Experimental Mathematics Leads to New Insights

Experimental

mathematics PROFESSOR JON BORWEIN leads to new insights

In this discussion, Professor Jonathan Borwein of the University of Newcastle, Australia explains how modern computer technology has greatly broadened the ability to discover new mathematical results

textbooks – on experimental mathematics and and mathematical software (Moore’s law related mathematical computation. The given and on). We do mathematics in a more grant proposal focused on this area and is entitled laboratory-like mode. Computer Assisted Research Mathematics and its Applications (CARMA), which is also the name of The mathematical research community is facing the centre I direct at Newcastle University. You a great challenge to re-evaluate the role of might say CARMA brought me to Australia! proof in light of the growing power of current computer systems, modern mathematical How would you describe the field of computing packages, and the growing capacity experimental mathematics? to data-mine on the Internet. Add to that the enormous complexity of many modern Experimental applied mathematics comprises the capstone results such as the Poincaré conjecture, use of modern computer technology as an active Fermat’s last theorem, and the classification agent of research for the purposes of gaining of finite simple groups. As the need and insight and intuition, discovering new patterns prospects for inductive mathematics blossom, and relationships, testing and conjectures, and the requirement to ensure the role of proof is confirming analytically derived results, much in properly founded remains undiminished both in the same spirit that laboratory experimentation research and teaching. is employed in the physical sciences. It is closely To start, what is your research speciality? related to what is known as ‘experimental Similarly, what have you found with analysis mathematics’ in pure mathematics, as has been into Giuga’s Conjecture? Could you explain I am a bit of a jack of all trades: I have a 1974 described elsewhere, including by the late Herb some of your methodology? Oxford DPhil in optimisation theory and have Wilf in the Princeton Companion to Mathematics. held professorships as a pure mathematician, Guiga’s is a 63-year-old conjecture about an applied mathematician, an operations Depending on the context, the role of rigorous prime numbers that no one has any idea researcher and a computer scientist. My current proof in experimental applied mathematics may how to prove. By clever computation based research interests span pure (analysis), applied be much reduced or may be unchanged from that on smart mathematics, we have shown that (optimisation), computational (numerical and of its pure sister. There are many complex applied the conjecture can only fail for counting computational analysis) mathematics, and high problems where there is little point to proving numbers with more than 20,000 digits – as my performance computing. the validity of a minor component, rather than collaborator David Bailey says, ‘computo ego finding strong evidence for the appropriateness of sum’. Such kinds of mathematical computations I have authored well over a dozen books, including the general method. have a long history of uncovering hardware and a 2010 publication on convex functions, which software bugs that more standard scientific was a Choice 2011 Outstanding Academic Book, In what way does your study appeal to both computational tests do not. A famous example and over 350 refereed articles. My most recent applied and pure mathematics? How will it is the so-called Pentium bug. co-authored book, Lattice Sums Then and Now, change the culture of mathematics? sits on the boundary between mathematical For that reason, some of my algorithms for Pi physics and number theory. Mathematics has traditionally been seen as a were run on every Cray Inc. supercomputer deductive science, as opposed to the inductive before it left the manufacturer, from around Which area of mathematics did your grant methods of the physical and biological 1986 until the company was sold. In 1986, Bailey proposal focus on? sciences. My colleagues and I are breaking had replicable hardware and software errors down this somewhat false barrier, which is that Cray was unaware of during a then-record Over the past decade I have (co-)authored half being rendered obsolete by the power and computation of 29 million decimal digits of Pi. a dozen books – both research monographs and versatility of modern computing hardware The record is now 10 trillion digits! WWW.RESEARCHMEDIA.EU 71 PROFESSOR JON BORWEIN

Number crunch!

A study being conducted at the University of Newcastle in Australia is building on previous research that shows how modern computers can discover completely unexpected relationships and formulas

