Mathematics by experiment - plausible reasoning in the 21st century

I shall talk generally about experimental and heuristic mathematics and give accessible, primarily symbolic, examples. The emergence of powerful mathematical computing environments like Maple and Matlab, the growing availability of correspondingly powerful (multi-processor) computers and the pervasive presence of the internet allow for research mathematicians, students and teachers, to proceed heuristically and ‘quasi-inductively’. We may increasingly use symbolic and numeric computation visualization tools, simulation and data mining. Many of the benefits of computation are accessible through low-end ‘electronic blackboard’ versions of experimental mathematics. This permits livelier classes, more realistic examples, and more collaborative learning. Moreover, the distinction between computing (HPC) and communicating (HPN) is increasingly blurred. The unique features of our discipline make this both more problematic and more challenging. For example, there is still no truly satisfactory way of displaying mathematical notation on the web; and we care more about the reliability of our literature than does any other science. The traditional role of proof in mathematics is arguably under siege. Illuminated by examples, I intend to pose questions—discussed at length in [4]—such as: • What constitutes secure mathematical knowledge? • When is computation convincing? Are humans less fallible? • What tools are available? What methodologies? • What about the ‘law of the small numbers’? • How is mathematics actually done? How should it be? • Who cares for certainty? What is the role of proof? And I shall offer some personal responses and assessments. References 1. Jonathan M. Borwein and Robert Corless, “Emerging Tools for Experimental Mathematics,” American Mathematical Monthly, 106 (1999), 889–909. 1 2. D.H. Bailey and J.M. Borwein, “Experimental Mathematics: Recent Developments and Future Outlook,” pp, 51-66 Vol. I of Mathematics Unlimited 2001 and Beyond, B. Engquist and W. Schmid (Eds.), Springer-Verlag, 2000. 3. J. Dongarra, F. Sullivan, “The top 10 algorithms,” Computing in Science & Engineering, 2 (2000), 22–23. 4. J.M. Borwein and D.H. Bailey,Mathematics by Experiment: Plausible Reasoning in the 21st Century, and Experimentation in Mathematics: Computational Paths to Discovery, (with R. Girgensohn), AK Peters Ltd, November 2003. The website with resources for, and an extended sample of, these two books is http://www.expmath.info. Implicitization and Commutative Algebra ECCAD’ 2004 Invited Talk, David A. Cox, Amherst Colege, USA This talk will discuss implicitization using resultants, moving surfaces, and relations between the two. I will illustrate how concrete questions in implicitization leads naturally to concepts in commutative algebra such as syzygies, free resolutions, regularity, and local complete intersections. The talk will be down-to-earth but the moral would be that serious commutative algebra is involved. Solving zero-dimensional systems of equations and inequations, depending on parameters ECCAD’ 2004 Invited Talk, Daniel Lazard, Université Paris 6, FRANCE Let us consider a system of polynomial equations and inequations and split the set of its variables in two subsets, the set of unknowns and the set of indeterminates (this partition may be given as input but may also be done as a first step of computation). The addressed problem consists in computing the number of real solutions of the systems as a function of the parameters, under the hypothesis that, for almost all value of the parameters, the equations have only a finite number of common complex solutions. We present a general algorithm for this problem. It has been used for solving several problems of big size, which are far outside the range of applicability of any other existing method. A noteworthy feature of this algorithm is that Gröbner bases, triangular sets and CAD (in the space of the parameters only) are all needed as subalgorithms. 1All references are available at http://www.cecm.sfu.ca/preprints/. 2 ECCAD’2004, Wilfrid Laurier University, Waterloo ON, Canada ECCAD’2004 Poster Abstracts (in alphabetical order)