Forum for Science, Industry and Business

Sponsored by:     3M 
Search our Site:

 

Computer Calculates Correct Conjecture

25.01.2011
A solution that counts: long-standing mathematical conjecture finally proved

A conjecture presented in 1985 - the Andrews and Robbins conjecture - has recently been proved for the first time. It is thus clear that the structure which goes by the name of "totally symmetric plane partitions" can be described using a single formula. Producing the proof required vast computer resources and was only possible after the formula had been prepared for computer-assisted calculation.

This finding by a Austrian Science Fund FWF supported research group based in Linz, Austria will be published in the Proceedings of the National Academy of Sciences today. The proof means that the last of a long list of famous mathematical conjectures relating to plane partitions has finally been proved.

Even mathematicians play with building blocks. At least if they are interested in so-called plane partitions, which are visualized with columns of "building blocks" (cubes) on a surface resembling a chessboard. When "building" such plane partitions, the mathematicians must adhere to certain rules: No column may be higher than the width of the surface, or than another column behind it or left of it. The question of how many column permutations may be built on a given surface area is easily answered, thanks to a specific formula. However, it becomes trickier if the permutations must follow stipulated symmetries, or if, instead of counting the permutations, you wish to count its constituents. Although formulas have been designed to do this too, the crux of the matter is that not all of these formulas have really been proved to be accurate. It is only conjectured.

THE PROOF IS IN THE COMPUTER
The proof that one of these formulas is correct has now been found by Dr. Christoph Koutschan and Dr. Manuel Kauers from the Research Institute for Symbolic Computation of the Johannes Kepler University Linz, Austria, in cooperation with Prof. Doron Zeilberger from the United States. It is a formula that is employed for calculating the individual components in totally symmetric plane partitions. Dr. Koutschan comments on the special method they used to find the proof: "We let the computer do the work! In some areas of mathematics this has long been a matter of routine." The underlying principle of such computer-assisted proof is simple. In order to prove A=B, the computer calculates an adjoint equation U=V with the following two properties: "If U=V is true, then A=B is also true" and "it is easy to verify that U=V".

Although it may sound easy, it represents a great challenge, according to Dr. Koutschan: "This method does not work for every equation. The most important step was for us to convert the Andrews-Robbins conjecture into a suitable form for the computer to be able to prove it." The fact that the adjoint equation was really somewhat more complex than "U=V", is illustrated by its size: if it were printed, it would cover approximately 1 million A4 pages, which makes it probably the longest equation ever used in a mathematical proof.

STANLEY`S LIST
In the end, the work that was spent on the "formulation" was well worth it. With the proof of the Andrews and Robbins conjecture, the scientists have managed to prove the last of a number of famous conjectures, which were presented by US mathematician Richard Stanley at a historic conference in Montreal in 1985. In the years following the conference, all of these conjectures were proved except for the Andrews and Robbins conjecture. Dr. Kauers comments: "As the last remaining item on Stanley s list, this conjecture attracted the attention of many experts. Still, it remained unproved for almost thirty years. The proof was finally obtained with an automatic method, which goes to show that modern computer programs can crack mathematical problems where traditional mathematicians fall short."

Granted, such successful results are still an exception. However, this FWF project underscores the potential of computer-based proof. Given the great pace at which computer performance is advancing, such methods will perhaps one day even offer answers to the great unsolved questions in mathematics.

Image and text will be available from Tuesday, 25th January 2011, 9 am CET onwards:

http://www.fwf.ac.at/en/public_relations/press/pv201101-2en.html

Original publication: A proof of George Andrews` and David Robbins` q-TSPP conjecture. C. Koutschan, M. Kauers, D. Zeilberger. DOI: 10.1073/pnas.1019186108