USING CONTEMPORARY COMPUTER of applied mathematicians and researchers have EVIDENCING GIUGA’S technology for active research is commonly actively integrated computer technology into their CONJECTURE referred to as experimental applied mathematics. studies. Characteristics of such computationally This method of study is utilised for insight and assisted, applied mathematical research includes: Computing the digits of Pi is just intuition: to discover correlations and relationships, computation and simulation for exploration and one example of the work carried test conjectures and confirm results derived by discovery, symbolic computing, high-precision out by Borwein in experimental analytics (similar to laboratory experiments in arithmetic, integer relation algorithms, graphics mathematics. Another area that has physical science). The practice is not far removed and visualisation, and connections with non- benefited from the use of modern from what is known in pure mathematics as traditional mathematics. computer technology and exploration experimental mathematics. is a mathematical conundrum called The key findings of Borwein’s study have thus far Giuga’s conjecture: a number theory Over the years, advances in computing have led the been nothing short of spectacular – particularly conjecture which postulates that, with way for progression in the field. This is particularly the results on the structure of short, random any positive integer n, we can confirm if true for Professor Jonathan Borwein of the walks and flights; randomness of the distribution n is a prime number by checking Guiga’s University of Newcastle, Australia. Over the past of digits of numbers; and other technical areas, condition. This is achieved through 25 years, Borwein has developed and cultivated including the creation of fast algorithms used calculating a sum, in which n is contained several series of results that were not possible via for hard image reconstruction issues. However, in the exponent of the summands. traditional methods. This was done across three it is the evolution of the study’s methodological research centres – in Vancouver, Canada and now underpinnings that Borwein finds most exciting. The sum would have a specific value – s Newcastle – all of which were built by Borwein. He calls it ‘experimental mathodology’, a for example – only if n is a prime number. name derived from a fortuitous misspelling of In other words, the sum would not have Among the plethora of highly technical findings ‘methodology’ (Borwein liked it and decided to the value of s if n is composite. gathered by Borwein and his group, perhaps keep it). These underpinnings are: gaining insight the most famous are the computer-generated and intuition, discovering new relationships, reverse-engineered results, as found by Borwein’s visualising math principles, testing (especially brother, Peter, along with two other researchers, falsifying conjectures), exploring a possible David Bailey and Simon Plouffe. They discovered a result to see if it merits formal proof, suggesting formula – the eponymous Bailey-Borwein-Plouffe approaches for formal proof, computing and (BBP) formula – that allows the binary digits of thereby replacing lengthy hand derivations, and Pi and other constants to be determined without confirming analytically derived results. Despite dating back to the 1950s, knowing the previous digits, as Borwein elaborates: the theory has never been “Last year, 25 hexadecimal digits (100 bits) of Pi The study has garnered much academic proven. It is often considered starting at the quadrillionth (10 to the power 15) insight, the lessons of which will develop too daunting for traditional position were computed by Ed Karrel at Nvidia, experimental mathematics in the classroom. mathematical methods. the graphics processing unit company. Until 1996 Borwein explains: “My postgraduate student Borwein, however, has it was thought to be impossible to ever compute Matt Skerritt and I have co-authored two made significant findings things like this. The discoverers of the BBP formula Springer-Verlag books: Modern Mathematical through the use of were the only mathematical finalists for the first Computation with Maple (2011) and Modern computers. With Edge of Computation Prize, and sat alongside the Mathematical Computation with Mathematica colleagues, he founders of Google, Netscape, Celera, among (2012) that introduce these tools into the was recently others”. The prize was ultimately won by the undergraduate classroom. We have even taught able to founder of quantum computing, David Deutsch. the course using modern collaboration tools to show a class with 15 students in Newcastle and 15 at EXPERIMENTAL MATH-ODOLOGY James Cook University in Northern Queensland. This experimental approach to mathematics There are several main areas of research in the field makes the subject lively and accessible to of experimental mathematics that are of students who are either proficient notable importance at present. or challenged in the field. For years, the majority

72 INTERNATIONAL INNOVATION VISUALISING THREE-STEP RANDOM WALKS. THE INTELLIGENCE THICK CIRCLE SHOWS COMPUTER ASSISTED RESEARCH THE EXPECTED DISTANCE MATHEMATICS AND ITS APPLICATIONS TRAVELLED OBJECTIVES This proposal has a pure and an applied part. The applied research is on a class of algorithms used successfully by many, but without real justification, for core signal reconstruction problems in astronomy, physics, bioscience, genomics, geoscience and medicine. Success that any number proving an exception to the enormous task as it requires much more robust will provide much better methods for such theory must have more than 4,771 distinct prime code than is currently available, which no one reconstructions and, equally importantly, an factors and be greater than 19,907 decimal digits wishes to pay for, and a series of culture changes.” understanding of why and when they work. long. This means that any shorter composite Success with the purer research, which still number cannot yield the value of s. Although this First, researchers need to be convinced that has immediate applications in other sciences, doesn’t confirm the theory, it provides fascinating ensuring reproducibility is worth their while, is also aimed at helping change the culture evidence in its favour. Indeed, it is only upon which will lead to increased productivity, less of mathematics at both a research and a realising small breakthroughs such as these that time spent recovering data or code, and a more pedagogical level. mathematicians are compelled and inspired to efficient conversion from data to published FUNDING seek full proof of a theory. papers. Second, the education system needs to provide more rewards at every institutional Over the four years of the ARC funded project THE FUTURE OF EXPERIMENTAL level, including departmental decisions, grant roughly AUS $550,000 came from ARC, MATHEMATICS funding and the publication of papers. Currently, with another $1 million from University of in the educational and industrial spheres, the Newcastle funding for CARMA (the research In terms of the future direction of his studies, main focus is on publication and project funding, centre) Borwein is clear: “I expect much more rather than reproducibility. However, this does CONTACT emphasis on advanced visualisation in two not favour those who spend much time following and three dimensions as illustrated in the or developing community standards. Finally, Laureate Professor Jon Borwein description of my recent work on Pi the standards set for peer review need to be School of Mathematical and Physical Sciences and other fundamental constants. I more robust: professionals, such as editors and Faculty of Science and Information Technology am heavily involved in attempts reviewers, need to emphasise the importance of University of Newcastle to make computational verification, validity testing and the full disclosure University Drive science research more of computational details. Indeed, some details Callaghan NSW 2308 reproducible and could be demoted to an online database or website, Australia reliable,” he providing the information remains accessible. reveals. “This T +612 4921 5535 is an Despite these challenges, the future remains bright E [email protected] for the computational experimental approach in applied mathematics – an approach that has JONATHAN BORWEIN is Laureate undeniably accelerated in the 21st Century, and Professor in the School of Mathematical will continue to do so. and Physical Sciences at the University of Newcastle (NSW) with other appointments at Dalhousie, Simon Fraser and Jeddah. He directs the university’s Priority Research Centre in Computer Assisted Research Mathematics and its Applications (CARMA). Borwein was Shrum Professor of Science (1993-2003), a Canada Research Chair in Information Technology (2001-08) at Simon Fraser University, and founding Director of the Centre for Experimental and Constructive Mathematics. From 2004-09 he worked in the Faculty of Computer Science at Dalhousie as a Canada Research Chair in Distributed and Collaborative Research, cross-appointed in mathematics.