-Rachford feasibility methods for matrix completion problems.’’ ANZIAM Journal. Galleys June 2014. See also http://carma.newcastle.edu.au/DRmethods/.
BSC J.M. Borwein, M. Skerritt and C. Maitland, ``Computation of a lower bound to Giuga's primality conjecture.’’ Integers 13 (2013). Online Sept 2013 at #A67, http://www.westga.edu/~integers/cgi-bin/get.cgi.
BS J.M. Borwein and A. Straub, ``Mahler measures, short walks and logsine integrals.’’ Theoretical Computer Science. Special issue on Symbolic and Numeric Computation. 479 (1) (2013), 4-21. DOI: http://link.springer.com/article/10.1016/j.tcs.2012.10.025.
BSWZ J.M. Borwein, A. Straub, J. Wan and W. Zudilin (with an Appendix by Don Zagier), ``Densities of short uniform random walks.’’ Can. J. Math. 64 (5), (2012), 961-990. http://dx.doi.org/10.4153/CJM-2011-079-2.
L J.E. Littlewood, A mathematician's miscellany, London: Methuen (1953); Littlewood, J. E. and Bollobas, Bela, ed., Littlewood’s miscellany, Cambridge University Press, 1986.