Scientific contact:
PD Dr. Manuel Kauers
Johannes Kepler University
Research Institute for Symbolic Computation Altenberger Straße 69 4040 Linz, Austria
T: +43/732/2468 9958
E: Manuel.Kauers@risc.jku.at
Austrian Science Fund (FWF)
Mag. Stefan Bernhardt
Haus der Forschung
Sensengasse 1
1090 Vienna, Austria
T: +43/1/5056740-8111
E: stefan.bernhardt@fwf.ac.at
W: http.//www.fwf.ac.at
Copy Editing & Distribution
PR&D - Public Relations for Research & Education Mariannengasse 8 1090 Vienna, Austria
T: +43 / 1 / 505 70 44
E: contact@prd.at
W: http://www.prd.at

Jacqueline Bogdanovic | PR&D
Further information:
http://www.fwf.ac.at

More articles from Information Technology:

nachricht Construction of practical quantum computers radically simplified
05.12.2016 | University of Sussex

nachricht UT professor develops algorithm to improve online mapping of disaster areas
29.11.2016 | University of Tennessee at Knoxville

All articles from Information Technology >>>

The most recent press releases about innovation >>>

Die letzten 5 Focus-News des innovations-reports im Überblick:

Im Focus: Electron highway inside crystal

Physicists of the University of Würzburg have made an astonishing discovery in a specific type of topological insulators. The effect is due to the structure of the materials used. The researchers have now published their work in the journal Science.

Topological insulators are currently the hot topic in physics according to the newspaper Neue Zürcher Zeitung. Only a few weeks ago, their importance was...

Im Focus: Significantly more productivity in USP lasers

In recent years, lasers with ultrashort pulses (USP) down to the femtosecond range have become established on an industrial scale. They could advance some applications with the much-lauded “cold ablation” – if that meant they would then achieve more throughput. A new generation of process engineering that will address this issue in particular will be discussed at the “4th UKP Workshop – Ultrafast Laser Technology” in April 2017.

Even back in the 1990s, scientists were comparing materials processing with nanosecond, picosecond and femtosesecond pulses. The result was surprising:...

Im Focus: Shape matters when light meets atom

Mapping the interaction of a single atom with a single photon may inform design of quantum devices

Have you ever wondered how you see the world? Vision is about photons of light, which are packets of energy, interacting with the atoms or molecules in what...

Im Focus: Novel silicon etching technique crafts 3-D gradient refractive index micro-optics

A multi-institutional research collaboration has created a novel approach for fabricating three-dimensional micro-optics through the shape-defined formation of porous silicon (PSi), with broad impacts in integrated optoelectronics, imaging, and photovoltaics.

Working with colleagues at Stanford and The Dow Chemical Company, researchers at the University of Illinois at Urbana-Champaign fabricated 3-D birefringent...

Im Focus: Quantum Particles Form Droplets

In experiments with magnetic atoms conducted at extremely low temperatures, scientists have demonstrated a unique phase of matter: The atoms form a new type of quantum liquid or quantum droplet state. These so called quantum droplets may preserve their form in absence of external confinement because of quantum effects. The joint team of experimental physicists from Innsbruck and theoretical physicists from Hannover report on their findings in the journal Physical Review X.

“Our Quantum droplets are in the gas phase but they still drop like a rock,” explains experimental physicist Francesca Ferlaino when talking about the...

All Focus news of the innovation-report >>>

Anzeige

Anzeige

Event News

ICTM Conference 2017: Production technology for turbomachine manufacturing of the future

16.11.2016 | Event News

Innovation Day Laser Technology – Laser Additive Manufacturing

01.11.2016 | Event News

#IC2S2: When Social Science meets Computer Science - GESIS will host the IC2S2 conference 2017

14.10.2016 | Event News

 
Latest News

Researchers identify potentially druggable mutant p53 proteins that promote cancer growth

09.12.2016 | Life Sciences

Scientists produce a new roadmap for guiding development & conservation in the Amazon

09.12.2016 | Ecology, The Environment and Conservation

Satellites, airport visibility readings shed light on troops' exposure to air pollution

09.12.2016 | Health and Medicine

VideoLinks
B2B-VideoLinks
More VideoLinks >>>